Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

Abstract

We study the nonlocal nonlinear problem \begin{equation}\label{ppp} \left\{ \begin{array}[c]{lll} (-\Delta)^s u = \lambda f(u) & \mbox{in }\Omega, \\ u=0&\mbox{on } \mathbb{R}^N\setminus\Omega, \end{array} \right. \tag{$P_{\lambda}$} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$\!,\,$N>2s$,\,$0<s<1$; $f:\mathbb{R}\rightarrow [0,\infty)$ is a nonlinear continuous function such that $f(0)=f(1)=0$ and $f(t)\sim |t|^{p-1}t$ as $t\rightarrow 0^+$, with $2<p+1<2^*_s$; and $\lambda$ is a positive parameter. We prove the existence of two nontrivial solutions $u_{\lambda}$ and $v_{\lambda}$ to (\ref{ppp}) such that $0\le u_{\lambda}< v_{\lambda}\le 1$ for all sufficiently large $\lambda$. The first solution $u_{\lambda}$ is obtained by applying the Mountain Pass Theorem, whereas the second, $v_{\lambda}$, 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 $L^{\infty}(\Omega)$.