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.