Related papers: Infinitely many positive solutions for nonlinear e…

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}…

In this article, we will prove the existence of infinitely many positive weak solutions to the following nonlocal elliptic PDE. \begin{align} (-\Delta)^s u&= \frac{\lambda}{u^{\gamma}}+ f(x,u)~\text{in}~\Omega,\nonumber…

We prove the existence of a ground state positive solution of Schr\"odinger-Poisson systems in the plane of the form $$ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\…

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

This paper treats the existence of positive solutions of $-\Delta u + V(x) u = \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We shall show the…

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -\Delta u - \Delta (u^2)u = |u|^{p-2}u & \mbox{ in } \Omega u= 0 &\mbox{ on }…

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 consider singularly perturbed nonlinear Schr\"odinger equations \be \label{eq:0.1} - \varepsilon^2 \Delta u + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &=…

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 investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu…

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 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.…

We build infinitely-many non-radial positive solutions to the Schr\"odinger system \begin{equation*} \left\{\begin{aligned} &-\Delta u_1+u_1=u_1^{{\mathfrak p} }-\Lambda u_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-\Delta…

In this paper we prove the existence of a positive solution to the equation $-\Delta u + V(x)u=g(u)$ in $R^N,$ assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem,…

We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u =…

In this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…