Related papers: Limit profiles for singularly perturbed Choquard e…

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer,…

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb…

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where…

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: \begin{equation}\label{0.1} -\Delta u+\varepsilon u=\big(I_{\alpha}\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N),…

In this paper, we study the existence, non-existence and asymptotic behavior of positive ground states for the nonlinear Choquard equation: \begin{equation}\label{0.1} -\Delta u+\varepsilon u=\big(I_{\alpha}\ast F(u)\big)F'(u),\quad u\in…

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{p\theta}…

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…

We study the nonlocal equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = \varepsilon^{-\alpha} \bigl(I_\alpha \ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in…

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$ -\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$…

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…

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…

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} :…

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -\Delta_{p} u + \varepsilon u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N},…

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 consider a $p$-fractional Choquard-type equation \[ (-\Delta)_p^s u+a|u|^{p-2}u=b(K\ast F(u))F'(u)+\varepsilon_g |u|^{p_g-2}u \quad\text{in $\mathbb{R}^N$}, \] where $0<s<1<p<p_g\leq p_s^*$, $N \geq \max\{2ps+\alpha,p^2 s\}$,…

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…

In this paper, we study the following coupled Choquard system in $\mathbb R^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast |v|^q\bigr)|u|^{p-2}u,\\ &-\Delta…

In the present paper, we consider the following singularly perturbed problem: \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u), & x\in \R^N; u\in H^1(\R^N), \end{array}…

We study the Schr\"{o}dinger-Poisson-Slater equation $$-\Delta u + u+\lambda(I_{2}*|u|^2)u=|u|^{p-2}u\quad\text{in $\mathbb R^3$},$$ where $p\in (3,6)$ and $\lambda>0$. By using direct variational analysis based on the comparison of the…

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon \Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, \qquad u|_{\partial\Omega}=0, \] where $1<p<q<\gamma<p^*$, $a\geq…