Related papers: On the non-autonomous Schr\"odinger-Poisson proble…

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: -\left(a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+u=f(x,u)+u^{5}, u\in H^1(\R^3), u>0, $x\in \R^3$ where a,b>0 are constants. Under…

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schr\"odinger equation $$ \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u =…

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta u+\lambda u=\frac{u^p}{v^q}+\rho(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -\Delta v+\mu…

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with…

Consider the Schr\"odinger--Bopp--Podolsky system \[ \begin{cases} -\epsilon^2\Delta u+(V+K\phi)u=u|u|^{p-1};\newline \Delta^2\phi-\Delta\phi=4\pi K u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $\epsilon>0$, where…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

In this paper we prove the existence of normalized solutions $(\lambda,u)\subset (0,\infty)\times H^1(\mathbb{R}^3)$ to the following Schr\"{o}dinger-Poisson equation $$ \begin{cases} -\Delta u+V(x)u+\lambda u+(|x|^{-1}\ast…

In this work we consider the following class of elliptic problems $$- \Delta_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are…

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and…

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1…

We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-\Delta u + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a…

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators $$\displaystyle -\Delta^2 u + \lambda\Delta u \geq (K_{\alpha, \beta} * u^p)u^q \quad\text{ in } \mathbb{R}^N (N\geq 1),$$…

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta…

In this paper, we study the following quasilinear {S}chr\"{o}dinger equation: $$\left\{ \begin{array}{l} - {\Delta u} - \frac{\kappa }{2}\Delta \left( {u}^{2}\right) u = h\left( u\right) \text{ in }{\mathbb{R}}^{N}, \\ u \in {H}^{1}\left(…

We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_\lambda$} \left\{ \begin{array}[c]{ll} \Delta^2 u- \Delta u + V(x)u -\lambda \Delta[\rho(u^2)]\rho'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in…

In this paper, we investigate the existence of nonnegative solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$ in $\mathbb R^n$, where $-\mathcal{L}_{K}$ is a integro-differential operator with measurable kernel $K$ and $V$ is a…

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the…

We consider a class of stationary Schr\"{o}dinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schr\"{o}dinger operator $-\Delta+V$ is indefinite. Previous results in this framework…

We prove the existence results for the Schr\"odinger equation of the form $$ -\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus…

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential.…