Related papers: The number of positive solutions to the Brezis-Nir…

This paper is concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u- \Delta(u^2)u =h(u), \ \ \mbox{in} \ \mathbb{R}^N, \end{equation*} where $N\geq 3$. Under appropriate assumptions on $V$ and $h$, we…

We study a class of critical Schr\"odinger-Poisson system of the form \begin{equation*} \begin{cases} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -\Delta \phi=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases}…

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…

In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} -\Delta \phi= u^{2} &\text{in…

In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schr\"odinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)-…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

We are concerned with the following class of equations with exponential nonlinearities: $$ \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two…

Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-\Delta)^s u=\varepsilon…

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array}…

We go further in the investigation of the Robin problem $(P_{\alpha})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u\geq0$ in $\Omega$, $\partial_{\nu}u=\alpha u$ on $\partial \Omega$; on a bounded domain $\Omega\subset\mathbb{R}^{N}$, with $a$…

In this paper, we are interested in the nonlinear Schr\"odinger problem $-\Delta u + Vu = \abs{u}^{p-2}u$ submitted to the Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $\Omega\subset\IR^N$…

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has the following…

This article is concerned with the quasilinear Schr\"odinger equation \[ \Delta u-\omega u+|u|^{p-1}u+\delta\Delta(|u|^2)u=0, \] where $\delta>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and…

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\Delta u + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the…

We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -\Delta u=\lambda ue^{u^2}~~&\mbox{in}~\Omega,\\[0.5mm] u>0 ~~ &{\text{in}~\Omega},\\[0.5mm] u=0~~&\mbox{on}~\partial \Omega, \end{cases} \end{equation*}…

We consider a nonautonomous semilinear elliptic problem where the power nonlinearity is multiplied by a discontinuous coefficient that equals one inside a bounded open set $\Omega$ and it equals minus one in its complement. In the slightly…

The multiplicity of positive weak solutions for a quasilinear Schr\"{o}dinger equations $-L_p u +(\lambda A(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$,…

We study a logarithmic fractional Schr\"odinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u+V(x)u+\phi u=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\…

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -\Delta u&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }\Omega,\\ u&=0, &\hbox{ on }\partial\Omega.…

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) & \text{in } \Omega, \frac{\partial u}{\partial \nu} + \gamma u=0 & \text{on }…