Existence of Nontrivial Solutions to Polyharmonic Equations with Subcritical and Critical Exponential Growth

The main purpose of this paper is to establish the existence of nontrivial 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 (−∆) u = f(x, u), subject to the Dirichlet boundary condition u = ∇u = ... = ∇m−1u = 0, on the bounded domains Ω ⊂ R2m 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 R2 and N−Laplacian in RN when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).