Related papers: Choquard equations under confining external potent…

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$ -\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$…

This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^su +Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$…

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation: $$ -\varepsilon^2\Delta v+V(x)v = \frac{1}{\varepsilon^\alpha}(I_\alpha*F(v))f(v) \quad \hbox{in}\ \mathbb{R}^N, $$ where $N\geq 3$,…

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha…

IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u=(I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array}…

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u +V u=(I_\alpha\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,\Omega$$ with Neumann or Dirichlet boundary conditions, when $\Omega$ is an annulus, or an…

We consider the critical Choquard system with both linear and nonlinear couplings $-\Delta v_1 + \mu_1 v_1 = ( I_\omega * |v_1|^{2_\omega^*} ) |v_1|^{2_\omega^* -2} v_1 + \theta p( I_\omega * |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad…

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost…

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_\alpha$…

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -\Delta u+ u =I_\alpha[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, %…

In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the…

We study the equation \begin{equation} (-\Delta)^{s}u+V(x)u= (I_{\alpha}*|u|^{p})|u|^{p-2}u+\lambda(I_{\beta}*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_\gamma(x)=|x|^{-\gamma}$ for any $\gamma\in (0,N)$, $p, q >0$,…

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla…

In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type $$ -\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N}, $$ where $N\geq 3$,\ $0<\alpha<N$,…

In this paper, we consider the existence of solutions for the linearly coupled Choquard system with potentials \begin{align*} \left\{\begin{aligned} &-\Delta u+\lambda_1 u+V_1(x)u=\mu_1(I_{\alpha}\star|u|^p)|u|^{p-2}u+\beta(x) v,\\ &-\Delta…

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u…

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu…

We study the following Choquard type equation in the whole plane $(C) -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr\"odinger potential and the nonlinearity…