Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\mathbb{R}^ N\) with nonlinearities in the critical growth range
This paper studies the homogeneous Dirichlet problem for equations of the form \(\text{div}(|Du|^{n-2}Du)= f(x,u)\) in a smooth bounded domain \(\Omega\) in \(\mathbb{R}^n\) with \(n\geq 2\). The critical growth range here is on \(f\), which is a nonnegative, continuous function. It is assumed that there is a constant \(\alpha_0\) such that \(f(x,u)= o(\exp(\alpha|u|^{n/(n-2)}))\) as \(u\to\infty\) for \(\alpha>\alpha_0\) and \(\exp(\alpha|u|^{n/(n-1)})= o(f(x,u))\) for \(\alpha<\alpha_0\). In addition, certain technical conditions are made on \(f\) which imply that the problem has a positive solution, but the important element here is that \(f\) grows faster than any polynomial as \(u\to\infty\). Thus the ``standard'' methods for analyzing critical growth problems must be modified. The analysis here is quite clever, and the results improve previous attempts by various authors to study this problem.
scientific article; zbMATH DE number 896469