Problem involving nonlocal operator

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.