Related papers: Infinitely many new solutions for singularly pertu…

In this paper, we consider the existence of normalized solutions for the following biharmonic nonlinear Schr\"{o}dinger system \begin{equation*} \begin{cases} \Delta^2u+\alpha_{1}\Delta u+\lambda u=\beta r_{1}|u|^{r_{1}-2}|v|^{r_{2}} u &…

In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)*|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad \text{in} \,\,\, \mathbb{R}^{2},…

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…

This paper is concerned with the bifurcation from infinity of the nonlinear Schr\"odinger equation $$-\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

Making use of an ${\it ansatz}$ for the eigenfunctions, we obtain an exact closed form solution to the non-relativistic Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$ in two dimensions, where the…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schr\"odinger equations with subcritical exponent. For some smooth bounded domain $\Omega\subset \mathbf R^n$, our boundary condition is given…

Consider nonlinear Schr\"odinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,u,x),\quad x\in \mathbb{T}^d.\eqno{(*)}\] Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis…

In this paper, we study the following semilinear Schr\"odinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite…

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4,…

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=\Delta u-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0,…

In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{equation*} -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+V(x)u=P(x)|u|^{p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{R}^3,…

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…

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…

We prove the existence of solutions \(u(t,x)\) of the Schr{\"o}dinger equation with a saturation nonlinear term \((u/|u|)\) having compact support, for each \(t>0,\) that expands with a growth law of the type \(C\sqrt{t}\). The primary tool…

By making use of an ${\it ansatz}$ for the eigenfunction, we obtain the exact solutions to the Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$, both in three dimensions and in two dimensions, where the…

In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in }…

We study a class of Schr\"{o}dinger-Kirchhoff system involving critical exponent. We aim to find suitable conditions to assure the existence of a positive ground state solution of Nehari-Poho\u{z}aev type $u_{\varepsilon}$ with exponential…

We propose a new, exactly solvable Schr\"{o}dinger equation. The potential partner is given by \[{ V=}-Bp\operatorname{csch}[px]^{2}-9p(B+p)\operatorname*{sech}[3px]^{2}+(B\coth[px]-3(B+p)\tanh[3px])^{2}.\] obtained using supersymmetric…