LAUSR

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