Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth

summary:Let $\Omega \subset \mathbb R^n$, $n\geq 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$ \displaylines { u\in W^1 L^{\Phi }(\Omega ), \quad -\operatorname {div}\Big (\Phi '(|\nabla u|)\frac {\nabla u}{|\nabla u|}\Big ) +V(x)\Phi '(|u|)\frac {u}{|u|}=f(x,u)+\mu h(x)\quad \text {in} \Omega ,\cr \frac {\partial u}{\partial {\bf n}}=0\quad \text {on} \partial \Omega ,\cr } $$ where $\Phi $ is a Young function such that the space $W^1 L^{\Phi }(\Omega )$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V(x)$ is a continuous potential, $h\in (L^{\Phi }(\Omega ))^*$ is a nontrivial continuous function, $\mu \geq 0$ is a small parameter and ${\bf n}$ denotes the outward unit normal to $\partial \Omega $