Related papers: A Nonlinear Elliptic PDE with Two Sobolev-Hardy Cr…

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial…

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -\Delta…

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N,…

We study the following nonlinear Schr\"odinger equation and we look for normalized solutions $(\mu,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -\Delta u + \mu u = g(u)\quad \text{in}\ {\bf R}^N, \qquad…

In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -\Delta…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…

In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-\Delta)^s u & =…

In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}_\gamma\left(|u|^{p_1}\right)|u|^{p_2}, p_1, p_2>0, \gamma \in[0, n)$, with initial…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schr\"odinger system \begin{align*} - \Delta u + \lambda_1 u &= \frac{\alpha u(\alpha…

On a compact Riemannian manifold, we study a singular elliptic equation with critical Sobolev exponent and critical Hardy potential. In a first part, we prove an $H^2_1$ type decomposition result for Palais-Smale sequences of the associated…

We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2…

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: (-\Delta)^{\alpha/2}u=\lambda u^q+u^{\frac{N+\alpha}{N-\alpha}}, \quad…

We study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully…

In this article, we deal with the following $p$-fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-\Delta)_{p, A}^{s} u+V(x)|u|^{p-2}…

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N,…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$…