Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth

Abstract

The author considers the semilinear elliptic equation $$ -\Delta^{m}u=g(x,u), $$ subject to Dirichlet boundary conditions $u=Du=\cdots=D^{m-1}u=0$, on a bounded domain $\Omega\subset\mathbb{R}^{2m}$. The notion of nonlinearity of critical growth for this problem is introduced. It turns out that the critical growth rate is of exponential type and the problem is closely related to the Trudinger embedding and Moser type inequalities. The main result is the existence of non trivial weak solutions to the problem.