Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth. (English) Zbl 0946.35026

Consider the Dirichlet problem (P) \((- \Delta) ^m u=g(x,u)\) in \(\Omega\) subject to the boundary condition \(u= D u= \ldots = D ^{m-1} u=0\) on \(\partial \Omega\) where \(\Omega\) is a bounded regular subset of \({\mathbb R}^{2m}\), \(m\geq 1\). Using a variational method, the author proves the existence of nontrivial weak solutions to (P) when \(g(x,t)\) belongs to \({\mathcal C} ^1(\overline{\Omega}\times{\mathbb R})\) and behaves like \(\exp(t^2)\) at infinity.


35J40 Boundary value problems for higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35J35 Variational methods for higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A15 Variational methods applied to PDEs