Related papers: Bubble solution for the critical Hartree equation …

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -\Delta u…

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu,…

In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u-\mu\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (\alpha)-2}u}{|x|^{\alpha}} + f(x,u), && \mbox{in} \ \Omega,\\ &u=0, && \mbox{in} \…

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(\mu x))^{2}u= \mu |u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ \Omega, \ \ \ \ u \in…

We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \begin{equation}\label{e051} \left\{ \begin{split} &\Delta…

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^su=\lambda u^q+u^{2_s^*-1},\ u>0\quad\text{in…

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] \ u= 0 & \mbox{on }\; \partial \Omega,…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has…

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases}…

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[\epsilon^2 \Delta u+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu}=0 \ \mbox{on}\ \partial \Omega \]…

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -\Delta u=\lambda_1\dfrac{V_1 e^{u}}{…

We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and…

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h…

Let $\Omega\subset \mathbb{R}^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\alpha,\beta,\lambda,\mu}(\Omega)…

We study the well-posedness for the inhomogeneous Hartree equation $i\partial_t u + \Delta u = \lambda(I_\alpha \ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $s\ge0$. Until recently, its well-posedness theory has been intensively…

We provide new conditions for the local existence of solutions to the time-weighted parabolic equation $ u_t - \Delta u = h(t)f(u) \mbox{ in } \Omega \times (0,T),$ where $ \Omega $ is a arbitrary smooth domain, $f\in C(\mathbb{R})$, $h\in…

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…

We prove the existence of positive and of nodal solutions for $-\Delta u = |u|^{p-2}u+\mu |u|^{q-2}u$, $u\in {\rm H_0^1}(\Omega)$, where $\mu >0$ and $2<q<p=2N(N-2)$, for a class of open subsets $\Omega$ of $\mathbb{R}^N$ lying between two…