Related papers: Existence and regularity for an entire Grushin-Cho…

In this paper, we investigate the following fractional Choquard type equation: \[ (- \Delta)_p^s\, u = \lambda\frac{|u|^{r-2}u}{|x|^\alpha}\,+\gamma \big(\int_\Omega \frac{|u|^q}{|x-y|^\mu}dy\big) |u|^{q-2}u \ \ \text{in } \Omega,\ \ u = 0…

We prove the existence of infinitely many solutions to a fractional Choquard type equation \[ (-\Delta)^s_p u+V(x)|u|^{p-2}u=(K\ast g(u))g'(u)+\varepsilon_W W(x)f'(u)\quad\text{in }\mathbb{R}^N \] involving fractional $p$-Laplacian and a…

In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)*|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad \text{in} \,\,\, \mathbb{R}^{2},…

In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-\Delta)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where…

In this paper, we study the following $p$-Laplacian equation $$ -\Delta_{p} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^N$, where $p\geq 2$, $\alpha \in(0,N)$ are constants and $R_{\alpha}$ is the…

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…

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-\Delta)^{\frac{1}{2}} u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0…

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…

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u…

We consider a semilinear elliptic problem [- \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_\alpha) is a Riesz potential and (p>1). This family of equations includes the Choquard or…

We investigate here the following weighted degenerate elliptic system \begin{align*} -\Delta_{s} u = \Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\frac{\alpha}{2(s+1)}} v^p, \quad -\Delta_{s} v =…

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…

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u…

We study the normalized solutions to the following Choquard equation \begin{equation*} \aligned &-\Delta u + \lambda u =\mu g(u) + \gamma (I_\alpha * |u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u & \text{in\ \ } \mathbb{R}^N…

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-\Delta u-\lambda u=\left(\frac{1}{|x|^{\mu}}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in…

In this paper, we study the mass-constrained fractional Choquard equation \( (-\Delta)^s u = \lambda u + \alpha (I_\mu * |u|^{\frac{2N-\mu}{N}})|u|^{\frac{2N-\mu}{N}-2}u + (I_\mu * |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the…

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 $\alpha\in(0,N)$,…

We study the existence of positive solutions with prescribed $L^2$-norm for the Schr\"odinger equation \[ -\Delta u-V(x)u+\lambda u=|u|^{p-2}u\qquad\lambda\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and…

We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$.…

In this paper, we study nonlinear Choquard equations \begin{equation}\label{eq 1a1-} (-\Delta+id)^{\frac{1}{2}}u=(I_\alpha*{|u|^p})|u|^{p-2}u\ \ {\rm in} \ \ \mathbb{R}^N, \ \ \ u\in H^{\frac{1}{2}}(\mathbb{R}^N), \end{equation} where…