Related papers: On the nonlinear Schr\"{o}dinger-Poisson systems w…

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where $a,$ $b>0$ are constants, $2<p<5$…

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schr\"{o}dinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr}…

We study the following doubly critical Schr\"{o}dinger system $$-\Delta u -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ \nu \al u^{\al-1}v^\bb, \quad x\in \RN, -\Delta v -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + \nu \bb u^{\al}v^{\bb-1}, \quad x\in \RN,…

In this paper we study the existence of ground state solutions for the asymptotically periodic Schr\"odinger-Poisson systems which are coupled by a Schr\"odinger equation of $p$-Laplacian and a Poisson equation of $q$-Laplacian. The method…

In this paper, we prove a multiplicity result of solutions for the following stationary Schr\"odinger-Poisson-Slater equations \begin{equation}\label{eq-abstract} -\Delta u - \lambda u + (\left | x \right |^{-1}\ast \left | u \right |^2) u…

This paper is dedicated to studying the following elliptic system of Hamiltonian type: $$\left\{ \begin{array}{ll} -\varepsilon^2\triangle u+u+V(x)v=Q(x)F_{v}(u, v), \ \ \ \ x\in \mathbb{R}^N,\\ -\varepsilon^2\triangle…

The paper considers the following nonhomogeneous Schr\"odinger-Maxwell system -\Delta u + u+\lambda\phi (x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -\Delta\phi = u^2, \ x\in \mathbb{R}^3, . \leqno{(SM)} where $\lambda>0$, $p\in(1,5)$ and…

In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad…

In this paper, we establish the existence of positive ground state solutions for a class of mixed Schr\"{o}dinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, \[ \left\{…

In this paper, we study the following Schr\"{o}dinger-Born-infeld system with a general nonlinearity $$ \left\{ \begin{array}{ll} -\triangle u+u+\phi u=f(u)+\mu|u|^4u\,\,&\mbox{in}\,\,\R^3,\\…

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 study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schr\"odinger equations (NLSEs), that is $-\epsilon^2 \Delta u + V(x) u= h_v(u,v), - \epsilon^2…

In this paper, we study the existence of normalized solutions for the nonautonomous Schr\"{o}dinger-Poisson equations \begin{equation}\nonumber -\Delta u+\lambda u +\left(\vert x \vert ^{-1} * \vert u \vert ^{2} \right)…

This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schr\"{o}dinger equation \begin{equation*} \begin{split} -\Delta_{H,p}u+V(x)|u|^{p-2}u-\Delta_{H,p}(|u|^{2\alpha})…

We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill -…

We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of…

Consider the following nonlinear Schr\"odinger--Bopp--Podolsky system in $\mathbb{R}^3$: \[ \begin{cases} - \Delta v + v + \phi v = v |v|^{p - 2}; \\ \beta^2 \Delta^2 \phi - \Delta \phi = 4 \pi v^2, \end{cases} \] where $\beta > 0$ and $3 <…

We prove the existence of ground states and high-energy solutions to the following Schr\"odinger-Poisson system \begin{align*} \begin{cases} - \Delta u + a(x) u + u v = 0,\newline \Delta v = u^2, \end{cases} \quad \text{in } \mathbb{R}^3,…

We consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u = f(u)\\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi = 4\pi\varepsilon…

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,…