Asymptotic Behavior of Least Energy Solutions of a Biharmonic Equation in Dimension Four

Abstract

In this paper we consider a biharmonic equation on a bounded domain in R 4 with large exponent in the nonlinear term. We study asymptotic behavior of positive solutions obtained by minimizing suitable functionals. Among other results, we prove that c p , the minimum of energy functional with the nonlinear exponent equal to p, is like ρ 4 e/p as p → +∞, where ρ 4 = 32ω 4 and ω 4 is the area of the unit sphere S 3 in R 4 . Using this result, we compute the limit of the L ∞ -norm of least energy solutions as p → +∞. We also show that such solutions blow up at exactly one point which is a critical point of the Robin function.