Related papers: On the Schrodinger equation in $R^N$ under the eff…

We establish the existence of a positive solution to the problem $$-\Delta u+V(x)u=f(u),\qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ for $N\geq3$, when the nonlinearity $f$ is subcritical at infinity and supercritical near the origin, and the…

In this paper we prove existence of ground state solutions of the modified nonlinear Schrodinger equation: $$ -\Delta u+V(x)u-{1/2}u \Delta u^{2}=|u|^{p-1}u, x \in \R^N, N \geq 3, $$ under some hypotheses on $V(x)$. This model has been…

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 +…

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where…

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…

We consider the following Schr\"{o}dinger equation $$ - \hslash ^2 \Delta u + V(x)u = \Gamma(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical…

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…

We prove the existence results for the Schr\"odinger equation of the form $$ -\Delta u + V(x) u = g(x,u), \quad x \in \mathbb{R}^N, $$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in $\mathbb{R}^N \setminus…

We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…

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

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -\Delta_\gamma u + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $\Delta_\gamma $ denotes the so-called…

We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u = g(x,u),$$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical…

We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Delta u+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

This paper is concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u- \Delta(u^2)u =h(u), \ \ \mbox{in} \ \mathbb{R}^N, \end{equation*} where $N\geq 3$. Under appropriate assumptions on $V$ and $h$, we…

We consider the problem {\Delta}u+V(x)u = f'(u) in RN. Here the nonlinearity has a double power behavior and V is invariant under an orthogonal involution, with V ({\infty}) = 0. An existence theorem of one pair of solutions which change…

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…

In the present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following elliptic equations with potential $\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\R^N, $ satisfying the normalization…

In this paper, we establish the existence of one solution for a Schr\"{o}dinger equation with jumping nonlinearities: $-\Delta u+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on…

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$…