Related papers: Schrodinger-Poisson-Slater equations with nonlinea…

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) =…

This paper is concerned with the following planar Schr\"{o}dinger-Poisson system \begin{equation*} \begin{cases} -\triangle{u}+V(x)u+\phi{(x)}|u|^{p-2}u=f(x,u),&\text{in $\mathbb{R}^{2}$}, \triangle{\phi}=|u|^{p},&\text{in…

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we…

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schr\"odinger} equations involving Stein-Weiss type convolution \begin{align*} -\Delta_N u -\Delta_N (u^{2})u +V(x)|u|^{N-2}u=…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega…

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class…

For a nonlinear partial differential equation for (pseudo)scalars in the bulk of Euclidean AdS_4, arising from a truncation of 11-dimensional supergravity over AdS_4 x S^7/Z_k, we use math tools and in particular Adomian Decomposition…

We construct in this paper global (for $t \geq 0$) and bounded solutions $u(t)$ for the nonlinear Schr\"odinger equation \[i \partial_t u + \Delta u + |u|^{p-1} u = 0, \quad t \in \mathbb{R}, x \in \mathbb{R}^d\] in mass sub-critical cases…

We consider nonnegative solutions to $-\Delta u=f(u)$ in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating$\&$sliding line technique, we prove symmetry and monotonicity properties of the solutions, under…

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}…

In this work we study the following fractional scalar field equation \begin{equation*}\label{P} \left\{ \begin{array}{ll} (-\Delta)^{s} u = g'(u) \mbox{ in } \mathbb{R}^{N} \\ u> 0 \end{array} \right. \end{equation*} where $N\geq 2$, $s\in…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

The paper deals with standing wave solutions of the dimensionless nonlinear Schr\"odinger equation \label{eq:abs1} i\Phi_t(x,t) = -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential…

We are interested in nonlinear fractional Schr\"odinger equations with singular potential of form \begin{equation*} (-\Delta)^su=\frac{\lambda}{|x|^{\alpha}}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$,…

We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert…

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$, which…

This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands…

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…