On the Dirichlet problem for the $n, \alpha$-Laplacian with the nonlinearity in the critical growth range
2011
Abstrakt
Let $\Omega\subset\er^n$, $n \geq 2$, be a bounded domain. Applying the Mountain Pass Theorem we prove the existence of a~non-trivial weak solution to the Dirichlet problem $$ %u\in W_0^1L^{\Phi}(\Omega)\qquad\text{and}\qquad -\divergence \Bigl(\Phi''(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Bigr) =f(x,u)\quad\text{ in }\Omega\ , $$ where $u$ is in the Sobolev-Orlicz space $W_0^1L^{\Phi}(\Omega)$ with a~Young function of the type $\Phi(t)\approx t^n\log^{\alpha}(t)$, $\alphan-1$, and $|f(x,t)|\approx \exp(\beta|t|^{\frac{n}{n-1-\alpha}})$, $\beta0$.