Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole

作者：D'Aprile Teresa^*

来源：Journal de Mathematiques Pures et Appliquees, 2016, 105(4): 558-602.

DOI：10.1016/j.matpur.2015.11.008

• Slightly supercritical exponent
• Multi-bubble solutions
• Finite-dimensional reduction
• Min-max argument

We are concerned with the existence and the asymptotic analysis when the parameter epsilon tends to 0 of solutions with multiple concentration for the following almost critical problem:
-Delta u = u(N+2/N-2+epsilon) in Omega, u > 0 in Omega, u = 0 on partial derivative Omega,
where Omega is a bounded domain in R-N with a smooth boundary and N >= 3. We are interested in concentration phenomena from the supercritical side epsilon -> 0(+). In particular we prove that, if Omega has a small and not necessarily symmetric hole, then for any fixed odd integer k >= 3 there exists a family of solutions which develops a multiple bubble-shape as epsilon -> 0(+), blowing up at k different points in Omega. This extends the previous result by Del Pino, Felmer and Musso [13], where solutions with a two-bubble profile are constructed.

• 出版日期2016-4