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

In this paper, we study the existence of nontrivial solution to a quasi-linear problem <graphic></graphic>where (-Δ)psu(x)=2limϵ→0∫RN\Bε(X)|u(x)-u(y)|p-2(u(x)-u(y))|x-y|N+spdy,<inline-graphic></inline-graphic>x∈RN<inline-graphic></inline-graphic> is a nonlocal and nonlinear operator and p∈(1,∞)<inline-graphic></inline-graphic>, s∈(0,1)<inline-graphic></inline-graphic>, λ∈R<inline-graphic></inline-graphic>, Ω⊂RN(N≥2)<inline-graphic></inline-graphic> is a bounded domain which smooth boundary ∂Ω<inline-graphic></inline-graphic>. Using the variational methods based on the critical points theory, together with truncation and comparison techniques, we show that there exists a critical value λ∗>0<inline-graphic></inline-graphic> of the parameter, such that if λ>λ∗<inline-graphic></inline-graphic>, the problem (P)λ<inline-graphic></inline-graphic> has at least two positive solutions, if λ=λ∗<inline-graphic></inline-graphic>, the problem (P)λ<inline-graphic></inline-graphic> has at least one positive solution and it has no positive solution if λ∈(0,λ∗)<inline-graphic></inline-graphic>. Finally, we show that for all λ≥λ∗<inline-graphic></inline-graphic>, the problem (P)λ<inline-graphic></inline-graphic> has a smallest positive solution. [ABSTRACT FROM AUTHOR]