Related papers: New type of solutions for Schr\"odinger equations …

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic…

We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{align*} \begin{cases} -\Delta u+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases}…

We find infinitely many positive non-radial solutions for a system of Schr\"odinger equations with critical growth in a fully attractive or repulsive regime in presence of an external radial trapping potential.

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…

In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\…

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…

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u…

This paper considers the fractional Schr\"{o}dinger equation \begin{equation}\label{abstract} (-\Delta)^s u + V(|x|)u-u^p=0, \quad u>0, \quad u\in H^{2s}(\R^N) \end{equation} where $0<s<1$, $1<p<\frac{N+2s}{N-2s}$, $V(|x|)$ is a positive…

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 paper, we are concerned with the following fractional relativistic Schr\"odinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \mbox{ in }…

We study the following nonlinear Schr\"odinger equation and we look for normalized solutions $(\mu,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -\Delta u + \mu u = g(u)\quad \text{in}\ {\bf R}^N, \qquad…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schr\"odinger} equations involving Stein-Weiss type convolution \begin{align*} -\Delta_N u -\Delta_N (u^{2})u +V(x)|u|^{N-2}u=…

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$…

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N,…

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u =…

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in…

This paper is devoted to the magnetic nonlinear Schr\"{o}dinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow…

In this paper, we investigate the following critical elliptic equation $$ -\Delta u+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), $$ where $V(y)$ is a bounded non-negative function in $\R^{N}.$ Assuming…

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.…