Related papers: Multi-bump positive solutions for a logarithmic Sc…

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…

In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\…

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log…

In this paper we will prove the existence of a positive solution for a class of Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\Delta u &+ u =Q(x)u\log u^2,\;\;\mbox{in}\;\;\Omega,\nonumber…

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 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…

This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands…

We consider the following system of Schr\"odinger equations \begin{equation*}\left.\begin{cases} -\Delta U + \lambda U = \alpha_0 U^3+ \beta UV^2 -\Delta V + \mu(y) V = \alpha_1 V^3+\beta U^2V \end{cases}\right. \text{in} \quad…

We study a linearly coupled Schr\"{o}dinger system in $\R^N(N\leq3).$ Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and the parameters in the…

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 consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and…

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert…

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

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$ where…

We prove the existence of infinitely many solutions $\lambda_1, \lambda_2 \in \mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schr\"odinger system \[ \begin{cases} -\Delta u - \lambda_1 u = \mu u^3+ \beta u v^2 & \text{in…

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…

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following…

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} +…