Related papers: Semi-classical states for the Choquard equation

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,\,\,…

The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*}…

In this paper, we study the following semilinear Schr\"odinger equation $$ -\epsilon^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an…

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has the following…

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\left(\int\limits_{\Omega}\frac{u^{p+1}(y)}{|x-y|^\alpha}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ \Omega,…

We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\mathrm{loc}$ and even $W^{2,…

This article addresses the microlocalization of eigenfunctions for the semiclassical Schr\"odinger operator $-h^2\Delta+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the…

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-\epsilon})u^{2_{s}^{\sharp}-2-\epsilon} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}\Omega,\\…

We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-\Delta)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $\Omega,$ } { \mathcal{N}_{s}u=0 }…

We study the Choquard equation with a local perturbation \begin{equation*} -\Delta u=\lambda u+(I_\alpha\ast|u|^p)|u|^{p-2}u+\mu|u|^{q-2}u,\ x\in \mathbb{R}^{N} \end{equation*} having prescribed mass \begin{equation*}…

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation with Neumann boundary condition \begin{equation*} \begin{aligned} -\Delta u &= \lambda \alpha(x)u +…

Consider the following nonlinear Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p-1}; \\ a^2 \Delta^2 \phi - \Delta \phi = 4 \pi u^2, \end{cases} $$ where $a,…

This work is devoted to the study of a nonlocal Ginzberg-Landau equation by the semigroup method and a nonlocal Fokker-Plank equation by the viscosity vanishing method. For the nonlocal Ginzberg-Landau equation, there exists a unique global…

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\…

In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon$ is…

In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-\Delta)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in…

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,\alpha}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian…

We consider the positive solutions for the nonlocal Choquard equation $- \Delta u + u - (|\cdot|^{-\alpha} * |u|^p) |u|^{p-2} u = 0$ in $\mathbb{R}^d$. Compared with ground states, positive solutions form a larger class of solutions and…

We study the sharp interface limit of the fractional Allen-Cahn equation $$ \varepsilon \partial_t u^{\varepsilon} = \mathcal{I}^s_n [u^{\varepsilon}] -\frac{1}{\varepsilon ^{2s}} W'(u^\varepsilon) \quad…