EXISTENCE OF NONTRIVIAL SOLUTIONS TO POLYHARMONIC EQUATIONS WITH SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH

摘要

The main purpose of this paper is to establish the existence of non-trivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation (-Δ)~m = f(x,u), subject to the Dirichlet boundary condition u -▽u = ... = ▽~(m-1)u = 0, on the bounded domains Ω (C) R~(2m) when the nonlinear term f satisfies exponential growth condition. We will study the above problem both in the case when f satisfies the well-known Ambrosetti-Rabinowitz condition and in the case without the Ambrosetti-Rabinowitz condition. This is one of a series of works by the authors on nonlinear equations of Laplacian in R~2 and N-Laplacian in R~N when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).