Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

Abstract

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