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Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\mathbb{R}^ N\) with nonlinearities in the critical growth range. (English) Zbl 0858.35043

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.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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