Related papers: Nonlinear fractional elliptic problem with singula…

We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in…

We consider the problem of multiplicity and uniqueness of radial solutions of a nonlinear elliptic equation of the form \begin{eqnarray*} \begin{gathered} \Delta u +f(u)=0,\quad x\in \mathbb{R}^N, N\geq 2, \lim\limits_{|x|\to\infty}u(x)=0.…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $\delta(x)=\text{dist}\,(x,\partial \Omega)$. Assume $\mu>0$, $\nu$ is a nonnegative finite measure on $\partial \Omega$ and $g \in C(\Omega \times \mathbb{R}_+)$. We study…

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber…

In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$ (-\Delta)^{\alpha} u-\mu\frac{u}{|x|^{2\alpha}}+a(x) u=|u|^{2^*-2}u+k(x)|u|^{q-2}u$$ $$ u\,\in\,H^\alpha({\mathbb R}^N),$$ where…

We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when…

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 establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…

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, some elliptic equation in a bounded open domain in $\mathbb{R}^N$ ($N\geq 2$) with $C^2$ boundary $\partial\Omega$ is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the…

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…

In this paper we investigate the existence of multiple solutions for the following two fractional problems \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\Omega})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \partial…

We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{%…

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n,…

We consider equation $-\Delta u+f(x,u)=0$ in smooth bounded domain $\Omega\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $\Omega\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partial\Omega$. We find the condition on the order of degeneracy…

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -\Delta u= h(u){f} \ \ \text{in}\,\ \Omega, $$ where $f$ is an irregular datum,…

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{…

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit…