Related papers: A mass supercritical problem revisited

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 nonlinear elliptic equation \begin{equation*} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega), \end{equation*} in an exterior domain $\Omega$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at…

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x…

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,…

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…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-\Delta+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded…

We prove the existence of a ground state solution for the following fractional scalar field equation $(-\Delta)^{s} u= g(u)$ in $\mathbb{R}^{N}$ where $s\in (0,1), N> 2s$,$ (-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1,…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a…

We study the non-scattering $L^{2}$ solution $u$ to the radial mass critical nonlinear Schr\"odinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly…

We investigate the problem of existence and uniqueness of ground states at fixed mass for two families of focusing nonlinear Schr\"odinger equations on the line. The first family consists of NLS with power nonlinearities concentrated at a…

We establish the existence and provide explicit expressions for the stationary states of the one-dimensional Schr\"odinger equation with a repulsive delta-prime potential and a focusing nonlinearity of power type. Furthermore, we prove…

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We…

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution,…

In this paper, we consider the $L_x^2$ solution $u$ to mass critical NLS $iu_t+\Delta u=\pm |u|^{\frac 4d} u$. We prove that in dimensions $d\ge 4$, if the solution is spherically symmetric and is \emph{almost periodic modulo scaling}, then…

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 study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to…

We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = \Delta u+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with…

We consider the nonlinear Choquard equation $$\begin{cases} & - \Delta u = (I_\alpha \ast F(u))F'(u) -\mu u \ \text{in}\ \mathbb{R}^N, & u \in \ H^1(\mathbb{R}^N), \ \int_{\mathbb{R}^N} |u|^2 dx=m, \end{cases} $$ where $\alpha\in(0,N)$,…