Positive solutions for a class semipositone quasilinear problem with Orlicz–Sobolev critical growth

In this work, we study the existence of positive solutions for the following class of semipositone quasilinear problems: −ΔΦu=λf(x,u)+b(u)−ainΩ,u>0inΩ,u=0on∂Ω,$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{rclcl}-\Delta _{\Phi } u & = & \lambda f(x,u)+b(u)-a & \mbox{in} & \Omega , \\[3pt] u& > & 0 & \mbox{in} & \Omega , \\[3pt] u & = & 0 & \mbox{on} & \partial \Omega , \end{array} \right.} \end{equation*}$$where Ω⊂RN$\Omega \subset \mathbb {R}^N$ is a bounded domain, N≥2$N\ge 2$, λ,a>0$\lambda ,a > 0$ are parameters, f(x,u)$ f(x,u)$ is a Caractheodory function, and b(t)$b(t)$ has a critical growth with relation to the Orlicz–Sobolev space W01,Φ(Ω)$W_0^{1,\Phi }(\Omega )$. The main tools used are variational methods, a concentration compactness theorem for Orlicz–Sobolev space and some priori estimates.