Related papers: Singularly perturbed critical Choquard equations

In this article, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects and in the presence of critical nonlinearity of Hartree type. To this end, we first investigate the corresponding…

We are concerned with a Brezis-Nirenberg type problem for a critical Choquard equation, in the sense of Hardy-Littlewood-Sobolev inequality, and with the Hardy potential in a smooth bounded domain. By exploiting variational methods we…

We study the following Choquard type equation in the whole plane $(C) -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr\"odinger potential and the nonlinearity…

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+…

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_t\psi = ic\hbar\sum_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a…

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad…

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha…

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-\Delta u=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu…

In this paper, we consider the following singularly perturbed fractional Kirchhoff problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N}…

We study the following singularly perturbed problem for a coupled nonlinear Schr\"{o}dinger system: {displaymath} {cases}-\e^2\Delta u +a(x) u = \mu_1 u^3+\beta uv^2, \quad x\in \R^3, -\e^2\Delta v +b(x) v =\mu_2 v^3+\beta vu^2, \quad x\in…

In this paper, an autonomous Choquard equation with the upper critical exponent is considered. By using the Poho\v{z}aev constraint method, the subcritical approximation method and the compactness lemma of Strauss, a groundstate solution in…

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 $\alpha\in(0,N)$,…

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha…

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schr\"odinger equations with subcritical exponent. For some smooth bounded domain $\Omega\subset \mathbf R^n$, our boundary condition is given…

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 the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*}…

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.

We study the following nonlinear Schr\"{o}dinger equation $$ iu_t=-\Delta u+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:=…