Related papers: Solution manifolds for some semilinear Schrodinger…

In this paper, we study the following semilinear Schr\"odinger equation $$ -\epsilon^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in…

In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} -\Delta \phi= u^{2} &\text{in…

We consider the stationary semilinear Schr\"odinger equation $-\Delta u + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In…

In this paper we analyze the existence of large positive radial solutions to some quasilinear elliptic systems. Also, a non-radially symmetric solution is obtained by using a lower and upper solution method. The equations are coupled by…

In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type $$ -\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N}, $$ where $N\geq 3$,\ $0<\alpha<N$,…

In this paper we analyze the existence of entire radially symmetric solutions for Schrodinger system type {\Delta}_{p_{i}}u_{i}+h_{i}(r)|\nabla u_{i}|^{p_{i}-1}=a_{i}(r)f_{i}(u_1,...,u_{d}) for i=1,...,d on R^{N} where p_{i}>1, d \in…

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},…

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…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

In this paper we consider the following quasilinear Schr\"odinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -\Delta \phi -\varepsilon^{4} \Delta_4…

In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4…

We are interested in nonlinear fractional Schr\"odinger equations with singular potential of form \begin{equation*} (-\Delta)^su=\frac{\lambda}{|x|^{\alpha}}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$,…

In this paper we investigate the existence of positive solutions to the following Schr\"odinger-Poisson-Slater system [c]{ll} - \Delta u+ u + \lambda\phi u=|u|^{p-2}u & \text{in} \Omega -\Delta\phi= u^{2} & \text{in} \Omega u=\phi=0 &…

In the paper we consider the following quasilinear Schr\"odinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p - 1} \newline - \Delta \phi - \beta \Delta_4 \phi = u^2,…

In this article we consider the system of equations {\Delta}u_{i}=p_{i}(x)f_{i}(u_{1},...,u_{d}) for i=1,...,d on R^{N}, N\geq3 and d\in{1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some…

We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above…

We are looking for solutions to nonlinear Schr\"odinger-type equations of the form $$ (-\Delta)^{\alpha / 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external…

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…

The existence of a positive solution to the following fractional semilinear equation is proven, in a situation where a ground state solution may not exist. More precisely, we consider for $0<s<1$ the equation $$ (-\Delta)^s u +…

In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$,…