Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity

Abstract

In this work, we study the existence and the multiplicity of non-negative solutions for the following problem

$$\begin{aligned} ({\mathrm{P}}_\uplambda ) \left\{ \begin{array}{ll} \mathcal {L} u = a(x) u^{q}+ \lambda b(x) u^p\quad \text {in }\Omega , \\ \\ u= 0 ,\;\; \text{ in } \,\mathbb {R}^n\setminus \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset \mathbb {R}^n \;(n\ge 2)$ , is a bounded smooth domain, $\lambda , p, q$ are positive real numbers, $s\in (0,1)$, $a,\, b$ are continuous functions, and $\mathcal {L}$ is a nonlocal operator defined later by (1.1). We establish the existence and we give a multiplicity of solutions by constrained minimization of the Euler-Lagrange functional corresponding to the problem $(P_\lambda )$, on suitable subsets of Nehari manifold and using the fibering maps. Precisely, we show the existence of $\lambda _0>0,$ such that for all $\lambda \in (0,\lambda _0)$, problem $(P_\lambda )$ has at least two non-negative solutions.