On fractional p-Laplacian problems with local conditions.

In this paper, we deal with fractional p-Laplacian equations of the form { (- Δ) p s ⁢ u = λ ⁢ f ⁢ (x , u) , x ∈ Ω , u ⁢ (x) = 0 , x ∈ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)_{p}^{s}u&\displaystyle=\lambda f% (x,u),&&\displaystyle x\in\Omega,\\ \displaystyle u(x)&\displaystyle=0,&&\displaystyle x\in\mathbb{R}^{N}\setminus% \Omega,\end{aligned}\right. where λ ∈ (0 , + ∞) {\lambda\in(0,+\infty)} , 0 < s < 1 < p < + ∞ {0<s<1<p<+\infty} and Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} , N ⩾ 2 {N\geqslant 2} , is a bounded domain with smooth boundary. With assumptions on f ⁢ (x , t) {f(x,t)} just in Ω × (- δ , δ) {\Omega\times(-\delta,\delta)} , where δ > 0 {\delta>0} is small, existence and multiplicity of nontrivial solutions are obtained via variational methods. [ABSTRACT FROM AUTHOR]