Related papers: Ground state of indefinite coupled nonlinear Schr\…

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schr\"odinger--Newton system with a point interaction: \[ \begin{cases} - \Delta_\alpha u = w u + \beta u |u|^{p - 2} &\text{on} ~…

We study the nonlinear Schr\"odinger system \[ \begin{cases} \displaystyle iu_t+\Delta u-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases} \]…

This paper is concerned with the following quasilinear Schr\"{o}dinger system in the entire space $\mathbb R^{N}$($N\geq3$): $$\left\{\begin{align} &-\Delta u+A(x)u-\frac{1}{2}\triangle(u^{2})u =…

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having…

We study the following doubly critical Schr\"{o}dinger system $$-\Delta u -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ \nu \al u^{\al-1}v^\bb, \quad x\in \RN, -\Delta v -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + \nu \bb u^{\al}v^{\bb-1}, \quad x\in \RN,…

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 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 establish the existence of positive ground state solutions for a class of mixed Schr\"{o}dinger systems with concave-convex nonlinearities in $\mathbb{R}^2$, subject to $L^2$-norm constraints; that is, \[ \left\{…

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having…

In this paper, we look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta…

We study the following class of linearly coupled Schr\"{o}dinger elliptic systems $$\left\{ \begin{array}{lr} -\Delta u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, &…

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…

We consider the following Scr\"odinger system $$\begin{cases}\displaystyle i\partial_t u + \Delta u +(|u|^2+\beta |v|^2) u= 0, \\ \displaystyle i\partial_t v + \Delta v +(|v|^2+\beta |u|^2) v = 0,\end{cases}$$ with initial data $(u_0,v_0)…

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 consider the Schr\"odinger type equation $-\Delta u+V(x)u=f(x,u)$ on the lattice graph $\mathbb{Z}^{N}$ with indefinite variational functional, where $-\Delta$ is the discrete Laplacian. Specifically, we assume that $V(x)$…

In this paper we prove the existence, regularity and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schr\"odinger operator.

By using variational methods, we study the existence of mountain pass solution to the following doubly critical Schr\"{o}dinger system: $$ \begin{cases} -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u &=h(x)\alpha|u|^{\alpha-2}|v|^\beta…

We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…

In this paper we are concerned with the fractional Schr\"{o}dinger equation $(-\Delta)^{\alpha} u+V(x)u =f(x, u)$, $x\in \rn$, where $f$ is superlinear, subcritical growth and $u\mapsto\frac{f(x, u)}{\vert u\vert}$ is nondecreasing. When…

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…