| Abstrakt: |
Abstract: Let $\Omega\subset\mathbb R^n$, $n\geq2$, 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 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, $\mu\geq0$ is a small parameter and $ n$ denotes the outward unit normal to $\partial\Omega$. |
