Related papers: Schr\"{o}dinger-type equations with sign-changing …

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad…

Using a new infinite-dimensional linking theorem, we obtained nontrivial solutions for strongly indefinite periodic Schr\"odinger equations with sign-changing nonlinearities.

We study the equation \begin{equation} (-\Delta)^{s}u+V(x)u= (I_{\alpha}*|u|^{p})|u|^{p-2}u+\lambda(I_{\beta}*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_\gamma(x)=|x|^{-\gamma}$ for any $\gamma\in (0,N)$, $p, q >0$,…

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…

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},…

We study the following problem \[ \begin{cases} -\Delta u = \lambda u + u^{2^*-2} v & \hbox{in} \Omega,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} & \hbox{in} \Omega,\\ u> 0,v> 0 & \hbox{in} \Omega,\\ u=v=0 & \hbox{on} \partial \Omega,…

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,$$ for $x \in \mathbb{R}^N$, $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing…

In this paper, we study the existence of localized sign-changing (or nodal) solutions for the following nonlinear Schr\"odinger-Poisson system \begin{equation*} \begin{cases} -\varepsilon^2 \Delta u+V(x)u+\phi…

We consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is…

In this paper, we study the following logarithmic Schr\"{o}dinger equation \[ -\Delta u+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $\Delta$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to…

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: -\left(a+b\ds\int_{\R^3}|D u|^2\right)\Delta u+u=f(x,u)+u^{5}, u\in H^1(\R^3), u>0, $x\in \R^3$ where a,b>0 are constants. Under…

In this paper, we study the following semilinear Schr\"odinger system $$ -\triangle u+u=(1+K_\alpha(\epsilon x))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $\epsilon>0$, $\alpha>0$ are small parameters.…

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…

In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: $$ \quad \left\{\begin{array}{lr}\ds \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dxdy =…

In this paper, we study radial solutions of $\Delta u + K(|x|) f(u)+\frac{ (N-2)^2 u}{|x|^{2+(N-2)\delta}} =0, \ 0<\delta<2$ in the exterior of the ball of radius $R>0$ in ${\mathbb R}^{N}$ where $f$ grows superlinearly at infinity and is…

Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of…

In this paper, we consider the following quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u-u\Delta(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,}…

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega,\\ -\Delta…

In this paper we consider the following problem $$\begin{cases} -\Delta_{g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{in }M\\ u\geq0, & \mbox{in }M\\ u\to0, & \mbox{as }d_{g}(x_{0},x)\to\infty \end{cases}$$where $(M,g)$ is a $N$-dimensional…

In this paper, we consider the following nonlinear critical Schr\"odinger system: \begin{eqnarray*}\begin{cases} -\Delta u=K_1(y)u^{2^*-1}+\frac{1}{2} u^{\frac{2^*}{2}-1}v^\frac{2^*}{2}, \,\,\,\,\,y\in\Omega,\,\,\,\,\,u>0,\cr -\Delta…