1. Introduction and main results
This article is concerned with the following Neumann problem for non-linear Schrödinger (NLS) equation:

with a mass constraint

where
$\Omega\subset \mathbb{R}^N(N\ge3)$ is a smooth bounded domain, c > 0 is a given constant, ν stands for the unit outer normal to
$\partial \Omega$, and λ serves as a Lagrange multiplier.
The features of problem (1.1)–(1.2) are the following:
(i) The presence of the Neumann boundary condition is quite rare in the analysis of solutions with prescribed mass.
(ii) The reaction exhibits a general mass supercritical growth at infinity.
(iii) The proof relies on powerful techniques, such as Morse theory and a new blow-up analysis for the NLS equation.
(iv) The analysis presented in this article can be extended to other classes of stationary problems, including biharmonic elliptic equations and Schrödinger–Poisson equations.
The analysis of solutions with prescribed mass is particularly significant from a physical point of view, in relationship with phenomena arising in non-linear optics, the theory of water waves, etc. Indeed, solutions with prescribed L 2-norm are especially relevant since this quantity is preserved along the time evolution. Moreover, the variational characterization of such solutions is often a strong help to analyse their orbital stability and instability properties, see [Reference Cazenave and Lions15, Reference Jeanjean, Jendrej, Le and Visciglia33, Reference Jeanjean and Le34, Reference Soave54, Reference Soave55].
The investigation of non-linear Neumann problem (1.1) finds applications in various fields. One of the main motivations stems from the analysis of standing waves in the form
$\psi(t,x) = e^{-i\lambda t}u(x)$, where
$\lambda \in \mathbb{R}$ and
$u:\mathbb{R}^N\to \mathbb{R}$, for the time-dependent NLS equation:

subject to Neumann boundary conditions. The model (1.3) plays an important role in non-linear optics and Bose–Einstein condensates (see [Reference Agrawal3, Reference Bao and Cai5, Reference Fibich and Merle23, Reference Frantzeskakis24, Reference Malomed, Condensation, Kevrekidis, Frantzeskakis and Carretero-Gonzalez40]). It is evident that
$\psi(t,x)$ is a solution to (1.3) if and only
$(u, \lambda)$ constitutes a coupled solution to (1.1) with
$f(u)=g(|u|)u$. We study Eq. (1.1) by searching for solutions satisfying the mass constraint
$\int_{\Omega}|u|^2 dx=c$. In this context, λ remains unknown and emerges as a Lagrange multiplier, a feature deemed meaningful from a physical standpoint due to mass conservation. The solutions under a L 2 constraint are commonly referred to as normalized solutions. These normalized solutions to (1.1) can be obtained as critical points of the energy functional
$J: H^1(\Omega)\to \mathbb{R}$ defined by:

on the L 2-sphere constraint:

where
$F(u)= \int_0^u f(t)dt$.
When
$\Omega= \mathbb{R}^N$, problem (1.1)–(1.2) is transformed into

Extensive studies have been conducted in recent years for this problem, particularly when considering non-linearities satisfying
$f(u)\sim|u|^{p-2}u$ as
$|u|\to+\infty$, where
$p\in (2_*, 2^*)$ with
$2_*:=2+\frac{4}{N}$ and
$2^*:=\frac{2N}{N-2}$. Within this L 2-supercritical range, the corresponding energy functional is unbounded from below on the constraint set
$\mathcal{S}_c$. The pioneering work in this direction was carried out by Jeanjean [Reference Jeanjean31], where a scaled functional and mountain pass arguments were introduced to address the L 2-supercritical problems. Bartsch and Soave [Reference Bartsch and Soave6] developed a natural constraint approach to investigate the L 2-supercritical NLS equations and systems on
$\mathbb{R}^N$. For more related results on
$\mathbb{R}^N$, we refer to [Reference Alves, Ji and Miyagaki4, Reference Bellazzini, Forcella and Georgiev7, Reference Ikoma and Tanaka30, Reference Jeanjean, Jendrej, Le and Visciglia33, Reference Jeanjean and Le34, Reference Wei and Wu56] and the associated references.
The exploration of normalized solutions for the NLS on bounded domains was initialled in [Reference Noris, Tavares and Verzini46]. When
$f(u)=|u|^{p-2}u$ with p being L 2-supercritical but Sobolev subcritical, Noris et al. [Reference Noris, Tavares and Verzini46] demonstrated the existence of a positive normalized solution on a unit ball with Dirichlet boundary conditions. The case for general bounded domains was addressed in [Reference Pierotti and Verzini49]. For NLS systems, one can refer to [Reference Noris, Tavares and Verzini47].
In [Reference Pellacci, Pistoia, Vaira and Verzini48], Pellacci et al. investigated normalized solutions of the NLS under both Dirichlet and Neumann boundary conditions, focusing on the concentration of solutions at specific points of Ω as the prescribed mass c varies. Specifically, for the Neumann problem (1.1)–(1.2) with
$f(u)=u^{p-1}$ for
$p\in (2_*, 2^*)$, they employed the Lyapunov–Schmidt reduction method to prove the existence of positive normalized solutions when
$c\in (0,c_0)$ for some
$c_0 \gt 0$. The solutions concentrate at a point
$\xi_0\in \overline\Omega$ as c → 0, where ξ 0 is either a non-degenerate critical point of the mean curvature H of the boundary
$\partial\Omega$ or the maximum point of the distance function from
$\partial \Omega$. However, the nature of these solutions—whether they correspond to local minimizers or mountain pass-type critical points of the associated functional—remains unclear.
Additionally, it is noteworthy that earlier works on L 2-subcritical Schrödinger–Poisson type systems under Neumann boundary conditions can be found in [Reference Afonso and Siciliano2, Reference Pisani and Siciliano50, Reference Pisani and Siciliano51], where the authors established the existence of infinitely many normalized solutions using the Ljusternik–Schnirelmann theory.
The methodologies used for bounded domains differ significantly from those utilized in the entire space. In fact, the approaches for
$\mathbb{R}^N$ heavily rely on the scaling transform
$(t\star u)(x)=t^{\frac{N}{2}}u(tx)$ and the associated Pohozaev identity. However, due to the lack of invariance under translations and dilation, as well as the emergence of uncontrollable boundary terms in the Pohozaev identity (particularly for non-convex domains), neither of these techniques is applicable to general bounded domains.
The normalized solutions are also explored within the framework of ergodic mean field game (MFG) systems, offering another key motivation for studying Eq. (1.1). MFGs were introduced in influential works by Huang, Caines, and Malham‘e [Reference Huang, Malhamé and Caines29] and Lasry and Lions [Reference Lasry and Lions37], independently. The primary objective of MFG is to establish a framework for characterizing Nash equilibria in differential games involving an infinite number of agents that are indistinguishable from one another. For more details, we refer interested readers to [Reference Cirant18, Reference Cirant, Cosenza and Verzini19, Reference Santambrogio53] and the references therein. From a mathematical perspective, such equilibria can be characterized by an elliptic system that combines a Kolmogorov equation and a Hamilton–Jacobi–Bellman equation. Moreover, this system has to satisfy normalization in
$L^1(\Omega)$ as follows:

The Neumann boundary conditions are based on the assumption that agents’ trajectories are restricted to Ω by bouncing off the boundary in a normal direction. For the quadratic Hamilton case
$H(\nabla v)=|\nabla v|^2$, by using a Hopf–Cole transformation
$\phi= e^{-v}/ \int e^{-v}= \sqrt{m}$, (1.5) is reduced to

which can be viewed as a single NLS equation with prescribed mass. In fact, we can check that (1.6) is equivalent to (1.1) by taking a simple transformation
$\phi= \frac{1}{\sqrt{c}}u$ and
$f(u)=h(\frac{u^2}{c})u$. In [Reference Cirant, Cosenza and Verzini19], Cirant et al. investigated the existence of the viscous ergodic MFG system with Neumann boundary conditions. They proved the existence of global minimizers in L 2-subcritical and critical cases, as well as local minimizers in L 2-supercritical cases.
When f satisfies the L 2-supercritical growth, the constraint functional
$J|_{\mathcal{S}_c}$ exhibits a mountain pass geometry structure. Hence, it is natural to seek mountain pass type normalized solutions to the Neumann problem (1.1)–(1.2). However, in this scenario, the functional J becomes unbounded from below on
$\mathcal{S}_c$, rendering the method in [Reference Cirant, Cosenza and Verzini19] ineffective. To the best of our knowledge, there are currently no references addressing this issue. The primary aim of this article is to develop a novel variational technique to explore the existence of mountain pass type normalized solutions for (1.1)–(1.2) under the L 2-supercritical growth conditions.
Furthermore, notice that the current existence results of normalized solutions for (1.4) mainly rely on the following global Ambrosetti–Rabinowitz type condition

(see [Reference Bartsch and Soave6, Reference Ikoma and Tanaka30, Reference Jeanjean31]) or combined non-linearities (see [Reference Alves, Ji and Miyagaki4, Reference Bellazzini, Forcella and Georgiev7, Reference Jeanjean, Jendrej, Le and Visciglia33, Reference Jeanjean and Le34, Reference Soave54–Reference Wei and Wu56]). Recently, normalized solutions of (1.4) have also been studied when (1.7) is replaced by certain global monotonicity conditions, as discussed in [Reference Bieganowski and Mederski10, Reference Chang, Liu and Yan17, Reference Jeanjean and Lu35, Reference Mederski and Schino42]. However, the exploration of normalized solutions on bounded domains with more diverse non-linearities beyond power non-linearity is currently limited. Our method is designed to be flexible and applies to a broad range of non-linearities, including
$f(u)=|u|^{p-2}u$,
$|u|^{q-2}u+ |u|^{p-2}u$, and
$-|u|^{q-2}u+|u|^{p-2}u$, where
$2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$, in a unified manner. Our results do not require the presence of condition (1.7) or the global monotonicity conditions. Additionally, motivated by the arguments in [Reference Pierotti and Verzini49], diverging from the prevalent focus on positive solutions in existing literature, we establish the existence of mountain pass type normalized solutions that are not necessarily positive.
Before stating our results, let us impose the following assumptions on f:
(f 1)
$f\in C^1({\mathbb{R}})$,
$\lim\limits_{|t|\to 0} \frac{f(t)}{t}=0$;
(f 2) there exist constants
$p\in (2_*, 2^*)$ and
$a_0 \gt 0$ such that
\begin{equation*}\lim_{|t|\to \infty} \frac{f(t)}{|t|^{p-2}t} =a_0;\end{equation*}
(f 3) there exist constants
$\mu\geq a_0(p-1)$ and M > 0 such that
\begin{equation*}\mu |t|^{p-2} \leq f'(t), ~\forall |t|\geq M. \end{equation*}
Our main result is the following theorem.
Theorem 1.1. Suppose
$(f_1)$–
$(f_3)$ hold. Then there exists a constant
$c^* \gt 0$ such that for any
$0 \lt c \lt c^*$, problem (1.1)–(1.2) has a normalized solution pair
$(u,\lambda)\in H^1(\Omega)\times \mathbb{R}$ of mountain pass type.
Remark 1.2. If
$f(t)t \gt 0$ for t ≠ 0, it allows us to establish the positivity of the Lagrange multiplier λ. In fact, under this condition, we can obtain a positive normalized solution u of (1.1). Subsequently, integrating Eq. (1.1) and using the Neumann boundary value condition, we can deduce that λ > 0.
A significant challenge in proving theorem 1.1 arises from establishing the boundedness of Palais–Smale sequences. To overcome this obstacle, we combine with a parameterized minimax principle with Morse index information for constrained functionals established recently in [Reference Borthwick, Chang, Jeanjean and Soave13] (see also [Reference Borthwick, Chang, Jeanjean and Soave12, Reference Chang, Jeanjean and Soave16]) and a new blow-up analysis for the NLS equation under Neumann boundary conditions. In applying this method, the sign condition
$\int_\Omega F(u) \geq 0$ for all
$u \in H^1(\Omega)$ is crucial to ensure the applicability of the monotonicity trick (see theorem 3.4 in §3). However, in our specific problem,
$\int_\Omega F(u)$ is not necessarily non-negative when u is not sufficiently large. Our strategy to tackle this challenge is to utilize a cut-off function and decompose f into two parts, ensuring that one of the parts satisfies this sign condition and allows for the application of the monotonicity trick.
To address the original Neumann problem after deriving solutions for approximation problems, we will propose an innovative blow-up analysis tailored for the NLS subject to Neumann boundary conditions. In contrast to the blow-up arguments applied in the Dirichlet case (see [Reference Esposito and Petralla20, Reference Pierotti and Verzini49]), the blow-up analysis for NLS with Neumann boundary conditions, subject to the L 2-constraint, proves to be considerably more intricate. In fact, in Neumann problems, the solution is not necessarily required to vanish at the boundary
$\partial\Omega$, and local extremum points of solutions can exist on the boundary. This is in stark contrast to Dirichlet problems where such extremum points are found in the interior of Ω.
We note that several studies have conducted blow-up analyses of the NLS under Neumann boundary conditions. It is well known that combining blow-up arguments with suitable Liouville-type theorems is highly effective for deriving a priori estimates. In [Reference Lin, Ni and Takagi39], Lin et al. delivered a comprehensive blow-up analysis to establish the boundedness for positive solutions to the following Neumann problem:

where d > 0 and g satisfies Sobolev subcritical growth conditions. The Sobolev critical case was subsequently addressed by Adimurthi et al. [Reference Adimurthi and Yadava1]. In [Reference Hu, Wang and Zhao28], Hu et al. used a blow-up argument for a non-linear Neumann problem involving the p-Laplacian to obtain a priori estimates. For further details on blow-up analyses of elliptic problems under Neumann boundary conditions, the reader is referred to [Reference Bonheure, Grumiau and Troestler11, Reference López-Gómez and Sampedro36].
Blow-up analysis can also establish a connection between the boundedness of solutions and their Morse indices. Consider the following problem:

where g satisfies superlinear and subcritical growth conditions. Harrabi et al. [Reference Harrabi, Ahmedou, Rebhi and Selmi27] proved that the
$L^\infty$ bounds for solutions to (1.9) are equivalent to bounds on their Morse indices.
In contrast to the arguments presented in [Reference Harrabi, Ahmedou, Rebhi and Selmi27], our discussion is conducted under a mass constraint and involves a family of non-linearities arising from varying Lagrange multipliers, along with a dense set used in the monotonicity trick. The most significant distinction, however, is that our blow-up analysis requires an exponential decay estimate for the solution sequence. This difference necessitates substantial modifications to the blow-up arguments in the Neumann case.
As we discussed earlier, we present a compelling argument to establish that the solution sequence not only exhibits the commonly observed exponential decay away from the boundary
$\partial\Omega$, as found in the fixed frequency NLS in the Neumann case (see [Reference Ni and Takagi43, Reference Ni and Takagi44]), but more significantly, it demonstrates uniform exponential decay away from the blow-up points.
To conduct the blow-up analysis and subsequently derive a contradiction, a critical step involves demonstrating that the solution sequence exhibits uniform exponential decay, especially when the blow-up points are situated on the boundary
$\partial\Omega$. However, in such cases, we can typically only establish that the solution sequence exponentially decays uniformly away from the blow-up points within a smaller domain in Ω, which is distant from the boundary. Specifically, as detailed in §4, through a comparison argument, we can only provide an estimation that the solution sequence exponentially decays uniformly away from the blow-up points in the domain
$\tilde{\Omega}_\theta \backslash \cup_{i=1}^k (B_{R\lambda_n^{-\frac{1}{2}}}(P_n^i)\cap \tilde{\Omega}_\theta)$ (
$\Omega_\theta$ is defined in §4,
$\tilde{\Omega}_\theta:= \overline{\Omega}\backslash \Omega_\theta $ and
$k\in\{1,2\}$). To be precise, it is demonstrated that there exist constants
$C_1 \gt 0$ and
$C_2 \gt 0$ such that

where un satisfies (4.1), R > 0 is a constant,
$P_n^i$ is a local extremum point,
$i\in \{1,\ldots, k\}$. However, this conclusion falls short of addressing our problem adequately. The limitation arises from the challenging nature of estimating the exponential decay of the solution in the neighbourhood near the boundary
$\partial\Omega$.
To overcome this limitation, we formulate a diffeomorphism, denoted as
$\Phi:\Omega^\theta\to \Omega_\theta$, in a manner that expands the equation to encompass a new domain, namely,
$\overline{\Omega}\cup \Omega^\theta$, which combines the original domain Ω with an adjoining tubular neighbourhood
$\Omega^\theta$ (see §4). Employing the comparison theorem, we deduce that the solution exhibits exponential decay away from the boundary of the newly defined domain
$\overline{\Omega}\cup \Omega^\theta$. Subsequently, through a similar argument as mentioned earlier, we observe that the solution exponentially decays uniformly away from the blow-up points in
$\overline{\Omega}\backslash \cup_{i=1}^k (B_{R\lambda_n^{-1/2}}(P_n^i)\cap \overline{\Omega})$. This establishes the intended conclusion we strive to attain.
This article is organized as follows. In §2, we provide the mountain pass geometry of the parameterized functionals Jρ uniformly for
$\rho\in[\frac{1}{2},1]$. Subsequently, in §3, we show the existence of a mountain pass critical point uρ with Morse index information for
$J_\rho|_{\mathcal{S}_c}$ for almost every
$\rho\in[\frac{1}{2},1]$. In §4, we perform a blow-up analysis of solutions with bounded Morse index for the Neumann problem. Finally, in §5, we prove theorem 1.1. For convenience, we denote by
$\|\cdot\|_r$ the norm of the spaces
$L^r(\Omega)(1\le r\le +\infty)$, by
$\|\cdot\|$ the norm in
$H^1(\Omega)$.
2. Mountain pass geometry
This section is dedicated to establishing the uniform mountain pass geometry for a parameterized functional. To achieve this goal, we first decompose f into two parts. By
$(f_1)$–
$(f_2)$, we deduce that there exists
$R_0 \gt 0$ such that
$f(t)t \gt 0$ for
$|t|\geq R_0$. Utilizing this observation, we define

where η is a smooth cut-off function such that

with
$|\eta'(t)|\leq 2$ for
$R_0 \lt |t| \lt R_0+1$. Clearly,
$f_1(t)t\geq 0$ for all
$t\not =0$.
Define
$J_{\rho}: H^1(\Omega)\to \mathbb{R}$ by

where
$F_1(u)= \int_0^u f_1(t)dt$ and
$F_2(u)= \int_0^u f_2(t)dt$.
We recall the Gagliardo–Nirenberg inequality (see [Reference Nirenberg45]): for every
$N\geq 3$ and
$p\in(2,2^*)$, there exists a constant
$C_{N,r,\Omega}$ depending on N, r and Ω such that

where

Define
$\mathcal{B}_\alpha:=\{u\in \mathcal{S}_c: \int_{\Omega} |\nabla u|^2\leq \alpha\}$,
$\forall \alpha \gt 0$. We have the following result.
Lemma 2.1. Assume
$(f_1)$–
$(f_3)$. Then there exists
$c^* \gt 0$ such that for any
$c\in (0, c^*)$, we can find
$\alpha^* \gt 0$ such that

Proof. By
$(f_1)$–
$(f_2)$, for any ϵ, δ > 0 and
$q\in (2, 2_*)$, there exist constants
$C^{\prime}_\epsilon, C_\delta \gt 0$ such that


Let
$u\in \mathcal{B}_\alpha$ and
$v\in \partial\mathcal{B}_{2\alpha}$, where α is to be determined. By (2.2)–(2.3), along with the Gagliardo–Nirenberg inequality, we obtain

Due to the arbitrariness of ϵ and δ, we conclude that for some
$c^* \gt 0$ and any
$c\in (0, c^*)$, taking
$\alpha^*= 4c$, we have

Hence, the conclusion follows.
For any
$c\in (0,c^*)$, we define

Then, following the approach outlined in [Reference Noris, Tavares and Verzini47, proposition 3.4] (see also [Reference Bellazzini and Jeanjean8, theorem 1.6]), we can derive the following result.
Lemma 2.2. Assume
$(f_1)$–
$(f_3)$ and
$c\in (0, c^*)$. Then mρ is achieved by some
$u^*_{\rho}\in \mathcal{B}_{2\alpha^*}\setminus \partial\mathcal{B}_{2\alpha^*}$.
Proof. Fix
$c\in (0, c^*)$. By lemma 2.1, for any
$\rho \in \left[\frac{1}{2},1 \right]$, we have

Let
$\{u_{\rho,n}\}\subset \mathcal{B}_{2\alpha^*}$ be a minimizing sequence for Jρ at the level mρ. Clearly,
$\{u_{\rho,n}\}$ is bounded in
$H^1(\Omega)$. Consequently, there exist a subsequence of
$\{u_{\rho,n}\}$, still denoted by
$\{u_{\rho,n}\}$, and some
$u^*_\rho\in H^1(\Omega)$ such that
$u_{\rho,n}\rightharpoonup u^*_\rho$ in
$H^1(\Omega)$ and
$u_{\rho,n}\to u^*_{\rho}$ strongly in
$L^r(\Omega)$ for
$r\in[1,2^*)$. This implies that
$u^*_{\rho}\in \mathcal{B}_{2\alpha^*}$ and thus
$J_\rho(u^*_{\rho})\ge m_{\rho}.$
By
$(f_1)$–
$(f_2)$, together with the Hölder inequality and the Lebesgue dominated convergence theorem, we have

Therefore, we obtain

Combining this with the previous inequality, we deduce that
$J_\rho(u^*_{\rho})=m_{\rho}$ and
$\|\nabla \left(u_{\rho,n}-u^*_{\rho}\right)\|_2^2\to0$. This implies that
$u_{\rho,n}\to u^*_{\rho}$ in
$H^1(\Omega)$ and hence
$u^*_{\rho}$ is a minimizer of Jρ. Thus, the desired conclusion follows.
Now we can show the mountain pass geometry of Jρ uniformly for
$\rho\in[\frac{1}{2}, 1]$.
Lemma 2.3. Assume
$(f_1)$–
$(f_3)$. Then, for any
$c\in (0,c^*)$, there exist
$w_1, w_2\in \mathcal{S}_c$ such that

where

Proof. Let
$B_r(x)$ denote a ball in
$\mathbb{R}^N$, centred at
$x\in \mathbb{R}^N$ with radius r > 0. Take
$\phi \in C^\infty_0(B_1(0))$ with ϕ > 0 in
$B_1(0)$ such that
$\int_{B_1(0)}\phi^2 =1$. For
$n\in \mathbb{N}$,
$x_0\in \Omega$, define

We can verify that
$\varphi_n\in\mathcal{S}_c$ and
$supp(\varphi_n)\subset B_\frac{1}{n}(x_0)\subset \Omega$ for sufficiently large n.
By
$(f_1)$–
$(f_2)$, taking
$R_0 \gt 0$ larger if necessary, there exist constants
$C_{R_0}, C^{\prime}_{R_0}, C_p \gt 0$ such that

Set
$\Omega_{n, 2R_0} :=\{x\in \Omega: |\varphi_n|\leq 2R_0 \}$. Then, for sufficiently large n such that
$\max\limits_{x\in \overline\Omega}|\varphi_n| \gt 2R_0$, we have for all
$\rho\in [\frac{1}{2},1]$,

Hence, there exists
$n_0 \gt 0 $ sufficiently large such that

Choose
$w_1=u^*_{\frac{1}{2}}$ and
$w_2=\varphi_{n_0}$. Clearly,
$u^*_{\frac{1}{2}}\in \mathcal{B}_{2\alpha^*}\setminus \partial\mathcal{B}_{2\alpha^*}, \varphi_{n_0}\not\in \mathcal{B}_{2\alpha^*}$. By continuity, for any
$\gamma \in \Gamma$, there exists
$t_\gamma \in [0,1]$ such that
$ \gamma(t_\gamma)\in \partial \mathcal{B}_{2\alpha^*}$. Thus, by lemma 2.1, it follows that

The proof is now complete.
3. Existence of MP solutions for a dense set
In this section, we establish the existence of a bounded Palais–Smale sequence at level cρ for almost every
$\rho\in [\frac{1}{2},1]$. Our approach involves applying a recently developed min-max principle on the L 2-sphere, as detailed in [Reference Borthwick, Chang, Jeanjean and Soave13]. This principle integrates the monotonicity trick presented in [Reference Jeanjean32] with the min-max theorem enriched by second-order insights from Fang and Ghoussoub [Reference Fang and Ghoussoub21], which is also elaborated upon in [Reference Ghoussoub25, Chapter 11].
For a domain
$D\subset \mathbb{R}^N$ and
$\phi, u\in H^1(D)$, we consider

where
$\lambda\in \mathbb{R}, \rho\in[\frac{1}{2},1]$. The Morse index of u, denote by m(u), is the maximum dimension of a subspace
$W\subset H^1(D)$ such that
$Q_{\lambda, \rho}(\phi;u;D) \lt 0$ for all
$\phi \in W\backslash \{0\}$.
To state the abstract minimax theorem, we recall a general setting introduced in [Reference Berestycki and Lions9]. Let
$(E, \langle\cdot,\cdot\rangle)$ and
$(H,(\cdot, \cdot))$ be two infinite dimensional Hilbert spaces such that

with continuous injections. The continuous injection
$E\hookrightarrow H$ has a norm at most 1 and E is identified with its image in H. For
$u\in E$, we denote
$\|u \|^2= \langle u,u \rangle$ and
$|u|^2= (u,u)$. For
$a\in (0, +\infty)$, we define
$S(a):= \{u\in E, |u|^2= a\}$. We denote by
$\|\cdot\|_*$ and
$\|\cdot\|_{**}$, respectively, the operator norm of
$\mathcal{L}(E, R)$ and of
$\mathcal{L}(E, \mathcal{L}(E, \mathbb{R}))$.
Definition 3.1. [Reference Borthwick, Chang, Jeanjean and Soave13]
Let
$\phi: E\to \mathbb{R}$ be a C 2-functional on E and
$\alpha\in (0,1]$. We say that
$\phi'$ and
$\phi^{\prime\prime}$ are α-Hölder continuous on bounded sets if for any R > 0 one can find
$M=M(R) \gt 0$ such that for any
$u_1, u_2\in B(0, R)$:

Definition 3.2. [Reference Borthwick, Chang, Jeanjean and Soave13]
Let ϕ be a C 2-functional on E, for any
$u\in E$ define the continuous bilinear map:

Definition 3.3. [Reference Borthwick, Chang, Jeanjean and Soave13]
For any
$u\in S(a)$ and θ > 0, we define an approximate Morse index by

If u is a critical point for the constrained functional
$\phi|_{S(a)}$ and θ = 0, then
$\tilde{m}_\theta(u)$ is the Morse index of u as a constrained critical point.
Theorem 3.4. ([Reference Borthwick, Chang, Jeanjean and Soave13], theorem 1.5)
Let
$I\subset(0, +\infty)$ be an interval and consider a family of C 2 functionals
$\Phi_\rho : E \to \mathbb{R}$ of the form:

where
$B(u)\geq 0$ for every
$u\in E$, and

Suppose moreover that
$\Phi^{\prime}_\rho$ and
$\Phi'^{\prime}_\rho$ are α-Hölder continuous on bounded sets for some
$\alpha \in (0,1]$. Finally, suppose that there exist
$w_1, w_2\in S(a)$ (independent of ρ) such that, set

we have

Then, for almost every
$\rho\in I$, there exist sequence
$\{u_n\}\subset S(a)$ and
$\zeta_n\to 0^+$ such that, as
$n \to \infty$,
(i)
$\Phi_\rho(u_n)\to c_\rho$;
(ii)
$\|\Phi^{\prime}_\rho|_{S(a)}(u_n)\|\to 0$;
(iii)
$\{u_n\}$ is bounded in E;
(iv)
$\tilde{m}_{\zeta_n}(u)\leq 1$.
Define
$h_{\rho}(t)=\rho f_1(t)+ f_2(t)$ for
$t\in \mathbb{R}$ and
$\rho\in [\frac{1}{2},1]$. In the following, we obtain the main result of this section.
Theorem 3.5. Assume
$(f_1)$–
$(f_3)$ and
$c\in (0, c^*)$. Then, for almost every
$\rho\in [\frac{1}{2}, 1]$, there exists a critical point uρ of Jρ on
$\mathcal{S}_c$ at level cρ, which solves the following problem

for some
$\lambda_\rho \in \mathbb{R}$. Moreover, the Morse index of
$\{u_{\rho}\}$ satisfies
$m(u_\rho)\le 2$.
Proof. We will apply theorem 3.4 to the family of functionals Jρ, where
$E=H^1(\Omega)$,
$H=L^2(\Omega)$,
$S(a)=\mathcal{S}_c$ and Γ is defined by (2.4). Specifically, we set

Thus, we have
$J_\rho(u)= A(u)-\rho B(u).$ Given that
$u\in \mathcal{S}_c$ and considering the boundedness of
$\int_\Omega F_2(u)dx$, we deduce that

Moreover, by assumptions
$(f_1)$–
$(f_2)$, it follows that
$J^{\prime}_\rho$ and
$J'^{\prime}_\rho$ are locally Hölder continuous on
$\mathcal{S}_c$. By lemma 2.3, we can apply theorem 3.4 to produce a bounded Palais–Smale sequence
$\{u_n\}\subset H^1(\Omega)$ for the constrained functional
$J_{\rho}|_{\mathcal{S}_c}$ at level cρ for almost every
$\rho\in [\frac{1}{2}, 1]$. Additionally, there exists a sequence
$\zeta_n \to 0^+$ such that
$\tilde{m}_{\zeta_n}(u_n)\leq 1$.
Since
$\|J^{\prime}_\rho |_{\mathcal{S}_c}(u_n) \|\to 0$, and by the boundedness of
$\{u_n\}$, there exists a sequence
$\{\lambda_n\}\subset \mathbb{R}$ such that for any
$\varphi\in H^1(\Omega)$, we have

This implies that

Using
$(f_1)$–
$(f_2)$ again, we deduce that
$\{\lambda_n\}$ is bounded. Therefore, up to a subsequence, we may assume that
$\lambda_n \to \lambda_\rho \in \mathbb{R}$ and
$u_n\rightharpoonup u_{\rho}$ weakly in
$H^1(\Omega)$. By (3.4), we obtain

which implies that uρ weakly solves (3.3). By the compact embedding
$H^1(\Omega)\hookrightarrow L^r(\Omega)$ for
$r\in[1,2^*)$ and standard arguments, we obtain that
$u_n\to u_{\rho}$ strongly in
$H^1(\Omega)$.
It remains to show that
$m(u_\rho)\le 2$. Since
$T_u\mathcal{S}_c$ has codimension 1, noting that
$d^2|_{\mathcal{S}_c}J_\rho$ and
$T_{u_n}\mathcal{S}_c$ vary with continuity, by
$\tilde{m}_{\zeta_n}(u_n)\leq 1$ it follows that
$\tilde{m}_0(u_\rho) \leq 1$. Then, we can use similar arguments as in [Reference Borthwick, Chang, Jeanjean and Soave12, proposition 3.5] to show
$m(u_\rho)\leq 2$. In fact, since the tangent space
$T_{u}\mathcal{S}_c$ has codimension 1, it suffices to show that
$u_\rho\in \mathcal{S}_c$ has Morse index at most 1 as a constrained critical point. If this were not the case, by definition 3.3, we may assume by contradiction that there exists a subspace
$W_0 \subset T_u\mathcal{S}_c$ with
$dim W_0 = 2$ such that

Since W 0 is finite-dimensional, there exists a constant β > 0 such that

Using the homogeneity of
$D^2J_\rho(u_\rho)$, we deduce that

Now, since
$J_\rho'$ and
$J_\rho^{\prime\prime}$ are α-Hölder continuous on bounded sets for some
$\alpha\in (0,1]$, it follows that there exists a sufficient small
$\delta_1 \gt 0$ such that, for any
$v\in \mathcal{S}_c$ satisfying
$\|v-u_\rho\|\leq \delta_1$,

Hence, using the fact that
$\|u_n-u_\rho\| \leq\delta_1$ for sufficiently large
$n\in \mathbb{N}$, and in view of (3.5), (3.6), and
$\zeta_n\to 0^+$, we obtain

for any such large n. Since dim
$W_0 \gt 1$, this provides a contradiction with theorem 3.4 (iv), recalling that
$\zeta_n\to 0$.
Remark 3.6. Note that for any
$\rho\in [\frac{1}{2}, 1]$, the constant function
$u_c:=\left(\frac{c}{|\Omega|}\right)^{\frac{1}{2}}$ is always a solution of (3.3) on
$\mathcal{S}_c$ for
$\lambda=\frac{h_{\rho}(u_c)}{u_c}$. Under the assumptions
$(f_1)$–
$(f_2)$, we can compute the constraint Morse index
$\tilde{m}_0(u_c)$ corresponding to Jρ, as demonstrated in [Reference Chang, Jeanjean and Soave16, proposition 2.1] (see also [Reference Cacciapuoti, Dovetta and Serra14, proposition 4.1]). Specifically, for any
$c\in(0,c^*)$ with some proper
$c^* \gt 0$, we have
$\tilde{m}_0(u_c)=0$ for all
$\rho\in[\frac{1}{2}, 1]$. This implies that uc is a local minimizer of Jρ for every
$\rho\in [\frac{1}{2}, 1]$. Based on this observation, we may select
$w_1=u_c$ in lemma 2.3 to construct the uniform mountain pass geometry for Jρ. Furthermore, the mountain pass type solution u obtained in theorem 1.1 cannot be a constant function, as this would contradict the fact that uc is a local minimizer.
4. Blow-up analysis
In this section, we develop a blow-up analysis for the sequence
$\{u_{\rho_n}\}$. The goal of this analysis is to prove that
$\{u_{\rho_n}\}$ is bounded in
$H^1(\Omega)$. Consequently, we aim to show that
$\{u_{\rho_n}\}$ converges strongly in
$H^1(\Omega)$ to a constrained critical point of J 1 as
$\rho_n\to 1^-$.
For simplicity, we denote
$u_n:= u_{\rho_n}$,
$\lambda_n:=\lambda_{\rho_n}$,
$c_n:= c_{\rho_n}$ in the following discussion. Here, un weakly solves the following problem

where
$\lambda_n\in \mathbb{R}$ and
$\rho_n \to 1^-$. By theorem 3.5, we have
$m(u_n)\le2$. Using standard regularity arguments, we obtain
$u_n\in C^2(\overline{\Omega})$.
Lemma 4.1. There exists a constant C such that
$\lambda_n\geq C$ for all n.
Proof. We assume by contradiction that
$\lambda_n\to -\infty$. Let V be a subspace of
$H^1(\Omega)$ with dimension k, where k > 2. Define
$\Omega_{M_0}:=\{x\in \Omega: |u_n(x)|\leq M_0\}$, where
$M_0=\max\{M, R_0+1\}$ and R 0 is given in §2. By assumptions
$(f_1)$ and
$(f_3)$, there exist constants
$C_{M_0}, C^{\prime}_{M_0} \gt 0$ such that for any
$\phi\in H^1(\Omega)$,

Taking
$\varphi\in V\setminus\{0\}$, we obtain

This implies that
$Q_{\lambda_n,\rho_n}(\varphi;u_n;\Omega)$ is negative definite on V for sufficiently large n, which contradicts the fact that
$m(u_n)\le 2$.
Lemma 4.2. If
$\lambda_n \to +\infty$, then
$\|u_n \|_{L^\infty} \to +\infty$.
Proof. By (4.1) and assumptions
$(f_1)$–
$(f_2)$, there exists a constant
$C_1 \gt 0$ such that

This implies that

In the following, we will analyse the asymptotic behaviour of the solution to (4.1) as
$\lambda_n \to +\infty$. For simplicity, we may assume without loss of generality that
$\max\limits_{x\in\overline{\Omega}}u_n(x) \gt 0$.
We begin by providing a local description of the blow-up points.
Lemma 4.3. Suppose that
$\lambda_n \to +\infty$. Let
$P_n\in \overline{\Omega}$ be such that, for some
$R_n\to \infty$,

Set
$\epsilon_n=\lambda_n^{-\frac{1}{2}}$. Then

Suppose moreover that

Then, passing to a subsequence if necessary, we have
(i)
$P_n\to P\in \Omega$;
(ii)
$\frac{dist(P_n, \partial\Omega)}{\epsilon_n}\to +\infty ~~as ~~n\to +\infty,$ and the scaled sequence
(4.4)\begin{equation} v_n(x):=a_0^{\frac{1}{p-2}} \epsilon_n^{\frac{2}{p-2}}u_n(\epsilon_n x +P_n) ~~{\rm for}~~ x\in \Omega_n:=\frac{\Omega-P_n}{\epsilon_n} \end{equation}
converges to some
$v\in H^1(\mathbb{R}^N)$ in
$C^2_{loc}(\mathbb{R}^N)$, where v satisfies
(4.5)\begin{equation} \left\{\begin{array}{ll} -\Delta v+ v=|v|^{p-2}v & {\mathrm{in}} \,~ \mathbb{R}^N,\\ |v(0)|=\max\limits_{x\in \mathbb{R}^N} v, \\ v(x)\to 0 ~~{\rm as}~~|x|\to +\infty; \end{array}\right. \end{equation}
(iii) there exists
$\phi_n\in C_0^{\infty}(\Omega)$, with supp
$\phi_n \subset B_{R\epsilon_n}(P_n)$ for some R > 0, such that
$Q_{\lambda_n,\rho_n}(\phi_n;u_n;\Omega) \lt 0$;
(iv) for all R > 0 and
$q\geq 1$,
\begin{equation*} \lim_{n\to \infty}\lambda_n^{\frac{N-2}{2}-\frac{q}{p-2}}\int_{B_{R\epsilon_n}(P_n)}|u_n|^q dx=\lim_{n\to \infty}\int_{B_{R}(0)}|v_n|^q dy =\int_{B_{R}(0)}|v|^q dy. \end{equation*}
Instead of (4.3), we suppose that

Then, passing to a subsequence if necessary, the following results hold:
(i)
$P_n\to P\in \partial\Omega$;
(ii)
$\frac{dist(P_n, \partial\Omega)}{\epsilon_n}\to d_0\geq 0 ~~as ~~n\to +\infty$, and the scaled sequence
$\{v_n\}$ defined in (4.4) converges to some
$v\in H^1(\mathbb{R}^N_+)$ in
$C^2_{loc}(\mathbb{R}^N_+)$ as
$n\to \infty$, where v satisfies
(4.7)\begin{equation} \left\{\begin{array}{ll} -\Delta v+ v=|v|^{p-2}v & {\mathrm{in}} \,~ \mathbb{R}^N_+,\\ \displaystyle\frac{\partial v}{\partial x_N}=0 \, &{\mathrm{on}}\,~\partial \mathbb{R}^N_+,\\ |v(0)|=\max\limits_{x\in \mathbb{R}^N_+} v, \\ v(x)\to 0 ~~{\rm as}~~|x|\to +\infty; \end{array}\right. \end{equation}
(iii) there exists
$\phi_n\in C_0^{\infty}(\Omega)$, with supp
$\phi_n \subset B_{R\epsilon_n}(P_n)\cap \overline{\Omega}$ for some R > 0, such that
$Q_{\lambda_n, \rho_n}(\phi_n;u_n;\Omega) \lt 0$;
(iv) for all R > 0 and
$q\geq 1$,
\begin{align*} \lim_{n\to \infty}\lambda_n^{\frac{N-2}{2}-\frac{q}{p-2}}\int_{B_{R\epsilon_n}(P_n)\cap \Omega}|u_n|^q dx&=\lim_{n\to \infty}\int_{B_{R}(0)\cap \Omega_n}|v_n|^q dy \\ &=\int_{B_{R}(0)\cap \mathbb{R}^N_+}|v|^q dy. \end{align*}
Proof. Since un may change sign, Pn can be either a positive local maximum or a negative local minimum point. For simplicity, we focus on the case where Pn is a positive local maximum point; the arguments for the negative local minimum case are analogous.
By (4.1), we get

Using
$(f_2)$ and lemma 4.1, we deduce that

Next, we show that
$\tilde{\lambda} \gt 0$.
Define the rescaled function

Clearly,
$\tilde{u}_n$ satisfies

Let
$d_n=dist(P_n, \partial \Omega)$. Then

where
$\mathbb{H}$ denotes a half-space such that
$0\in \overline{\mathbb{H}}$ and
$d(0, \partial \mathbb{H})=L$. By regularity arguments, up to a subsequence,
$\tilde{u}_n\to \tilde{u}$ in
$C^2_{loc}(\overline{D})$, where
$\tilde{u}$ solves

where D is either
$\mathbb{R}^N$ or
$\mathbb{H}$.
We claim that
$m(\tilde{u})\le 2$. To see this, suppose for contradiction that there exists k > 2 such that there are k positive functions
$\phi_1, \ldots, \phi_k \in H^1(D)$, orthogonal in
$L^2(\Omega)$, satisfying

for every
$i\in \{1,\ldots, k\}$.
Define the rescaled functions

Additionally, let

where M 0 is as given in lemma 4.1.
By direct computations, there exist constants
$\tilde{C}_{M_0}, \tilde{C}^{\prime}_{M_0} \gt 0$ such that the following estimates hold:

and

Note that
$a_0^{-\frac{1}{p-2}}\tilde{u}_n\leq \tilde{\epsilon}_n^{\frac{2}{p-2}} M_0$ for
$x\in \tilde{\Omega}_{n, M_0}$, by (4.10), it follows that

This implies that
$m(u_n)\geq k \gt 2$ for sufficiently large n, thereby yielding a contradiction. Thus, the claim is valid.
Having established that
$\tilde u$ is a finite Morse non-trivial solution of (4.9), we can apply either [Reference Farina22, theorem 2] or [Reference Yu57, proposition 2.1]. This allows us to conclude that the occurrence of
$\tilde{\lambda}=0$ is ruled out, regardless of whether D is a half-space or a whole space. Consequently, we can assert that
$\tilde{\lambda}\in (0, a_0]$.
In the sequel, we consider the sequence
$\{v_n\}$ defined by (4.4). Clearly, vn satisfies

Up to a subsequence, we have
$v_n\to v$ in
$C^2_{loc}(\overline{D})$, where D is either
$\mathbb{R}^N$ or a half space
$\mathbb{H}$, and v solves

Arguing as above implies that
$m(v)\leq 2$.
More precisely, we distinguish the following three cases:
(1) If
$\limsup\limits_{n\to +\infty}\frac{dist(P_n, \partial\Omega)}{\tilde{\epsilon}_n}= +\infty$, then
$D=\mathbb{R}^N$. Using
$m(v)\le2$, by standard regularity arguments,
$v\in C^2(\mathbb{R}^N)$,
$|v(x)|\to0$ as
$|x|\to+\infty$. If
$v\ge0$, then using the strong maximum principle and [Reference Esposito and Petralla20, theorem 1.1] we know that
$v \gt 0, m(v)=1$, and it coincides with the unique radial ground-state solution U 0 to
$-\Delta u+u=u^{p-1}$ in
$\mathbb{R}^N$. If v is sign-changing, then
$m(v)=2$ and v admits exactly one local maximum point and one local minimum point in Ω.
(2) If there exists a constant a > 0 such that
\begin{eqnarray*} 0 \lt a\leq\limsup_{n\to +\infty}\frac{dist(P_n, \partial\Omega)}{\tilde{\epsilon}_n} \lt +\infty, \end{eqnarray*}
we may assume that
$\lim\limits_{n\to +\infty} \frac{dist(P_n, \partial\Omega)}{\tilde{\epsilon}_n}= d \gt 0$. Then, up to subsequences,
$v_n\to v $ in
$C^2_{loc}(\{x_N \gt -d\})$, and v weakly solves
\begin{equation*} \left\{\begin{array}{ll} -\Delta v+ v=|v|^{p-2}v & {\mathrm{in}} \,~ \{x_N \gt -d\},\\ |v(x)|\leq |v(0)|= \tilde{\lambda}^{-\frac{1}{p-2}} & {\mathrm{in}} \,~ \{x_N \gt -d\}, \\ \displaystyle \frac{\partial v}{\partial x_N}=0 \, &{\mathrm{on}}\,~\{x_N=-d\}. \end{array}\right. \end{equation*}
Let
$\tilde{v}(x)=v(x',x_N-d)$, where
$x'=(x_1,x_2,\ldots,x_{N-1})$. It is straightforward to verify that
$\tilde{v}$ is a finite Morse index solution to the following system:
\begin{equation*} \left\{\begin{array}{ll} -\Delta \tilde{v}+ \tilde{v}=|\tilde{v}|^{p-2}\tilde{v} & {\mathrm{in}} \,~ \{x_N \gt 0\},\\ |\tilde{v}(x)|\leq |\tilde{v}(0,\ldots,0,d)|= \tilde{\lambda}^{-\frac{1}{p-2}} & {\mathrm{in}} \,~ \{x_N \gt 0\}, \\ \displaystyle \frac{\partial \tilde{v}}{\partial x_N}=0 \, &{\mathrm{on}}\,~ \{x_N=0\}. \end{array}\right. \end{equation*}
Next, we extend
$\tilde{v}$ by reflection with respect to
$\{x_N= 0\}$. Specifically, for
$(x',x_N)\in \mathbb{R}^N$, we define
\begin{equation*} \hat{v}(x',x_N):= \begin{cases} &\tilde{v}(x',x_N) ~~~~~~~~{\rm if}~x_N\geq 0,\\ &\tilde{v}(x',-x_N)~~~~~~{\rm if}~x_N \lt 0. \end{cases} \end{equation*}
Consequently,
$\hat{v}$ satisfies the equation
\begin{equation*} \left\{\begin{array}{ll} -\Delta \hat{v}+ \hat{v}=|\hat{v}|^{p-2}\hat{v} & {\mathrm{in}} \,~ \mathbb{R}^N,\\ |\hat{v}(x)|\leq |\hat{v}(0,\ldots,0,-d)|= |\hat{v}(0,\ldots,0,d)|=\tilde{\lambda}^{-\frac{1}{p-2}} & {\mathrm{in}} \,~ \mathbb{R}^N, \\ \displaystyle \frac{\partial \tilde{v}}{\partial x_N}=0 \, &{\mathrm{on}}\,~ \{x_N=0\}. \end{array}\right. \end{equation*}
Thus,
$\hat{v}$ is a bounded function that solves the equation
$-\Delta \hat{v}+ \hat{v}=|\hat{v}|^{p-2}\hat{v}$ in
$\mathbb{R}^N$ in the weak sense. By applying Schauder interior and boundary estimates (see [Reference Gilbarg and Trudinger26]), we conclude that
$\hat{v}\in C^{2,\alpha}_{loc}({\mathbb{R}^N})$ for
$\alpha\in (0,1)$, and the regularity extends up to the hyperplane
$\{x_N=0\}$. Since
$\hat{v}$ is symmetric across
$\{x_N=0\}$, it follows that the second derivatives of
$\hat{v}$ are continuous everywhere, including at the boundary
$\{x_N=0\}$. Consequently, by applying elliptic regularity theory and considering that
$\hat{v}$ is a solution with finite Morse index, we can show that
$\hat{v}\in C^2(\mathbb{R}^N)$ and
$|\hat{v}(x)|\to0$ as
$|x|\to+\infty$. Therefore,
$\tilde{v}$ must be a sign-changing solution, a situation that can occur according to [Reference McLeod, Troy and Weissle41, theorem 1]. In fact, if
$\hat{v}$ is a positive solution, then by applying [Reference Esposito and Petralla20, theorem 1.1] once again, we deduce that
$\hat{v}$ coincides with the unique radial ground-state solution U 0 to
$-\Delta u+u=u^{p-1}$ in
$\mathbb{R}^N$. It is well known that U 0 has a unique global maximum and is a radially strictly decreasing function. This would contradict the fact that
$\max\limits_{x\in\mathbb{R}^N} \hat{v}=|\hat{v}(0,\ldots,0,-d)|= |\hat{v}(0,\ldots,0,d)|$.
(3) If
$\limsup\limits_{n\to +\infty}\frac{dist(P_n, \partial\Omega)}{\tilde{\epsilon}_n}= 0$, then, up to a subsequence,
$\Omega_n\to \mathbb{R}^N_+(\{x_N \gt 0\})$. By similar arguments as above,
$v_n\to v$ in
$C^2_{loc}(\mathbb{R}^N_+)$, where v is a finite Morse index of the following equation
\begin{equation*} \left\{\begin{array}{ll} -\Delta v+ v=|v|^{p-2}v & {\mathrm{in}} \,~ \{x_N \gt 0\},\\ \displaystyle \frac{\partial v}{\partial x_N}=0 \, &{\mathrm{on}}\,~\{x_N=0\},\\ |v(y)|\leq |v(0)|= \tilde{\lambda}^{-\frac{1}{p-2}}. \end{array}\right. \end{equation*}
Extending v by reflection with respect to
$\{x_N=0\}$ and defining
\begin{equation*} \hat{v}(x',x_N):= \begin{cases} v(x',x_N)& ~{\rm if}~~x_N\geq 0,\\ v(x',-x_N)&~{\rm if}~~x_N \lt 0. \end{cases} \end{equation*}
Then
$\hat{v}$ satisfies
\begin{equation*} \left\{\begin{array}{ll} -\Delta \hat{v}+ \hat{v}=|\hat{v}|^{p-2}\hat{v} & {\mathrm{in}} \,~ \mathbb{R}^N,\\ |\hat{v}(x)|\leq |\hat{v}(0)| & {\mathrm{in}} \, \mathbb{R}^N, ~\\ \displaystyle \frac{\partial v}{\partial x_N}=0 \, &{\mathrm{on}}\,~\{x_N=0\}. \end{array}\right. \end{equation*}
Using arguments analogous to those in case (1), we deduce that if
$\hat{v}$ is positive, then
$m(\hat{v})=1$, and thus
$\hat{v}$ coincides with U 0. Using [Reference McLeod, Troy and Weissle41, theorem 1], it is established that
$\hat{v}$ may exhibit sign-changing behaviour, and in such instance,
$m(v)=2$.
All in all, we get that (ii) holds. Employing similar reasoning to [Reference Esposito and Petralla20, theorem 3.1], we conclude that (iii) and (iv) are also satisfied.
We now proceed to provide a comprehensive global blow-up analysis. The following result offers a detailed global description of the asymptotic behaviour of
$\{u_n\}$ as
$\lambda_n\to+\infty$.
Lemma 4.4. There exists
$k\in \{1,2\}$ and sequences of points
$\{P_n^1\},\ldots,\{P_n^k\}$, such that


and moreover,

where
$d_n(x)=\min\{|x-P_n^i|:i=1,\ldots,k\}$ is the distance function from
$\{P_n^1,\ldots, P_n^k\}$ for
$x\in \overline{\Omega}$.
Proof. Take
$P_n^1\in \overline{\Omega}$ such that
$u_n(P_n^1)=\max\limits_{\overline{\Omega}}|u_n(x)|$. If (4.15) is satisfied for
$P_n^1$, then we get k = 1. It is evident that
$P_n^1$ satisfies (4.13).
Otherwise, if
$P_n^1$ does not satisfy (4.15), we suppose that there exists δ > 0 such that

For sufficiently large R, up to a subsequence, it holds

Let
$P_n^2\in \overline{\Omega}\backslash B_{R\lambda_n^{-1/2}}(P_n^1)$ such that

Then, (4.17) yields that
$|u_n(P_n^2)|\to +\infty$ as
$n\to +\infty$.
We claim that

If (4.18) is not true, then up to subsequence
$\lambda_n^{\frac{1}{2}}|P_n^1-P_n^2|\to R'\geq R$. Define

As in lemma 4.3, we can deduce that
$ v_{n,1}\to v$ in
$C^2_{loc}(\overline{D})$, where
$D=\mathbb{R}^N$ or D is a half-space. Then, up to subsequences,

Since
$v(x)\to 0$ as
$|x|\to \infty$, taking R larger if necessary, it follows that

This contradicts to (4.17), which proves the claim (4.18).
In the following, we shall show that

Let
$\tilde{\epsilon}_{n,2}=a_0^{-\frac{1}{2}}|u_n(P_n^2)|^{-\frac{p-2}{2}}$. Clearly,
$\tilde{\epsilon}_{n,2}\to 0$. By (4.17), we get
$\tilde{\epsilon}_{n,2}\leq (2\delta)^{-\frac{p-2}{2}}\lambda_n^{-\frac{1}{2}}$. From (4.18), we can assert that

On the other hand, for any
$x\in B_{\tilde{R}_{n,2}\tilde{\epsilon}_{n,2}}(P_n^2)$ and R > 0, we have

for arbitrarily large n. Consequently,

which implies that

We define

Set
$d_{n,2}={\rm dist} (P_n^2, \partial \Omega)$. Then, passing to subsequences if necessary,

Then
$\tilde{u}_{n,2}$ satisfies the following equation

Since
$P_n^2$ is a local maximum or a local minimum point, by lemma 4.1, we get

Using the similar argument as in lemma 4.3, we deduce
$\tilde{\lambda}^{(2)} \gt 0$, namely

Set

where
$\epsilon_n=\lambda_n^{-\frac{1}{2}}$ is defined in (4.4). Up to a subsequence, there exists a function
$v^{(2)}\in H^1(D)$ such that
$v_{n,2}\to v^{(2)}$ in
$C_{loc}^2(\overline{D})$, where D is either
$\mathbb{R}^N$ or a half space. Moreover,
$v^{(2)}$ solves the following problem

Then, by a similar discussion as above, we conclude that either
$\frac{d_{n,2}}{\epsilon_n}\to +\infty$ or
$\frac{d_{n,2}}{\epsilon_n}$ remains bounded. Define
$R_{n,2}= \tilde{R}_{n,2}\lambda_n^{\frac{1}{2}}\tilde{\epsilon}_{n,2}$. Clearly,
$ R_{n,2}\to +\infty$ as
$n\to+\infty$. Hence, (4.19) holds.
If (4.15) does not hold, we can apply similar arguments as before to show that there exists
$P_n^3$ such that (4.13)–(4.14) are satisfied. For
$P_n^i,i=1,2,3$, applying lemma 4.3 again, we can find
$\phi^i_n\in C^\infty_0(\Omega)$ with supp
$\phi^i_n\subset B_{R\varepsilon_n}(P_n^i)\cap \overline{\Omega}$ for some R > 0, such that

In light of (4.14), we observe that
$\phi^1_n,\phi^2_n,\phi_n^{3}$ are mutually orthogonal for sufficiently large n, which implies
$\lim\limits_{n\rightarrow+\infty}m(u_n)\ge3$. This leads to a contradiction with the fact that
$m(u_n)\leq2$.
In the subsequent analysis, we show that un exhibits exponential decay away from the blow-up points.
Lemma 4.5. Let
$\{P_n^1\},\ldots,\{P_n^k\}$ be given in lemma 4.4. Then there exist constants
$C_1 \gt 0$,
$C_2 \gt 0$ such that, for some R > 0,

Proof. The proof is inspired by [Reference Li and Zhao38, lemma 2.1] and [Reference Esposito and Petralla20, theorem 3.2]. For any θ > 0, we set

and define the inner normal bundle

where νx is the unit outer normal of
$\partial\Omega$ at x.
Since
$\partial\Omega$ is a smooth compact submanifold of
$\mathbb{R}^N$, by the tubular neighbourhood theorem, it follows that there exists a diffeomorphism
$\Phi_{Ib}$ from
$\Omega_\theta$ onto
$(\partial\Omega)_\theta$. More precisely, for any
$x\in \Omega_\theta$, it is easily seen that there exists a unique
$\bar{x}\in \partial\Omega$ such that
${\rm dist}(x, \bar{x})= {\rm dist}(x, \partial\Omega)$. Hence, we can define
$\Phi_{Ib}(x):=(\bar{x}, -{\rm dist}(x, \bar{x})\nu_{\bar{x}})$ for any
$x\in \Omega_\theta$. Clearly,
$\Phi_{Ib}(x)=x$ for
$x\in \partial\Omega$.
Similarly, let

and define the outer normal bundle

Then there exists a diffeomorphism
$\Phi_{Ob}:\Omega^\theta \to (\partial\Omega)^\theta$ defined by
$\Phi_{Ob}(x):=(\hat{x}, dist(x, \hat{x})\nu_{\hat{x}}), \forall x\in \Omega^\theta $. Here
$\hat{x}\in \partial\Omega$ is the unique element in
$\partial \Omega$ such that
${\rm dist}(x, \hat{x})= {\rm dist}(x, \partial\Omega)$. Moreover,
$\Phi_{Ob}|_{\partial\Omega}= {\rm Identity}$.
Consider the reflection
$\Phi_C:(\partial\Omega)_\theta\to (\partial\Omega)^\theta$ defined by
$\Phi_C((x,y)):=(x,-y)$. Then,
$\Phi:=\Phi_{Ib}^{-1}\circ \Phi_C^{-1}\circ \Phi_{Ob}$ is a diffeomorphism from
$\Omega^\theta$ onto
$\Omega_\theta$ and
$\Phi |_{\partial\Omega}={\rm Identity}$. Furthermore, we take
$x=\Phi(z)=(\Phi_1(z),\ldots,\Phi_N(z))$ for
$z\in\Omega^\theta$,
$z=\Psi(x)=\Phi^{-1}(x)=(\Psi_1(x),\ldots,\Psi_N(x))$ for
$x\in \Omega_\theta$, and

Then
$g_{ij}|_{\partial \Omega}= g^{ij}|_{\partial\Omega}=\delta_{ij}$, where δij is the Kronecker symbol. For simplicity, denote
$G=(g^{ij})$,
$g(x)=det(g_{ij})$ and
$\hat{u}_n(x)=u_n(\Phi(x))$ for
$x\in \Omega^\theta$. Then
$\hat{u}_n$ satisfies

where

Next we define

and
$\bar{g}:= det(\bar{g}_{ij})$. Let
$A(x, \xi)=(A_1(x, \xi),\ldots,A_N(x, \xi))$ for
$\xi=(\xi_1,\ldots,\xi_N)$ with

Then
$\bar{u}_n$ weakly solves

Given that Φ is a diffeomorphism, the functions
$g_{ij}(x), \frac{\partial g_{ii}}{\partial x_i}$ and
$\frac{\partial g_{ij}}{\partial x_i}$ are all smooth on the domain
$\Omega^\theta$. Consequently, there exists a constant
$C_0 \gt 0$ such that
$|\frac{\partial g_{ii}}{\partial x_i}|, |\frac{\partial g_{ij}}{\partial x_i}|\leq C_0$ are both bounded by C 0 for
$x\in \Omega^\theta$. Moreover, according to the Taylor expansion, as θ approaches 0, the functions gij and g ij tend to 0, while gii and g ii tend to 1, and the determinant g(x) converges to 1.
For any
$\tilde{x}\in \overline{\Omega}$, there exists β > 0 such that
$B_\beta (\tilde{x})\subset \overline{\Omega}\cup \Omega^\theta$. For
$x\in B_\beta(\tilde{x})\backslash \tilde{x}$, denote
$\sigma= |x-\tilde{x}|$. Then, for any smooth increasing function
$\phi(\sigma)$, we have

for some
$C_{\Phi ,\Omega} \gt 0$, where
$\phi'=\frac{d\phi(\sigma)}{d\sigma}$. Moreover, for θ > 0 small enough, there exist small constants
$\delta_0, \delta_1$ such that

and

Then

From now, let us fix
$\theta_*\in (0, \theta)$ such that
$\frac{1}{2}\leq\sqrt{g}\leq \frac{3}{2}$ and (4.24) holds. Define

In view of (4.5), for every
$\epsilon\in (0,1)$ small, to be chosen later, there exist
$R^* \gt 0$ and
$n_R\in \mathbb{N}$ large such that

Let
$A_n: =\{x\in \overline{\Omega}: d_n(x)\geq R\lambda_n^{-1/2}\}$. Clearly,

Note that there exists a constant
$n_\theta \gt 0$ large enough such that for
$n \gt n_\theta$, for any
$i\in \{1,\ldots, k\}$, we have
$B_{\lambda_n^{-1/2}R}(P_n^i)\subset \Omega^{\theta_*}\cup \Omega_{\theta_*}$. Then we can deduce that
$A_n\subset A_n^*$, for
$n \gt \max\{n_\theta, n_R\}:= n_{\theta R}$. From the definition of
$\bar{u}_n$, we get

Using
$(f_1)$–
$(f_2)$, we conclude that

holds for sufficiently small ϵ > 0.
For any
$x_0\in A_n$ such that
$B_r(x_0)\subset A_n^*$, consider the function

Clearly,
$\phi'(\rho) \gt 0$ and
$\phi^{\prime\prime}(\rho) \gt 0$. By direct computations, we obtain

for any
$\gamma\in (0, \gamma_*]$, where
$\gamma_*=(4N+2C_{\Phi,\Omega})^{-\frac{1}{2}}$. Then, fixing
$\gamma\in (0, \gamma_*]$, by (4.24) and (4.26) it follows that

In addition, for
$x\in \partial B_r(x_0)$, we have
$\left(\phi_n -\bar{u}_n \right) \geq \lambda_n^{\frac{1}{p-2}}(1-\epsilon) \gt 0$. Then, together with (4.25) and (4.27), by the comparison principle ([Reference Pucci and Serrin52, theorem 10.1]) it follows that
$\bar{u}_n\leq \phi_n~~{\mathrm{in}}~~B_r(x_0)$, which implies that
$u_n(x_0)\leq \lambda_n^{\frac{1}{p-2}} e^{-\gamma\lambda_n^{\frac{1}{2}}r}$.
Take
$r={\rm dist}(x_0, \partial A_n^*)$. We distinguish the following two cases:
(i)
$dist(x_0, \partial A_n^*)= dist(x_0, B_{R\lambda_n^{-1/2}}(P_n^i))$ for some
$i\in \{1,\ldots,k\}$;
(ii)
$dist(x_0, \partial A_n^*)= dist(x_0, \partial(\overline{\Omega}\cup \Omega^{\theta_*}))$.
For case (i), we have
$|x_0- P_n^i|= r+ R\lambda_n^{-\frac{1}{2}}$. Then

For case (ii), we deduce that

Hence, since x 0 is arbitrary, we conclude that there exist
$C_1, C_2 \gt 0$ such that

This completes the proof.
5. Proof of theorem 1.1
In this section, we complete the proof of theorem 1.1. We have previously established a sequence of mountain pass type critical points
$\{u_{\rho_n}\}$ of
$J_{\rho_n}$ on
$\mathcal{S}_c$, which have uniformly bounded Morse indices. These critical points are constructed for a sequence
$\rho_n\to 1^-$. Building on the blow-up analysis conducted in §4, we now present the following proposition.
Proposition 5.1. Let
$\{u_n\}\subset H^1(\Omega)$ be a sequence of solutions to (4.1), corresponding to some
$\{\lambda_n\}\subset \mathbb{R}$ and
$\rho_n\to 1^-$. Suppose that

for some c > 0, and the energy levels
$\{c_n:=J_{\rho_n}(u_n)\}$ are bounded. Then, the sequence of solution pairs
$\{(u_n, \lambda_n)\}$ is bounded in
$H^1(\Omega)\times \mathbb{R}$. Moreover,
$\{u_n \}$ is a bounded Palais–Smale sequence for the functional J constrained on
$\mathcal{S}_c$ at level c 1.
Proof. First, we show that
$\{\lambda_n\}$ is bounded. Suppose, by contradiction, that
$\lambda_n\to +\infty$. By lemma 4.4, there exist at most k blow-up limits
$\{P_n^1\},\ldots,\{P_n^k\}$ with
$k\leq 2$ in
$\overline{\Omega}$.
In the following, we denote by
$\{v_n^i \}$ the scaled sequence around
$\{P_n^i\}$, and by vi the limits of
$\{v_n^i\}$. Note that for these blow-up points
$\{P_n^1\},\ldots,\{P_n^k\}$, it may hold

Without loss of generality, assume there exists an integer
$k_1\in \{0, 1,\ldots, k\}$ such that

and

On the one hand, we can deduce that for any R > 0,

In fact, since
$p\in (2_*, 2^*)$, the first term satisfies

By lemma 4.4, we have

and

which imply that (5.1) holds.
On the other hand, by lemma 4.5, there exist constants
$C, C' \gt 0$ such that

Taking
$n\to +\infty$, we obtain a contradiction to (5.1).
Hence,
$\{\lambda_n\}$ is bounded.
Now, we show that
$\{u_n\}$ is bounded in
$H^1(\Omega)$. By contradiction, we suppose that
$\|u_n\|\to \infty$. Using standard arguments, we have
$\|u_n\|_{L^\infty} \to \infty.$ Take
$P_n\in \overline{\Omega}$ such that
$|u_n(P_n)|=\|u_n\|_{L^{\infty}}$. Let
$\tilde{\epsilon}_n=a_0^{-\frac{1}{2}}|u_n(P_n)|^{-\frac{p-2}{2}}$ and define

Clearly,
$\tilde{u}_n$ satisfies (4.8). Using the boundedness of
$\{\lambda_n\}$, we get
$\lambda_n \tilde{\epsilon}_n^2 \to 0$ as
$n\to \infty$. Then, up to a subsequence,
$\tilde{u}_n\to \tilde{u}$ in
$C^2_{loc}(\overline{D})$, where
$\tilde{u}$ is a finite Morse index solution of

where D is either
$\mathbb{R}^N$ or a half space. By invoking [Reference Farina22, theorem 2] and [Reference Yu57, proposition 2.1], respectively, we conclude that
$\tilde{u}\equiv 0$. This contradicts to the fact that
$|\tilde{u}(0)|=1$. Therefore, we deduce that
$\{u_n\}$ is a bounded sequence in
$H^1(\Omega)$. Consequently, employing standard arguments, we establish that
$\{u_n \}$ is a bounded Palais–Smale sequence for
$J|_{\mathcal{S}_c}$ at level c 1.
Completion of proof of theorem 1.1
Let
$\{u_n\}$ be the sequence given in proposition 5.1 for some
$c\in (0, c^*)$. By the compact embedding
$H^1(\Omega)\hookrightarrow L^r(\Omega)$ for
$r\in [1,2^*)$, and using similar arguments as in §3, we can deduce that
$u_{n}\to u$ strongly in
$H^1(\Omega)$. This, in turn, implies that u is a mountain pass type normalized solution of (1.1).
Acknowledgements
Xiaojun Chang was partially supported by NSFC (12471102) and the Research Project of the Education Department of Jilin Province (JJKH20250296KJ). The research of V.D. Rădulescu was supported by the grant ‘Nonlinear Differential Systems in Applied Sciences’ of the Romanian Ministry of Research, Innovation and Digitization, within PNRR-III-C9-2022-I8/22.