1. Introduction and main results

In 1983, Brézis and Nirenberg in [1.1] studied the following problem:

(1.1)\begin{equation} \left\{ \begin{array}{rllll} -\Delta u & = & \lambda u^{q-1} + u^{2^*-1} & \mbox{in} & \Omega, \\ u & \gt & 0 & \mbox{in} & \Omega, \\ u&=&0 & \mbox{on} & \partial \Omega,\\ \end{array} \right. \end{equation}

where $\Omega \subset \mathbb{R}^N$ is a bounded domain, $N \geq 3$, λ is a fixed real parameter, $q \in [2,2^{*})$ and $2^*=2N/(N-2)$ is the critical exponent in the sense of Sobolev’s embedding.

Brézis and Nirenberg proved the following results:

1. (a) For q = 2 and $ N \geq 4 $, problem (1.1) has a solution for every $ \lambda \in (0, \lambda_1) $, where λ ₁ denotes the first eigenvalue of $ -\Delta $. Moreover, it has no solution if $ \lambda \not\in (0, \lambda_1) $ and Ω is star-shaped.

2. (b) When q = 2, N = 3, and Ω is a ball, problem (1.1) has a solution if and only if $ \lambda \in \left(\frac{\lambda_1}{4}, \lambda_1\right) $.

3. (c) For $ q \in (2, 2^*) $ and $ N \geq 4 $, problem (1.1) has a solution for every λ > 0.

4. (d) When N = 3 and $ 4 \lt q \lt 6 $, problem (1.1) has a solution for every λ > 0.

5. (e) When N = 3 and $ 2 \lt q \leq 4 $, problem (1.1) has a solution only for sufficiently large values of λ.

Recently, do Ó, Ruf, and Ubilla in [Reference do Ó, Ruf and Ubilla5] studied the following problem:

(1.2)\begin{equation} \left\{ \begin{array}{rllll} -\Delta u & = & u^{2^* + r^{\alpha}-1} & \mbox{in} & B, \\ u & \gt & 0 & \mbox{in} & B, \\ u&=&0 & \mbox{on} & \partial B,\\ \end{array} \right. \end{equation}

where $ B \subset \mathbb{R}^N $ is the unit ball centred at the origin, $N\geq 3$, $r=|x|$, and $\alpha \in (0,\min\{N/2,N-2\})$.

The authors demonstrated that problem (1.2) has a radial solution, which is surprising because it corresponds to a supercritical perturbation of the equation $-\Delta u = u^{2^* - 1}$, which has no solution due to the known Pohozaev identity. In this same line of reasoning, in the context of the situation of item (b), we studied the effect of a supercritical perturbation for the case of non-existence $\lambda \in (0,\frac{\lambda_1}{4 }]$, which also generated the existence of a positive solution. We will also have the same conclusion for situation (e), in which, due to the supercritical perturbation, we will obtain a solution for all positive λ and not just for sufficiently large λ. Motivated by the results of [1.1] and [Reference do Ó, Ruf and Ubilla5], we studied this problem in a more general context, more precisely, let us consider the following problem:

(1.3)\begin{equation} \left\{ \begin{array}{rllll} -\Delta u & = & \lambda u^{q-1} + u^{2^*+ r^\alpha -1} & \mbox{in} & B, \\ u & \gt & 0 & \mbox{in} & B, \\ u&=&0 & \mbox{on} & \partial B,\\ \end{array} \right. \end{equation}

where $B \subset \mathbb{R}^N$ is a unit ball centred at the origin, $N\geq 3$, $r=\vert x \vert$, and $\alpha \in (0,\min\{N/2,N-2\})$ and λ is a fixed real parameter and $q\in [2,2^*]$.

We will now present the main result of this article.

We would like to highlight that in the case N = 3 we obtain a solution for the perturbed problem for each $\lambda \in [0,\lambda_1)$, that is, the perturbation solves the non-existence interval $[0,\lambda_1/4]$.

Let $H_0^1(B) := \{u \in L^{2}(B) \colon \nabla u \in L^2(B) \colon u=0 \, \text{on} \, \partial B \} $ be the usual Sobolev space equipped with the gradient norm, or let $\| u \|_{H^1_0(B)} = \|\nabla u \|_{L^2(B)}$. We say that $u\in H_0^1(B)$ is a weak solution to problem (1.3) if u > 0 in B and it holds:

(1.4)\begin{align} \int_B \nabla u \nabla\varphi \, \text{d}x = \lambda \int_B u^{q-1}\varphi \, \text{d}x + \int_B u^{2^*-1+r^{\alpha}}\varphi \, \text{d}x, \, \forall\, \varphi \in H_0^1(B). \end{align}

Theorem 1.1 shows (see (b) and (e)) that there are critical equations without solutions that have a solution when a non-negative term is added to them, converting them into supercritical equations. Note that this phenomenon was already observed in [Reference do Ó, Ruf and Ubilla5].

We also consider some perturbations of problem (1.1) that become superlinear on the ball and subcritical for $ r \in (0, \delta) $, for some small δ. However, it can be supercritical away from $r= 0,$ as in the following equation:

(1.5)\begin{equation} \left\{ \begin{array}{rllll} -\Delta u & = & \lambda u^{q-1} + u^{2^* + f(r)-1} & \mbox{in} & B, \\ u & \gt & 0 & \mbox{in} & B, \\ u&=&0 & \mbox{on} & \partial B,\\ \end{array} \right. \end{equation}

where $B \subset \mathbb{R}^N$ is a unit ball centred at the origin, $N\geq 3$, $r=\vert x \vert$, λ is a fixed real parameter, $q\in [2,2^*)$ and $f\colon [0,1) \rightarrow \mathbb{R}$ is a continuous function satisfying:

1. (f) $ f(0) \lt 0$ and $\displaystyle\inf_{r\in [0,1)}(2^* + f(r)) \gt 2$.

The next result involves the assumption (f):

The definition of a weak solution for problem (1.5) is analogous to the one we defined in Eq. (1.4). The case $0 \lt q \lt 2$, which corresponds to a concave-convex problem, was studied in [Reference Clemente, Marcos do Ó and Ubilla3] under a subcritical assumption. Therefore, theorem 1.3 complements the result in [Reference Clemente, Marcos do Ó and Ubilla3].

The article is organized as follows: In §2, we present preliminary results, in §3, we prove theorem 1.1, and in §4, we prove theorem 1.3.

2. Preliminaries

First, we define the Sobolev space of radial functions $H_{0,\mathrm{rad}}^1(B) := \{u \in H_0^1(B) \colon u(x) = u(|x|) \}$ equipped with the usual standard $\|u\|=\|\nabla u \|_2$. We will now present the ‘radial lemma’, which can be found in [Reference do Ó, Ruf and Ubilla5, Reference Strauss8].

Lemma 2.1. Let $u \in H_{0,\mathrm{rad}}^1(B)$. Then

(2.1)\begin{equation} |u(r)| \leq \frac{1}{(N-2)^{1/2}}\frac{\|\nabla u\|_2}{r^{(N-2)/2}} \end{equation}

and

(2.2)\begin{equation} |u(r)| \leq \frac{(1-r)^{1/2}}{r^{(N-2)/2}}\|\nabla u\|_2 . \end{equation}

For the next result, we refer [Reference do Ó, Ruf and Ubilla5]

Proposition 2.2. Let α > 0; then

(2.3)\begin{equation} \sup \Big\{\int_{B} |u(x)|^{2^* + r^\alpha} \, \text{d} x \colon u \in H^1_{0, \mathrm{rad}}(B), \, \Vert \nabla u \Vert_2 = 1 \Big\} \lt +\infty. \end{equation}

Corollary 2.3. The following embedding is continuous:

(2.4)\begin{equation} H^1_{0,\mathrm{rad}}(B)\hookrightarrow L^{2^*+r^{\alpha}}(B) \ , \end{equation}

where $L^{2^*+r^{\alpha}}(B)$ is defined as follows (see, e.g., [Reference Diening, Harjulehto, Hästö and Rocircuvzivcka4])

\begin{equation*} L^{2^*+r^{\alpha}} (B) := \Big\{u\colon B \to \mathbb R\ \mathrm{measurable} \colon \int_B|u(x)|^{2^*+r^{\alpha}} \; \mathrm{d} x \lt \infty \Big\} \end{equation*}

with norm

\begin{equation*}\ \|u\|_{2^*+r^{\alpha}} = \inf \Big\{\lambda \gt 0 \ , \ \int_B\Big|\frac {u(x)}{\lambda}\Big|^{2^*+r^{\alpha}} \mathrm{d} x \le 1 \Big\}. \end{equation*}

The following proposition follows directly from the definition:

3. Proof of theorem 1.1

To establish a weak solution of problem (1.3), we define the functional $J \colon H^{1}_{0, \text{rad}}(B) \to \mathbb{R}$ given by

(3.1)\begin{equation} J(u) = \frac 1 2 \Vert \nabla u \Vert^2_2 -\frac{\lambda}{q}\Vert u^+ \Vert_q^q - \int_B \frac 1{2^* + r^\alpha}\, (u^+)^{2^* + r^{\alpha}} \text{d} x, \end{equation}

where $u^+(x)=\operatorname{max}\{u(x),0\}$. By proposition 2.2 and by corollary 2.3, it follows that the functional J is well defined. We also note that J is a functional of class C ¹. If u > 0 is a critical point of the functional then u is a weak solution to problem (1.3) thanks to the symmetric criticality principle (see [Reference Palais7, Reference Willem10]). The strategy then consists of obtaining positive critical points of the functional J. For this, we will use the Mountain Pass Lemma, due to Ambrosetti and Rabinowitz [Reference Ambrosetti and Rabinowitz1].

In the next lemmas, we will demonstrate that the functional J has the geometry of the Mountain Pass Theorem.

Lemma 3.1. There exist ρ > 0 and θ > 0 such that

\begin{align*} J(u) \geq \rho \gt 0, \, \, \text{if} \, \, \Vert \nabla u\Vert_2 = \theta. \end{align*}

Proof. Note that

\begin{align*} J(u) & = \frac{1}{2} \Vert \nabla u \Vert_{2}^2 - \frac{\lambda}{q}\Vert u^+ \Vert_q^{q} - \int_{B} \dfrac{( u^+ )^{2^* + r^\alpha}}{2^*+r^\alpha}\, \mathrm{d} x \\ & \geq \frac{1}{2} \Vert \nabla u \Vert_{2}^2 - \frac{\lambda}{q}\Vert u \Vert_q^{q} - \frac{1}{2^*}\int_{B}\vert u \vert^{2^* + r^\alpha}\, \mathrm{d} x. \end{align*}

Let $u \in H_{0, \mathrm{rad}}^1(B)$ be such that $\Vert \nabla u \Vert_2 = \theta$ where $\theta \in (0,1)$ will be chosen. By proposition 2.4 and corollary 2.3 follow that

\begin{align*} \int_{B}\vert u \vert^{2^* + r^\alpha}\, \mathrm{d} x \leq \Vert u \Vert^{2^*}_{2^* + r^\alpha} \leq C \Vert \nabla u \Vert^{2^*}_2. \end{align*}

Therefore,

(3.2)\begin{align} J(u) \geq \frac{1}{2} \Vert \nabla u\Vert^2_2 - \frac{\lambda}{q}\Vert u \Vert_q^q - \frac{C}{2^*} \Vert \nabla u\Vert^{2^*}_2. \end{align}

We observe that when q = 2, we consider $\lambda \in [0, \lambda_1)$, then the expression $ \sqrt{\Vert \nabla u \Vert^2 - \lambda \Vert u \Vert^2}$ defines a norm in $H_{0}^1(B)$ equivalent to norm $\|\nabla u \|_2$. Since $\| \nabla u \|_2 = \theta$, we have

\begin{align*} J(u) \geq \frac{C}{2} \theta^2 - \frac{C}{2^*} \theta^{2^*}. \end{align*}

So, choosing $\theta_1 \in (0,1)$ small enough we have that for $\theta \in (0,\theta_1)$ fixed there is $\rho_1 \gt 0$ such that $J(u) \geq \rho_1 \gt 0$.

If $q \in (2, 2^*]$, by using (3.2) and Sobolev inequality, we get

\begin{align*} J(u) &\geq \frac{1}{2} \Vert \nabla u\Vert^2_2 - \frac{C_1\lambda}{q}\Vert\nabla u \Vert_q^q - \frac{C}{2^*} \Vert \nabla u\Vert^{2^*}_2 \\ &= \frac{1}{2} \theta^2 - \frac{\lambda C_1}{q} \theta^q - \frac{C}{2^*} \theta^{2^*}. \end{align*}

Since $2^* \geq q \gt 2$, we can choose $\theta_2 \in (0,1)$ small enough such that for any fixed $\theta \in (0, \theta_2)$, there exists $\rho_2 \gt 0$ such that $J(u) \geq \rho_2 \gt 0$.

Now, we will state the second condition of the mountain pass geometry.

Proof. Let $u \in H_{0, \mathrm{rad}}^1(B) \setminus \{0 \}$ such that u > 0 in B. We have for t > 1 that

\begin{align*} J(tu) &= \frac{t^2}{2} \Vert \nabla u^+ \Vert^2_2 - \frac{\lambda t^q}{q}\Vert u \Vert^q_q - \int_{B} \dfrac{t^{2^* + r^\alpha }( u^+)^{2^* + r^\alpha}}{2^* + r^\alpha}\, \mathrm{d} x \\ & \leq \frac{t^2}{2} \Vert \nabla u \Vert^2_2 - \frac{\lambda t^q}{q}\Vert u^+ \Vert^q_q - \frac{t^{2^*}}{{2^* + 1}}\int_{B} (u^+)^{2^* + r^\alpha}\, \mathrm{d} x. \end{align*}

Therefore, since $2 \leq q \leq 2^*$ we get

\begin{align*} \lim_{t \to + \infty} J(tu) = - \infty, \end{align*}

which proves the lemma.

We now define S_N as the best constant in the Sobolev embedding $H^1(\mathbb{R}^N) \hookrightarrow L^{2^*}(\mathbb{R}^N)$, that is,

(3.3)\begin{align} S_N := \inf \left\{\frac{\Vert \nabla u \Vert^2_{L^2(\mathbb{R}^N)}}{\Vert u \Vert_{L^{2^*}(\mathbb{R}^N)}^2} \, \colon u\in L^{2^*}(\mathbb{R}^N)\setminus\{0\}; \,\, \nabla u \in L^2(\mathbb{R}^N) \right\}. \end{align}

We consider

\begin{align*} \bar{u}(x) = C\left( 1+ \vert x \vert^{2}\right)^{-\frac{(N-2)}{2}} \end{align*}

the standard Sobolev instantons, which satisfy the equation (see [Reference Talenti9])

\begin{equation*} -\Delta u = u^{2^*-1} \ , \ \hbox{on } \ \mathbb R^N. \end{equation*}

We also consider $u^*(x)=\bar{u}(x/S_N^{1/2})$ and $U_\varepsilon (x) = \varepsilon^{-\frac{(N-2)}{2}}u^*(x/\varepsilon)$. As in [Reference Talenti9] and also [Reference Willem10], we know that,

\begin{equation*} \int_{\mathbb{R}^N} \vert \nabla U_\varepsilon \vert^2 \,\text{d}x = S_N^{N/2} \ \hbox{and } \ \int_{\mathbb R^N} |U_\varepsilon|^{2^*} \text{d}x = S_N^{N/2}. \end{equation*}

Taking a suitable cut-off function η and setting $u_\varepsilon = \eta\, U_\varepsilon$, it is known that

(3.4)\begin{align} \int_B \vert \nabla u_{\varepsilon} \vert^2 \,\text{d}x = S_N^{N/2} + O(\varepsilon^{N-2})\ \ , \ \int_B|u_\varepsilon(x)|^{2^*}\; \mathrm{d} x = S_N^{N/2} + O(\varepsilon^N). \end{align}

Do Ó, Ruf, and Ubilla, in [Reference do Ó, Ruf and Ubilla5], demonstrated the following lemma:

Lemma 3.3. There exists a constant C > 0 such that for all ɛ > 0 small

\begin{equation*} \int_{B} |u_{\varepsilon}(x)|^{2^*+r^\alpha} \; \mathrm{d} x \ge \int_B|u_\varepsilon (x)|^{2^*}\; \mathrm{d} x + C\, |\log \varepsilon | \varepsilon^{\alpha} + O(\varepsilon^{N/2}) + O(\varepsilon^{N-2}) . \ \end{equation*}

Now let’s control the min-max level of the mountain pass theorem.

Proof. By lemmas 3.1 and 3.2, J has the geometry of the Mountain Pass lemma. We consider u_ɛ as before and set

\begin{equation*} c = \inf_{\gamma \in \Gamma} \max_{u \in \gamma} J(u) \end{equation*}

where

\begin{equation*}\Gamma := \Big\{\gamma : [0,R] \to H_0^1(B) \hbox{continuous }, \gamma(0) = 0, \gamma(1) = R\, u_\varepsilon \Big\}, \end{equation*}

with R > 0 sufficiently large such that $J(R\, u_\varepsilon) \le 0$. By (3.4) and lemma 3.3, we note that R can be chosen independent of ɛ. The path $\gamma_\varepsilon(t) = t u_\varepsilon, \, t \in [0,R]$, belongs to Γ, and

(3.5)\begin{equation} c \le \max_{t \in [0,R]} J(t\,u_\varepsilon) := J(t_\varepsilon u_\varepsilon) . \end{equation}

We have also that $\frac d{dt} J(t\, u_\varepsilon)\Big|_{t = t_\varepsilon} = 0$ and by J satisfying the geometric conditions of the Mountain Pass lemma, we can assume that $ t_\varepsilon \in (\delta, R]$ with δ > 0 because if $t_\varepsilon \rightarrow 0$ by (3.4) and lemma 3.3 we obtain that $J(t_\varepsilon u_\varepsilon) \rightarrow 0$. So, for ɛ > 0 small enough, we have $J(t_\varepsilon u_\varepsilon) \lt S_N^{N/2}/N$.

Now, let’s consider the following auxiliary functional:

\begin{align*} J_A(u) = \frac{1}{2}\Vert \nabla u \Vert_2^2 - \int_B \frac{(u^+)^{2^*+r^\alpha}}{2^*+r^\alpha}\,\mathrm{d}x, \,\, u \in H_{0,\mathrm{rad}}^1(B). \end{align*}

So, for $ t_\varepsilon \in (\delta, R]$ we have the estimate

(3.6)\begin{align} J(t_\varepsilon u_\varepsilon) & = \frac{t_\varepsilon^2}{2} \Vert \nabla u_\varepsilon \Vert^2_2 -\frac{\lambda t_\varepsilon^q}{q}\Vert u_\varepsilon \Vert_q^q - \int_B \frac {t_\varepsilon^{2^*+r^{\alpha}}}{2^* + r^\alpha} (u_\varepsilon)^{2^* + r^\alpha} \, \mathrm{d} x \nonumber \\ & \leq J_A(t_\varepsilon u_\varepsilon) \nonumber \\ & = \frac{t_\varepsilon^2}{2} \Vert \nabla u_\varepsilon \Vert^2_2 - \frac{t_\varepsilon^{2^{*}}}{2^*}\int_B u_\varepsilon^{2^*} \, \mathrm{d} x \nonumber \\ &\quad + t_\varepsilon^{2^*}\int_B \left(\frac 1{2^*} - \frac 1{2^* + r^\alpha}\right) u_\varepsilon^{2^*} \, \mathrm{d} x \nonumber \\ &\quad+ \int_B \frac 1{2^* + r^\alpha}\left( (t_\varepsilon u_\varepsilon)^{2^*} - (t_\varepsilon u_\varepsilon)^{2^* + r^\alpha}\right) \, \mathrm{d} x \nonumber \\ & \leq \max_{t \in [0,R]}\left( \frac{t^2}{2} \Vert \nabla u_\varepsilon \Vert^2_2 - \frac{t^{2^{*}}}{2^*}\Vert u_{\varepsilon}\Vert^{2^*}_{2^*} \right) + c\varepsilon^{\alpha} - c\varepsilon^{\alpha}\vert \log \varepsilon \vert \nonumber \\ &\quad + O(\varepsilon^{N/2}) + O(\varepsilon^{N-2})\nonumber \\ & = \frac{1}{2} \bigg( \frac{\Vert \nabla u_\varepsilon \Vert^2_2}{\Vert u_{\varepsilon} \Vert^{2^*}_{2^*}} \bigg)^{2/(2^*-2)} \Vert\nabla u_\varepsilon \Vert^{2}_2 - \frac{1}{2^*}\bigg( \frac{\Vert \nabla u_\varepsilon \Vert^2_2}{\Vert u_{\varepsilon}\Vert_{2^*}^{2^*}} \bigg)^{2^*/(2^*-2)}\Vert u_{\varepsilon}\Vert_{2^*}^{2^*} \nonumber \\ & \quad + c\varepsilon^{\alpha} - c\varepsilon^{\alpha}\vert \log \varepsilon \vert \nonumber \\ & = \frac{1}{N} \frac{\big( \Vert \nabla u_\varepsilon \Vert^2_2\big)^{2^*/(2^*-2)}}{\big( \Vert u_{\varepsilon}\Vert^{2^*}_{2^*} \big)^{2/(2^*-2)}} + c\varepsilon^{\alpha} - c\varepsilon^{\alpha}\vert \log \varepsilon \vert, \end{align}

where we use that $\alpha \in (0,\min \{N/2,N-2\})$, the lemma 3.3, and the estimate

(3.7)\begin{equation} \begin{aligned} \int_B\Big( \frac 1{2^*}- \frac{1}{2^* + r^\alpha}\Big) |u_\varepsilon|^{2^*} \mathrm{d} x & = \int_B \dfrac{r^\alpha}{2^*(2^* + r^\alpha)} |u_\varepsilon|^{2^*}r^{N-1} \; \mathrm{d} x \\ & \leq c \int_0^\varepsilon r^\alpha \varepsilon^{-N}r^{N-1}\; \mathrm{d} r + c \int_\varepsilon^1 r^\alpha \frac {\varepsilon^N}{r^{2N}}r^{N-1}\; \mathrm{d} r \\ & \leq c\, \varepsilon^\alpha + c\, (\varepsilon^\alpha - \varepsilon^N) = c\, \varepsilon^\alpha. \end{aligned} \end{equation}

Therefore, by using (3.4) and (3.6), we obtain

\begin{align*} J(t_\varepsilon u_\varepsilon) \leq J_A(t_\varepsilon u_\varepsilon) & \leq \frac{1}{N} \frac{\big( S_N^{N/2} + O(\varepsilon^{N-2})\big)^{2^*/(2^*-2)}}{\big( S_N^{N/2} + O(\varepsilon^{N})\big)^{2/(2^*-2)}} + c\varepsilon^{\alpha} - c\varepsilon^{\alpha}\vert \log \varepsilon \vert, \\ &=\frac{1}{N}S_N^{N/2} + O(\varepsilon^{N-2}) + c\varepsilon^{\alpha} - c\varepsilon^{\alpha}\vert \log \varepsilon \vert, \\ & \lt \frac{1}{N}S_N^{N/2}, \; \text{for}\; \varepsilon \gt 0 \; \text{small enough and for all}\, N \geq 3. \end{align*}

3.1. Proof of theorem 1.1

By lemmas 3.1 and 3.2, we know that the functional J satisfies the geometric conditions of the Mountain Pass lemma; by lemma 3.4, it follows that there is a sequence of Palais-Smale $\{u_n\}\subset H_{0,\text{rad}}^1(B)$ such that:

\begin{align*} J(u_n) \rightarrow c \lt \frac 1N S_N^{N/2}, \, \text{and}\, J'(u_n) \rightarrow 0. \end{align*}

Let’s show that the sequence $\{u_n\}$ is bounded in $H_{0,\text{rad}}^1(B)$. Indeed, for n sufficiently large and $q \in (2,2^*] $, we have:

\begin{align*} c + 1 + \Vert \nabla u_n \Vert_2 & \geq J(u_n) - \frac{1}{q} J'(u_n)u_n \\ & = \left( \frac{1}{2} - \frac{1}{q}\right)\Vert \nabla u_n \Vert_2^2 + \int_B \left( \frac{1}{q}-\frac{1}{2^*+r^\alpha}\right)(u_n^+)^{2^*+r^\alpha}\,\mathrm{d}x \\ & \geq \left( \frac{1}{2} - \frac{1}{q}\right)\Vert \nabla u_n \Vert_2^2. \end{align*}

It follows that $\{u_n\}$ is bounded in $H_{0,\mathrm{rad}}^1(B)$. If q = 2, we recall that $\lambda \in [0, \lambda_1)$ and in this case the expression $(\Vert \nabla u \Vert^2_2 - \lambda \Vert u \Vert^2_2)^{1/2}$ defines a norm in $H^1_{0,\mathrm{rad}}(B)$ equivalent to the usual norm $\Vert \nabla u \Vert_2$. Thus, we will also have for n sufficiently large that:

\begin{align*} c + 1 + \Vert \nabla u_n \Vert_2 & \geq J(u_n) - \frac{1}{2^*} J'(u_n)u_n \\ & \geq \left( \frac{1}{2} - \frac{1}{2^*}\right)\left( \Vert \nabla u_n \Vert_2^2 - \lambda \Vert u \Vert_2^2\right) \\ & \quad + \int_B \left( \frac{1}{2^*}-\frac{1}{2^*+r^\alpha}\right)(u_n^+)^{2^*+r^\alpha}\,\mathrm{d}x \\ & \geq c_1 \Vert \nabla u_n \Vert_2^2. \end{align*}

It follows that $\{u_n\}$ is bounded in $H_{0,\mathrm{rad}}^1(B)$. So there exists $ u \in H_{0,\text{rad}}^1(B) $ such that $ u_n \rightharpoonup u $ in $ H_{0,\text{rad}}^1(B) $. We have two possibilities:

If $ u \not\equiv 0 $, then u is a non-trivial non-negative solution to problem (1.3). By the maximum principle, we guarantee that u is positive, thus proving the theorem.

If u = 0, we have $ u_n \rightharpoonup 0 $, and for every ɛ > 0 and n sufficiently large, the following inequality holds:

(3.8)\begin{align} \int_B |u_n|^{2^* + r^{\alpha}} \, \text{d}x - \int_B |u_n|^{2^*} \, \text{d}x \leq \varepsilon. \end{align}

Indeed, note that for all η > 0, we have $H^1_{0,\mathrm{rad}}(B \setminus B_\eta) \subset\subset L^s(B \setminus B_\eta)$ for all $s \geq 1$. Therefore,

\begin{align*} u_n \rightarrow 0 \text{in } L^s(B \setminus B_\eta) \text{for all } s \geq 1, \end{align*}

and consequently,

(3.9)\begin{align} \int_{B \setminus B_\eta} |u_n|^{2^*+r^{\alpha}} \, \text{d} x \rightarrow 0 \text{and } \int_{B \setminus B_\eta} |u_n|^{2^*} \, \text{d} x \rightarrow 0 \text{as } n \rightarrow \infty. \end{align}

By (3.9), we can write

(3.10)\begin{align} \int_{B_\eta} |u_n|^{2^*+r^{\alpha}} \, \text{d} x &= \omega_{N-1} \int_0^{\eta} |u_n(r)|^{2^*+r^{\alpha}} \; r^{N-1} \text{d} r \\ &= \omega_{N-1} \int_0^{\eta} |u_n(r)|^{2^*} \left( |u_n(r)|^{r^{\alpha}} - 1 \right) \; r^{N-1} \text{d} r \tag{(3.11)} \\ &\quad + \omega_{N-1} \int_0^{\eta} |u_n(r)|^{2^*} \; r^{N-1} \text{d} r \nonumber \\ & = \omega_{N-1} \int_0^{\eta} |u_n(r)|^{2^*} \left( |u_n(r)|^{r^{\alpha}} - 1 \right) \; r^{N-1} \text{d} r \tag{(3.12)}\\ &\quad + \int_B |u_n(x)|^{2^*} \, \text{d}x + o(1).\tag{(3.13)} \end{align}

Using lemma 2.1 (Radial Lemma), we can estimate

\begin{align*} & \int_0^{\eta} |u_n(r)|^{2^*} \left( |u_n(r)|^{r^{\alpha}} - 1 \right) \; r^{N-1} \text{d} r\\ & \quad \leq \int_0^{\eta} |u_n(r)|^{2^*} \left[ \left( \frac{1}{r^{(N-2)/2}} \right)^{r^{\alpha}} - 1 \right] \; r^{N-1} \text{d} r \\ & \quad \leq \int_0^{\eta} |u_n(r)|^{2^*} \left[ \exp\left( r^{\alpha} \log \left( \frac{1}{r^{(N-2)/2}} \right) \right) - 1 \right] \; r^{N-1} \text{d} r \\ & \quad \leq \int_0^{\eta} |u_n(r)|^{2^*} r^{\alpha} \left| \log r^{(N-2)/2} \right| \; r^{N-1} \text{d} r \\ & \quad \leq C_1 \eta^{\alpha} \left| \log \eta \right| \int_0^1 |u_n(r)|^{2^*} \; r^{N-1} \text{d} r \\ & \quad \leq C_2 \eta^\alpha \left| \log \eta \right|, \end{align*}

where C ₁ and C ₂ are constants.

Therefore, for all ɛ > 0, we can choose $\eta = \eta(\varepsilon) \gt 0$ sufficiently small such that

\begin{equation*} C_2 \eta^{\alpha} \left| \log \eta \right| \leq \frac{\varepsilon}{2}, \end{equation*}

which implies

(3.14)\begin{equation} \omega_{N-1} \int_0^{\eta} |u_n(r)|^{2^*} \left( |u_n(r)|^{r^{\alpha}} - 1 \right) \; r^{N-1} \text{d} r \leq \frac{\varepsilon}{2}. \end{equation}

From (3.9), (3.10), and (3.14), we obtain that for sufficiently large n and for all ɛ > 0,

\begin{align*} \int_B |u_n|^{2^*+r^{\alpha}} \, \text{d} x - \int_B |u_n|^{2^*} \, \text{d}x \leq \varepsilon. \end{align*}

Therefore, we have proven (3.8).

Now, for sufficiently large n, we obtain the inequality

\begin{align*} -\frac{1}{2^*} \int_B |u_n|^{2^*} \, \text{d}x \leq - \int_B \frac{|u_n|^{2^* + r^\alpha}}{2^* + r^\alpha} \, \text{d}x. \end{align*}

Thus, for sufficiently large n, we get

\begin{align*} J_0(u_n) \leq J_A(u_n), \end{align*}

where

\begin{align*} J_0(u) &= \frac{1}{2} \| \nabla u \|_2^2 - \frac{\lambda}{q} \| u^+ \|_q^q - \frac{1}{2^*} \int_B (u^+)^{2^*} \, \text{d}x, \quad u \in H_{0,\mathrm{rad}}^1(B). \end{align*}

Then, we have

\begin{align*} J_0(u_n) \rightarrow d \leq c \lt \frac{1}{N}S^{N/2}. \end{align*}

Since $ u_n \rightharpoonup 0 $ also in $ L^{2^*}(B) $, it follows that $ \langle J_0'(u_n), \varphi \rangle \rightarrow 0 $ for all $ \varphi \in H_{0,\text{rad}}^1(B) $. Indeed, by the embedding $ H_{0,\mathrm{rad}}^1(B) \hookrightarrow L^s(B) $ for all $ s \in [1,2^*] $, we have

\begin{align*} & \int_B \nabla u_n \cdot \nabla \varphi \, \text{d}x \rightarrow 0, \quad \int_B (u_n^+)^{q-1} \varphi \, \text{d}x \rightarrow 0, \quad \text{and} \\ & \int_B (u_n^+)^{2^*-1} \varphi \, \text{d}x \rightarrow 0 \text{as } n \rightarrow \infty. \end{align*}

Therefore, $\{u_n\}$ is a Palais-Smale sequence for the functional J ₀ at the level $d \lt \frac{1}{N} S^{N/2}$. According to [1.1, Reference Willem10], the functional J ₀ satisfies the Palais–Smale condition for levels $d \lt \frac{1}{N} S^{N/2}$. Thus, we have $u_n \rightarrow 0$ strongly in $H_{0,\text{rad}}^1(B)$, and by the continuity of the functional J, it follows that $J(u_n) \rightarrow 0$, which leads to a contradiction.

Therefore, we have $u \not\equiv 0$. Choosing $\varphi = u^-$ as the test function in the equation $\langle J'(u), \varphi \rangle = 0$, we get that $u = u^+ \geq 0$. By the strong maximum principle (see [Reference Evans6, theorem 4, pp. 333]), it follows that u > 0 in B. Therefore, u is a weak solution of the problem (1.3), and this completes the proof of theorem 1.1.

4. Proof of theorem 1.3

For problem (1.5), we will follow a similar strategy to the one we used in the proof of theorem 1.1. We define the functional $J\colon H_{0, \text{rad}}^1(B) \rightarrow \mathbb{R}$ given by

(4.1)\begin{align} J(u) = \frac 12 \Vert \nabla u \Vert^2_2 - \frac{\lambda}{q}\Vert u^+ \Vert_{q}^q - \int_B \frac{(u^+)^{2^* + f(r)}}{2^*+f(r)}\, \text{d}x, \end{align}

where $u^{+}(x) = \max \{u(x),0 \}$, $f\colon [0,1) \rightarrow \mathbb{R}$ is a continuous function satisfying condition (f) and $q \in [2,2^*)$. The parameter λ is considered in two cases: if q = 2 then $\lambda \in [0,\lambda_1)$, if $q \in (2,2^*)$ then $\lambda \geq 0$. We will show in the following lemma that the functional J is well defined and by standard arguments, we will obtain that J is of class C ¹. We also know that positive critical points of J are weak solutions to the problem (1.5).

Proof. We only have to demonstrate that the variable integral is finite. Let $u \in H^{1}_{0, \mathrm{rad}}(B)$, then we write

(4.2)\begin{align} \int_B |u|^{2^* + f(r)}\, \text{d}x & = \int_{B_{\rho_1}} |u|^{2^* + f(r)}\, \text{d}x + \int_{B_{\rho_2}\setminus B_{\rho_1}} |u|^{2^* + f(r)}\, \text{d}x \nonumber\\ & \quad + \int_{B \setminus B_{\rho_2}} |u|^{2^* + f(r)}\, \text{d}x \end{align}

where ρ ₁ and ρ ₂ will be chosen later. By hypothesis (f), it follows that there exists $\rho_1 \gt 0$ such that $2 \lt 2^* + f(r) \lt 2^*, \forall r \in [0, \rho_1]$. From Hölder’s inequality and proposition 2.4, it follows that

(4.3)\begin{align} \int_{B_{\rho_1}} |u|^{2^* + f(r)}\, \text{d}x \leq C( \| u \|_{2^*}^{F_+} + \| u \|_{2^*}^{F_-}) \lt + \infty \end{align}

where $F_+ = \sup_{r \in [0,\rho_1]}( 2^* +f(r))$ and $F_-= \inf_{r\in [0,\rho_1]}(2^*+f(r))$. Now, we consider $\rho_2 \gt 0$ sufficiently close to 1. By lemma 2.1, we know that, for $r \in [\rho_1 , \rho_2]$

(4.4)\begin{align} |u(r)| \leq \frac{(1-r)^{1/2}}{r^{(N-2)/2}}\| \nabla u \|_2 \leq \frac{(1-\rho_1)^{1/2}}{\rho_1^{(N-2)/2}}\| \nabla u\|_2 := C_{\rho_1}\|\nabla u\|_2. \end{align}

Since f is continuous in $[\rho_1,\rho_2]$ it follows that $f\in L^\infty[\rho_1,\rho_2]$ and therefore the second integral in (4.2) is finite. For $r \in [\rho_2 , 1)$, again by lemma 2.1, we get

(4.5)\begin{align} |u(r)| \leq \frac{(1-r)^{1/2}}{r^{(N-2)/2}}\| \nabla u \|_2 \leq \frac{(1-\rho_2)^{1/2}}{\rho_2^{(N-2)/2}}\| \nabla u\|_2 := C_{\rho_2}\|\nabla u \|_2 \leq 1 \end{align}

since ρ ₂ was chosen sufficiently close to 1. Therefore, we obtain that the third integral in (4.2) is also finite. Therefore, we conclude that the J functional is well-defined.

As previously mentioned, we must ensure that the functional J has a positive critical point, for this, we will use the Mountain Pass Theorem, due to Ambrosetti and Rabinowitz [Reference Ambrosetti and Rabinowitz1]. We will show now that the functional J has the geometry of the Mountain Pass Theorem.

Lemma 4.2. There exist ρ > 0 and θ > 0 such that

\begin{align*} J(u) \geq \rho \gt 0, \, \, \text{if} \, \, \Vert \nabla u\Vert_2 = \theta. \end{align*}

Proof. Let $u \in H^1_{0 \mathrm{rad}}(B)$ be such that $\| \nabla u \|_2 = \theta \lt 1$. By (4.3) and Sobolev’s inequality, we have for ρ ₁ small enough that

(4.6)\begin{align} \int_{B_{\rho_1}} |u|^{2^* + f(r)}\, \text{d}x \leq C( \| u \|_{2^*}^{F_+} + \| u \|_{2^*}^{F_-}) \leq C_1 (\| \nabla u \|_2^{F_+} + \| \nabla u \|_2^{F_-}) \leq C_2 \| \nabla u \|_2^{F_-}, \end{align}

where $F_+ = \sup_{r \in [0,\rho_1]}( 2^* +f(r))$ and $F_-= \inf_{r\in [0,\rho_1]}(2^*+f(r))$. Let $\rho_2 \gt 0$ be sufficiently close to 1 as in lemma 4.1. By (4.4) and (4.5), and choosing θ > 0 small enough such that $\operatorname{Max}\{C_{\rho_1},C_{\rho_2}\} \| \nabla u \|_2 \lt 1$, we obtain

(4.7)\begin{align} \int_{B\setminus B_{\rho_1}} |u|^{2^*+f(r)}\,\text{d}x \leq \int_{B} \left( \operatorname{Max}\{C_{\rho_1},C_{\rho_2}\} \| \nabla u \|_2\right)^{2^* + f(r)} \, \text{d}x \leq C_3 \|\nabla u \|_2^{F_-} \end{align}

where $C_3 =|B| \left( \operatorname{Max}\{C_{\rho_1},C_{\rho_2}\}\right)^{F_-} $. Then, by (4.6) and (4.7), we get

(4.8)\begin{align} \int_B |u|^{2^* + f(r)}\, \text{d}x \leq C \| \nabla u \|_2^{F_-}, \end{align}

where $C= \operatorname{Max}\{C_2,C_3\}$. Therefore, we have

\begin{align*} J(u) & = \frac{1}{2}\|\nabla u \|_2^2 - \frac{\lambda}{q}\|u\|_q^q - \int_B \frac{|u|^{2^*+f(r)}}{2^*+f(r)}\mathrm{d}x\\ & \geq \frac{1}{2}\|\nabla u \|_2^2 - \frac{\lambda}{q}\|u\|_q^q - C \| \nabla u \|_2^{F_-}. \end{align*}

When q = 2, we consider $\lambda \in [0,\lambda_1)$ and then the expression $ \sqrt{\Vert \nabla u \Vert^2 - \lambda \Vert u \Vert^2}$ define a norm in $H_{0}^1(B)$ equivalent to usual norm. Since $2^* \gt q \geq 2$ and $F_- \gt 2$ due to the above inequality and by Sobolev inequality follows that for $\|\nabla u \|_2 = \theta$ with θ sufficiently small, that there exists ρ > 0 such that $J(u) \geq \rho \gt 0$.

Proof. Let $u \in H_{0, \mathrm{rad}}^1(B) \setminus \{0 \}$ such that u > 0 in B. We have for t > 1 that

\begin{align*} J(tu) &= \frac{t^2}{2} \Vert \nabla u \Vert^2_2 - \frac{\lambda t^q}{q}\Vert u \Vert^q_q - \int_{B} \dfrac{t^{2^* + f(r) }\vert u \vert^{2^* + f(r)}}{2^* + r^\alpha}\, \text{d} x \\ & \leq \frac{t^2}{2} \Vert \nabla u \Vert^2_2 - \frac{\lambda t^q}{q}\Vert u \Vert^q_q - \frac{t^{F_-}}{{2^* + F_+}}\int_{B}\vert u \vert^{2^* + f(r)}\, \text{d} x. \end{align*}

Therefore, since $F_- \gt 2$ and $q\in [2,2^*)$, we get

\begin{align*} \lim_{t \to + \infty} J(tu) = - \infty, \end{align*}

which proves the lemma.

Now we will show that the functional J satisfies the (PS) condition.

Proof. Let $\{u_n\} \subset H^1_{0,\text{rad}}(B)$ be a Palais–Smale sequence. So, we get

(4.9)\begin{align} J(u_n) \rightarrow c \gt 0 \,\, \text{and} \,\, J'(u_n) \rightarrow 0. \end{align}

Since $2^* + f(r) \gt 1$ for $r \in [0,1)$ by standard calculations we know that the sequence $\{u_n\}$ is bounded in $H^1_{0, \text{rad}}(B)$, So, up to a subsequence, there exists $u \in H_{0,\text{rad}}^1(B)$ such that

(4.10)\begin{equation} \begin{aligned} & u_n \rightharpoonup u \,\, \text{in} \,\, H_{0,\text{rad}}^1(B), \\ & u_n \rightarrow u \,\, \text{in} \,\, L^s(B) \,\, \forall \,\, s \in [1,2^*). \end{aligned} \end{equation}

From $J'(u_n) \rightarrow 0$, we can choose $\varphi = u_n -u$ as the test function and obtain the following inequality:

(4.11)\begin{equation} \begin{aligned} \bigg|\int_B \nabla u_n \nabla (u_n-u) \, \text{d}x - &\lambda \int_B (u_n^+)^{q-1}(u_n-u)\,\text{d}x - \int_B(u_n^+)^{2^*-1+f(r)}(u_n-u)\, \text{d}x \bigg|\\ & \leq \varepsilon_n \Vert \nabla(u_n-u)\Vert_2 \leq C \varepsilon_n, \end{aligned} \end{equation}

where $\varepsilon_n \rightarrow 0$. As $q \in [2,2^*)$, by Hölder inequality and (4.10), we obtain

\begin{align*} \int_B (u_n^+)^{q-1}(u_n-u)\,\text{d}x & \leq \left(\int_B \vert (u_n^+)^q\,\text{d}x\right)^{(q-1)/q}\left( \int_B \vert u_n - u \vert^q\, \text{d}x\right)^{1/q} \\ & \leq C \Vert u_n -u \Vert_{q} \rightarrow 0. \end{align*}

Therefore, by (4.11), the lemma will be proved if we check that

(4.12)\begin{align} \int_B(u_n^+)^{2^*-1+f(r)}(u_n-u)\, \text{d}x \rightarrow 0. \end{align}

Indeed,

\begin{align*} \frac {1}{\omega_{N-1}} \int_B(u_n^+)^{2^*-1+f(r)}(u_n-u)\, \text{d}x & = \int_0^{\rho_1}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}r \\ &\quad + \int_{\rho_1}^{\rho_2}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}x \\ & \quad + \int_{\rho_2}^{1}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}r, \end{align*}

where ρ ₁ and ρ ₂ will be chosen later. We will estimate each integral above separately. First, for r > 0 small enough, we know that $2-2^* \lt f(r) \lt 0$ because f is continuous at r = 0 and $f(0) \lt 0$. Therefore, for r small enough, we have that $2 \lt 2^*+f(r) \lt 2^*$. So, we can choose $\rho_1 \gt 0$ small enough such that $2 \lt 2^*+f_+(\rho^1) \lt 2^*$, where $f_+(\rho^1) = \displaystyle\sup_{r \in [0,\rho_1]}f(r)$. Then, we get

\begin{align*} & \omega_{N-1}\int_0^{\rho_1}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}r \\ & \quad \leq 2 \Vert (u_n^+)^{2^*-1+f(r)}\Vert_{\frac{2^*+f(r)}{2^*-1+f(r)}}\Vert u_n-u \Vert_{2^*+f(r)}\\ & \quad \leq C \Vert u_n-u \Vert_{2^*+f_+(\rho^1)} \rightarrow 0. \end{align*}

To estimate the second integral, we need to choose $\rho_2 = 1-\rho^{N-2}_1$ sufficiently close to 1. So, by inequality (2.1) of the lemma 2.1, we get

\begin{align*} \int_{\rho_1}^{\rho_2}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}x &\leq \left( \frac{1}{\rho_1^{(N-2)/2}}\right)^{2^*-1+f_+(\rho)}\int_{\rho_1}^{\rho_2}(u_n-u)r^{N-1}\,\text{d}r \\ &\leq C \Vert u_n-u \Vert_{L^1(B)} \rightarrow 0, \end{align*}

where $f_{+}(\rho)= \displaystyle\sup_{r\in [\rho_1,\rho_2]}f(r)$.

To estimate the last integral. Note that for $\rho_2 = 1-\rho^{N-2}_1$ and $\rho_2 \lt r \lt 1$, we have

\begin{align*} \frac{(1-r)^{1/2}}{\rho_1^{(N-2)/2}} \leq 1. \end{align*}

By inequality (2.2) from lemma 2.1, we get

\begin{align*} \int_{\rho_2}^{1}(u_n^+)^{2^*-1+f(r)}(u_n-u)r^{N-1}\, \text{d}r &\leq \int_{\rho_1}^1\left( \frac{(1-r)^{1/2}}{\rho_1^{(N-2)/2}}\right)^{2^*-1+f(r)}(u_n-u)r^{N-1}\,\text{d}r\\ &\leq \Vert u_n-u \Vert_{L^1(B)} \rightarrow 0. \end{align*}

Therefore, (4.12) is verified and the proof of the lemma is concluded.

From lemmas 4.2, 4.3, and 4.4, we conclude that the functional J has a non-trivial critical point u. Using $\varphi = u_-$ as a test function in equation $\langle J'(u),\varphi \rangle =0$, we obtain that $u = u_+\geq 0$ and by the strong maximum principle (see [Reference Evans6, theorem 4, pp. 333]) it follows that u is positive, thus finishing the proof of the theorem 1.3.