Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth. (English) Zbl 1249.35085

The authors study the existence of a nontrivial solution of the quasilinear elliptic problem \(-\Delta _N u + | u | ^{N-2} u = a(x) g(u)\) in \(\Omega \), \(u = 0\) on \(\partial \Omega \), where \(\Delta _{N}\) is the \(N\)-Laplacian operator, \(\Omega \subset {\mathbb R}^N\) is an exterior domain with smooth boundary, \(a(x)\) is a continuous function changing sign in \(\Omega \) and the nonlinearity \(g(u)\) has an exponential critical growth at infinity. Variational techniques are applied, in particular, the mountain pass theorem together with the Moser’s function and the Trudinger-Mose inequality which enables to overcome the difficulties created by the lack of compactness in the exterior domain.


35J35 Variational methods for higher-order elliptic equations
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations