Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

Abstract We study the nonlocal nonlinear problem $$\begin{aligned} \left\{ \begin{array}[c]{lll} (-\Delta )^s u = \lambda f(u) &{} \text{ in } \Omega , \\ u=0&{}\text{ on \mathbb {R}^N{\setminus }\Omega \quad (P_{\lambda }) \end{array} \right. \end{aligned}$$ ( - ? ) s u = ? f in ? , 0 on R N \ P where $$\Omega $$ is a bounded smooth domain $$\mathbb {R}^N$$ $$N>2s$$ > 2 $$0<s<1$$ < 1 ; $$f:\mathbb {R}\rightarrow [0,\infty )$$ : ? [ ? continuous function such that $$f(0)=f(1)=0$$ and $$f(t)\sim |t|^{p-1}t$$ t ? | p as $$t\rightarrow 0^+$$ + with $$2<p+1<2^*_s$$ ? $$\lambda positive parameter. prove existence of two nontrivial solutions $$u_{\lambda }$$ $$v_{\lambda v to ( $$P_{\lambda ) $$0\le u_{\lambda }< v_{\lambda }\le 1$$ ? for all sufficiently large . The first solution obtained by applying Mountain Pass Theorem, whereas second, via sub- super-solution method. point out our results hold regardless behavior nonlinearity f at infinity. In addition, we obtain these belong $$L^{\infty }(\Omega L