Related papers: Limit profiles for singularly perturbed Choquard e…

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -\Delta u +V(x) u &=& (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda…

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 the present work we shall consider the existence and multiplicity of solutions for nonlocal elliptic singular problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following…

We study the fractional Schr\"odinger equations with a vanishing parameter: $$ (-\Delta)^s u+u =|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq…

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 study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of…

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 study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the…

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{q-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 consider the following linearly coupled Kirchhoff--Choquard system in $\mathbb{R}^3$: \begin{align*} \begin{cases} -\left(a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u = \mu (I_{\alpha} * |u|^p)…

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…

We are mainly concerned with the nonlinear $p$-Laplace equation \begin{equation*} -\Delta_pu+\rho|u|^{p-2}u=\psi(x,u) \end{equation*} on a locally finite graph $G=(V,E)$, where $p$ belongs to $(1, +\infty)$. We obtain existence of positive…

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential.…

In this study, we consider the following extended attraction chemotaxis system of two species parabolic-parabolic-elliptic type with nonlocal terms \[ \begin{cases} u_t=d_1\Delta u-\chi_1\nabla (u\cdot \nabla…

We consider a nonlinear Schr\"odinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are…

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…

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball $B_R$. First, we prove the radial symmetry of the positive ground state solutions by using…

We focus on the ground state of the cubic-quintic nonlinear Schr\"{o}dinger energy functional \begin{gather*} \begin{aligned} {E}(\varphi)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla \varphi|^2+V(x)|\varphi|^2\right)\,dx…

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 $\gamma\in (0,N)$, $p, q >0$,…