On fractional p-Laplacian type equations with general nonlinearities

Authors: ADEL DAOUAS, MOHAMED LOUCHAICH

Abstract: In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Keywords: Fractional p-Laplacian problem, variational method, fractional Sobolev space, Palais-Smale condition

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