Related papers: Solutions to discrete nonlinear Kirchhoff-Choquard…

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\,…

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 $\gamma\in (0,N)$, $p, q >0$,…

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) \Delta u+\mu V(x)u=\lambda…

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(\theta+\int_{\Omega} |\nabla u| dx\right)\Delta u =u^p \quad{\rm in}\quad \Omega\setminus…

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential…

We consider the general Choquard equations $$ -\Delta u + u = (I_\alpha \ast |u|^p) |u|^{p - 2} u $$ where $I_\alpha$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{N -…

We consider the stationary magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u=\bigg(\frac{1}{|x|^{\alpha}}*F(|u|)\bigg)\frac{f(|u|)}{|u|}{u},\] where $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a vector potential, $V$ is…

We consider the following nonlinear fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}, $$ where…

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…

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\}, %…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in…

In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: \begin{equation*}\begin{cases} -\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\right)\Delta u+\lambda…

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and…

We consider the nonlinear Choquard equation $$ -\Delta u+V u=(I_\alpha \ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_\alpha$ is the Riesz potential integral operator of order $\alpha \in…

We study the following class of nonlinear Choquard equation, $$ -\Delta u +V(x)u =\Big( \frac{1}{|x|^\mu}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<\mu<N$, $N \geq 3$, $V$ is a continuous real function and $F$ is the…

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 existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}}…

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 study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\lambda|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where…