Related papers: Entire solutions to nonlinear scalar field equatio…

This article concerns the fractional elliptic equations \begin{equation*}(-\Delta)^{s}u+\lambda V(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-\Delta)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $\lambda…

In this paper we study the following nonlinear Choquard equation $$ -\Delta u+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$…

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 work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p}…

We study the direct and inverse scattering problem for the semilinear Schr\"{o}dinger equation $\Delta u+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point…

In this paper, we aim to study the existence of ground state normalized solutions for the following quasilinear Schr\"{o}dinger equation $-\Delta u-\Delta(u^2)u=h(u)+\lambda u,\,\, x\in\R^N$, under the mass constraint…

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions,…

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schr\"odinger equation of the form \begin{equation*} -\Delta u+\lambda u=g(u), \quad u \in H^1(\mathbb{R}^N), \, N \geq 1. \end{equation*} Our…

We consider the following nonlinear fractional Schr\"{o}dinger equation $$ (-\Delta)^su+u=K(|x|)u^p,\ \ u>0 \ \ \hbox{in}\ \ R^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$. Under some…

We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\"odinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schr\"odinger operator with a magnetic…

In this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…

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…

We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - \Delta u + (V - \omega) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\…

We are concerned with singular elliptic equations of the form $-\Delta u= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{align*} \begin{cases} -\Delta u+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases}…

Considering the radial nonlinear Schrodinger equation - \Delta u + V(x)u = g(x,u) in R^N, N \geq 3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring this problem is indefinite and g is an asymptotically…

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…