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*} (\(p'\) 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.