Related papers: Multiple positive normalized solutions for nonline…

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 are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…

We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\"{o}dinger equation \begin{equation}\label{1} {\Delta}^{2}u+\mu \Delta u-{\lambda}u={|u|}^{p-2}u, ~~~~x \in \R^{N}\\…

In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial…

This article concerns with the existence of multiple positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\…

This paper deals with the existence of positive solution for the singular quasilinear Schr\"odinger equation $-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)~\mbox{in} ~ \Omega,$ where $\gamma > 1$, $\Omega \subset \mathbb{R}^{N},…

We are concerned with the following nonlinear Schr\"odinger equation $$-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,~u\in H^1(\R^N),$$ where $N\geq 3$, $2<p<\frac{2N}{N-2}$. For $\varepsilon$ small enough and a class of $V(x)$, we show the…

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 =…

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized…

We prove the existence of infinitely many non-radial positive solutions for the Schr\"{o}dinger-Newton system $$ \left\{\begin{array}{ll} \Delta u- V(|x|)u + \Psi u=0, &x\in\mathbb{R}^3,\newline \Delta \Psi+\frac12 u^2=0, &x\in\mathbb{R}^3,…

We investigate the structure of nodal solutions for coupled nonlinear Schr\"{o}dinger equations in the repulsive coupling regime. Among other results, for the following coupled system of $N$ equations, we prove the existence of infinitely…

This paper focuses on the existence of multiple normalized solutions to Schr\"{o}dinger equations with general nonlinearities in bounded domains via variational methods. We first obtain two positive normalized solutions, one is a normalized…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

We find infinitely many positive non-radial solutions for a nonlinear Schrodinger-Poisson system.

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 paper, we consider the following weakly coupled nonlinear Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} -\epsilon^{2}\Delta u_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + \beta|u_1|^{p - 2}|u_2|^pu_1, & x\in…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

We study the existence and nonexistence of normalized solutions $(u_a, \lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schr\"{o}dinger equation with mixed nonlocal nonlinearities. This study can be viewed as a…

We study the following coupled Schr\"odinger system \ds -\Delta u+u=u^{2^*-1}+\be u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}+\la_1u^{\al-1}, &x\in \R^N, \ds -\Delta v+v=v^{2^*-1}+\be u^{\frac{2^*}{2}}v^{\frac{2^*}{2}-1}+\la_2v^{r-1}, &x\in \R^N,…

The nonlinear Schr\"{o}dinger-Newton system \begin{equation*} \begin{cases} \Delta u- V(|x|)u + \Psi u=0, &~x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by…