The positive solutions to a quasi-linear problem of fractional p-Laplacian type without the Ambrosetti–Rabinowitz condition

Abstract

In this paper, we study the existence of nontrivial solution to a quasi-linear problem where $$ (-\Delta )_{p}^{s} u(x)=2\lim \nolimits _{\epsilon \rightarrow 0}\int _{\mathbb {R}^N \backslash B_{\varepsilon }(X)} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{| x-y | ^{N+sp}}dy, $$ $$ x\in \mathbb {R}^N$$ is a nonlocal and nonlinear operator and $$ p\in (1,\infty )$$ , $$ s \in (0,1) $$ , $$ \lambda \in \mathbb {R} $$ , $$ \Omega \subset \mathbb {R}^N (N\ge 2)$$ is a bounded domain which smooth boundary $$\partial \Omega $$ . Using the variational methods based on the critical points theory, together with truncation and comparison techniques, we show that there exists a critical value $$\lambda _{*}>0$$ of the parameter, such that if $$\lambda >\lambda _{*}$$ , the problem $$(P)_{\lambda }$$ has at least two positive solutions, if $$\lambda =\lambda _{*}$$ , the problem $$(P)_{\lambda }$$ has at least one positive solution and it has no positive solution if $$\lambda \in (0,\lambda _{*})$$ . Finally, we show that for all $$\lambda \ge \lambda _{*}$$ , the problem $$(P)_{\lambda }$$ has a smallest positive solution.