Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four
Mohamed AyedMassimo GrossiKhalil Mehdi
35J6035J65biharmonic operatorleast energy solution
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.
Indiana University Mathematics Journal
2006
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10.1512/iumj.2006.55.2723
10.1512/iumj.2006.55.2723
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Indiana Univ. Math. J. 55 (2006) 1723 - 1750
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