IUMJ

Title: Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four

Authors: Massimo Grossi, Mohamed Ben Ayed and Khalil El Mehdi

Issue: Volume 55 (2006), Issue 5, 1723-1750

Abstract:

In this paper we consider a biharmonic equation on a bounded domain in $\mathbb{R}^4$ 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 $\rho_{4}e/p$ as $p \to +\infty$, where $\rho_{4} = 32\omega_{4}$ and $\omega_{4}$ is the area of the unit sphere $S^{3}$ in $\mathbb{R}^{4}$. Using this result, we compute the limit of the $L^{\infty}$-norm of least energy solutions as $p \to +\infty$. We also show that such solutions blow up at exactly one point which is a critical point of the Robin function.