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}^Nsetminus 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,,,,fin L^{p}(Omega), u & = 0,, mbox{on},, mathbb{R}^Nsetminus Omega end{split} end{align*} ($p$ being the conjugate of $p$), exists in a weak sense, for $qin(p, p_s^*)$ under certain condition on $lambda$, where $-mathscr{L}_Phi $ is a general nonlocal integrodifferential operator of order $sin(0,1)$ 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}^Nsetminus Omega end{split} end{align*} depending upon the capacity.