Related papers: Normalized Solutions for nonlinear Schr\"{o}dinger…

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…

In this paper, we study the following Schr\"odinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ & \int |u|^2dx=\Theta, \end{aligned}…

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schr\"odinger equation $$ \left\{ \begin{array}{ll} -\Delta u = \lambda u + f(u)\quad & \text{in…

We study the normalized solutions to the following Choquard equation \begin{equation*} \aligned &-\Delta u + \lambda u =\mu g(u) + \gamma (I_\alpha * |u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u & \text{in\ \ } \mathbb{R}^N…

In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation \begin{equation}\label{1}\nonumber - \Bigl(a+b\int_{{\R^3}} {{{\left| {\nabla u} \right|}^2}}\Bigl) \Delta u =\lambda u+ {|…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an…

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…

We consider the Schr\"odinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -\Delta u+V(x) u+|u|^{p-2}u=\lambda \phi u, & \mbox{in}\mathbb{R}^{3},\\ -\Delta\phi= u^{2}, & \mbox{in}\mathbb{R}^{3}. \end{array} \right.\nonumber…

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

In this paper, we investigate the following nonlinear Schr\"odinger equation with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u+ \lambda u= f(u) & {\rm in} \,~ \Omega,\\ \displaystyle\frac{\partial u}{\partial…

In this article we are concerned with the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\…

Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in }…

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}, \\…

We study the existence of solutions $(\underline u,\lambda_{\underline u})\in H^1(\mathbb{R}^N; \mathbb{R}) \times \mathbb{R}$ to \[ -\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N \] with $N \ge 3$ and prescribed $L^2$ norm, and…

In this paper, we study the existence of nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\…

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…

Consider the Schrodinger equation -\Delta u =(k+V) u in an infinite slab S= \R^{n-1}x (0,1), where V is a bounded potential supported on a set D of finite measure. We prove necessary conditions for the existence of nontrivial admissible…

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…

In this paper, we study a class of fractional Schr\"{o}dinger equation \begin{equation} \label{eq0} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in H^{s}(\mathbb{R}^{N}),…

We are concerned with solutions of the following quasilinear Schr\"odinger equations \begin{eqnarray*} -{\mathrm{div}}\left(\varphi^{2}(u) \nabla u\right)+\varphi(u) \varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u), \quad x \in…