Existence of positive solutions for a class of semipositone quasilinear problems through Orlicz-Sobolev space

Abstract:

In this paper we show the existence of weak solutions for a class of semipositone problems of the type \begin{equation}\tag {P} \left \{ \begin {array}{rclcl} -\Delta _{\Phi } u & = & f(u)-a & \mbox {in} & \Omega , \\ u(x)& > & 0 & \mbox {in} & \Omega , \\ u & = & 0 & \mbox {on} & \partial \Omega , \\ \end{array} \right . \end{equation} where $\Omega \subset \mathbb {R}^{N}$, $N \geq 2$, is a smooth bounded domain, $f:[0,+\infty ) \to \mathbb {R}$ is a continuous function with subcritical growth, $a>0$, and $\Delta _{\Phi } u$ stands for the $\Phi$-Laplacian operator. By using variational methods, we prove the existence of a solution for $a$ small enough.