Related papers: Weighted Choquard equation perturbed with weighted…

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 consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where…

In this paper, we are interested in the least energy nodal solutions to the following nonlocal Choquard equation with a local term \begin{equation*}\left\{\begin{array}{rll} -\Delta u&=\lambda|u|^{p-2}u+\mu \phi(x)|u|^{q-2}u\\ -\Delta…

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\ $is a real valued vector potential, $V$ is a real…

We consider nonlinear Choquard equation $$ - \Delta u + V u = \bigl(I_\alpha \ast |u|^{\frac{\alpha}{N}+1}\bigr) |u|^{\frac{\alpha}{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external…

In this paper, we study the nonlocal Choquard equation $$ -\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon= (I_\alpha * |u_\varepsilon|^p)|u_\varepsilon|^{p-2}u_\varepsilon $$ where $N\geq 1$, $I_\alpha$ is the Riesz potential of order…

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -\Delta u+\lambda u=\gamma…

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…

We consider a nonlinear Choquard equation $$ -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p =2$…

In this paper we are interested in the following nonlinear Choquard equation $$ -\Delta u+(\lambda V(x)-\beta)u =\big(|x|^{-\mu}\ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$…

We study the Choquard equation involving mixed local and nonlocal operators $$-\Delta u+(-\Delta)^{s}u+V(x)u=(\frac{1}{|x|^{\mu}}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $\mu\in(0,2)$, $F(t)=\int_{0}^{t} f(\tau)\,d\tau$, and…

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N}…

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 present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-\Delta)^s u+V(x)u=\varepsilon^{\mu-N}(\frac{1}{|x|^\mu}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$…

In this article, we study the following problem $$-{\rm div} (\omega(x)|\nabla u|^{N-2} \nabla u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R^{N}}$,…

For the Choquard equation, which is a nonlocal nonlinear Schr\"odinger type equation, $ -\Delta u+V_{\mu,\nu} u=(I_\alpha\ast |u|^{\frac{N+\alpha}{N}}){|u|}^{\frac{\alpha}{N}-1}u$, in $\mathbb{R}^N$ where $N\ge 3$, $V_{\mu, \nu} :…

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_{\mu}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu}-2}u, \quad…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad…

We look for a solutions to a nonlinear, fractional Schr\"odinger equation $$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic…