In this paper, we are interested in positive solutions of

$$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$

$\Omega $

${\mathbb {R}}^N (N \ge 3)$

$ a(x), b(x)$

$\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $

$\frac {N}{2}$

$ a(x)=b(x)=1$

$\Omega $

$$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$

$\lambda _H$

$\Omega .$

$\lambda _H$

$q.$

Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

We consider the radially symmetric positive solutions to quasilinear problem

\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*}

having prescribed mass $\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities

\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*}

where $\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.

By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac {da^2}{h(a^2)}+a^2g_{S^n}$ for some function h where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb {R}^n$ for any $n\ge 2$. More precisely, for any $\lambda \ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions ${h\in C^2((0,\infty );\mathbb {R}^+)}$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.

In this paper, we study the singular boundary value problem

$$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:

\[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \]

$b,\, \omega >0$

$p>2$

In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media

\[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \]

$\Omega \subset \mathbb {R}^{3}$

$\mathcal {C}^{1,1}$

$\nu$

$\lambda <0$

$2^{\ast }_{\alpha }=3+\alpha$

$0<\alpha <3$

$W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$

The paper is concerned with positive solutions to problems of the type

\[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \]

$\mathbb {B}^N$

$1< p<2^*-1:=\frac {N+2}{N-2}$

$\;\lambda < \frac {(N-1)^2}{4}$

$f \in H^{-1}(\mathbb {B}^{N})$

$f \not \equiv 0$

$a\in L^\infty (\mathbb {B}^N)$

$\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$

$d(x,\, 0)$

$a(x) \leq 1$

$a(x) \geq 1$

$\mu ( \{ x : a(x) \neq 1\}) > 0.$

$a(x) \equiv 1$

In this paper, we consider the following non-linear system involving the fractional Laplacian0.1

\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}

$0< s<1$

$x_n$

We study minimizers of the Allen–Cahn system. We consider the $\varepsilon$-energy functional with Dirichlet values and we establish the $\Gamma$-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $\varepsilon$-energy functional as $\varepsilon \rightarrow 0$.

In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

We consider the Dirichlet problem for $p(x)$-Laplacian equations of the form

$$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$

The odd nonlinearity $f(x,u)$ is $p(x)$-sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $. Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$.

For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators:

\[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \]

$\Omega \subset \mathbb {R}^d$

$\alpha,\,\beta$

$(\alpha,\, \beta )$

$p$

$q$

We consider Calderón's problem for the connection Laplacian on a real-analytic vector bundle over a manifold with boundary. We prove a uniqueness result for this problem when all geometric data are real-analytic, recovering the topology and geometry of a vector bundle up to a gauge transformation and an isometry of the base manifold.

In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

Let $\Omega \subset \mathbb {R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$. Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, where $\mu \in {\mathbb {R}}$. We study boundary value problems ($P_\pm$) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$, where $p>1$, $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$. Different critical exponents for the existence of a solution to ($P_\pm$) appear according to concentration of $\nu$. The solvability for problem ($P_+$) was proved in [3, 29] in subcritical ranges for $p$, namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$, we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ($P_+$) in supercritical ranges for $p$, i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ($P_-$) under a smallness assumption on $\nu$.

The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$—where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}

$a, \epsilon, \eta \gt 0$

$\lambda\in \mathbb{R}$

$p=2N/(N-2)$

We study the possible singularities of an m-subharmonic function $\varphi $ along a complex submanifold V of a compact Kähler manifold, finding a maximal rate of growth for $\varphi $ which depends only on m and k, the codimension of V. When $k < m$, we show that $\varphi $ has at worst log poles along V, and that the strength of these poles is moreover constant along V. This can be thought of as an analogue of Siu’s theorem.