Equações elípticas com o Laplaciano fracionário e não linearidades indefinidas

Abstract

In this work, we investigate the existence, non-existence and multiplicity of positive solutions of the problem below, \begin{equation*}\left\lbrace \begin{array}{rll} (-\Delta)^s u -\lambda u & = f(x)g(u) ,& \textrm{em } Omega\\u & = 0 &\textrm{sobre } \partial \Omega\end{array} light.end{equation*} where $\Omega$ is a bounded smooth domain of $\R^N$, $f$ is a continuous and bounded function that changes sign in $\Omega$, and $g$ is a real function and can be subcritical or critical. The operator $(-\Delta)^s$ is the Fractional Laplacian, $N \geq 2s$, $s \in (0,1)$ e $\lambda \geq \lambda_1$, where $\lambda_1$ is the first eigenvalue of operator $(-\Delta)^s$. Ours results will be obtained through variational sub-super solution methods, mountain pass theorem and linking theorem.