Problem involving nonlocal operator

TL;DR

This paper investigates elliptic PDEs with nonlinear nonlocal operators, establishing existence results for solutions under various conditions, including the presence of measure data and fractional operators.

Contribution

It extends the theory of nonlocal elliptic PDEs by proving existence of weak solutions with measure data and fractional operators, covering degenerate cases and nonlinear integrodifferential operators.

Findings

Existence of weak solutions for nonlinear nonlocal PDEs.

Characterization of solutions with measure data based on capacity.

Conditions on parameters for solution existence.

Abstract

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the weak sense to the problem \begin{align*} \begin{split} -\mathscr{L}_\Phi u & = \lambda |u|^{q-2}u\,\,\mbox{in}\,\,\Omega,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus \Omega \end{split} \end{align*} if and only if a weak solution to \begin{align*} \begin{split} -\mathscr{L}_\Phi u & = \lambda |u|^{q-2}u +f,\,\,\,f\in L^{p'}(\Omega),\\ u & = 0\,\, \mbox{on}\,\, \mathbb{R}^N\setminus \Omega \end{split} \end{align*} ($p'$ being the conjugate of $p$), exists in a weak sense, for $q\in(p, p_s^*)$ under certain condition on $\lambda$, where $-\mathscr{L}_\Phi $ is a general nonlocal integrodifferential operator of order $s\in(0,1)$ and $p_s^*$ is the fractional Sobolev conjugate of $p$. We further prove the existence of a measure $\mu^{*}$ corresponding to which a weak solution exists to the problem \begin{align*} \begin{split} -\mathscr{L}_\Phi u & = \lambda |u|^{q-2}u +\mu^*\,\,\,\mbox{in}\,\, \Omega,\\ u & = 0\,\,\, \mbox{in}\,\,\mathbb{R}^N\setminus \Omega \end{split} \end{align*} depending upon the capacity.