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}_Φu & = λ|u|^{q-2}u\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} if and only if a weak solution to \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +f,\,\,\,f\in L^{p'}(Ω),\\ u & = 0\,\, \mbox{on}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} ($p^{\prime }$ being the conjugate of $p$), exists in a weak sense, for $q\in (p,p_s^{\ast })$ under certain condition on $\lambda$, where $-{\mathcal{L}}_{\Phi }$ is a general nonlocal integrodifferential operator of order $s\in (0,1)$ and $p_s^{\ast }$ is the fractional Sobolev conjugate of $p$. We further prove the existence of a measure ${\mu }^{\ast }$ corresponding to which a weak solution exists to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +μ^*\,\,\,\mbox{in}\,\, Ω,\\ u & = 0\,\,\, \mbox{in}\,\,\mathbb{R}^N\setminus Ω\end{split} \end{align*} depending upon the capacity.