Problem involving nonlocal operator

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*} ($p^{\mathrm{\prime }}$ being the conjugate of $p$), exists in a weak sense, for $q\in (p,p_s^{\ast })$ under certain condition on $\lambda$, where $-{\mathcal{L}}_{\mathrm{\Phi }}$ 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}_\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.