Related papers: Singularly perturbed critical Choquard equations

In this work, we study the semiclassical limit of cubic Nonlinear Schr\"odinger equations for mixed states. We justify the limit to a singular Vlasov equation (in which the force field is proportional to the gradient of the density), for…

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u =…

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases}…

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing…

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…

In this paper we consider a fractional $p$-Laplacian equation in the entire space $\mathbb{R}^{N}$ with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the…

We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}\Delta u + u = f(u){\text{ in }}\Omega ,{\text{ }}u > 0{\text{ in }}\Omega ,{\text{ }}u = 0{\text{ on }}\partial \Omega , \] where $\Omega$ is a domain in…

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of…

In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^\eta u =|x|^\eta f(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\,…

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in…

We prove the existence of a solution to the semirelativistic Hartree equation $$\sqrt{-\Delta+m^2}u+ V(x) u = A(x)\left( W * |u|^p \right) |u|^{p-2}u $$ under suitable growth assumption on the potential functions $V$ and $A$. In particular,…

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…

This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case…

In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i \nabla+A(x))^2u + \mu g(x)u = \lambda u + (|x|^{-\alpha} * |u|^{2^*_\alpha})|u|^{2^*_\alpha-2}u ,\; u>0 \;\text{in} \;…

We look for multiple solutions $\mathbf{U}\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem \[ \nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3, \] with a nonlinear function…

We consider the following nonlinear fractional Choquard equation, \begin{equation}\label{e:introduction} \begin{cases} (-\Delta)^{s} u + u = (1 + a(x))(I_\alpha \ast (|u|^{p}))|u|^{p - 2}u\quad\text{ in }\mathbb{R}^N,\\ u(x)\to 0\quad\text{…

In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases}…

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue…

We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schr\"odinger operators, the asymptotic repartition of eigenvectors is the same as in the regular…

In this paper we consider the higher order Lioville-type equation $(-\Delta)^{m} u=\rho^{2m} V(x) e^{u}$ in $\Omega\subseteq\mathbb{R}^{2m}$ with $V\neq0$ a given smooth potential, $\rho\in\mathbb{R}^{+}$ a small parameter which tends to…