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.