Multiple peak solutions for polyharmonic equation with critical growth

Yuxia Guo and Ting Liu

Mathematische Nachrichten, 2021, vol. 294, issue 2, 310-337

Abstract: This paper is concerned with the following elliptic problem: P (−Δ)mu=u+m∗−1+λu−s1φ1,inB1,u∈D0m,2(B1),(P)where (−Δ)m is the polyharmonic operator, m∗=2NN−2m is the critical Sobolev embedding exponent. B1 is the unit ball in RN, s1 and λ>0 are parameters, φ1>0 is the eigenfunction of (−Δ)m,D0m,2(B1) corresponding to the first eigenvalue λ1 with maxy∈B1φ1(y)=1, u+=max(u,0). By using the Lyapunov–Schimit reduction method combining with the minimax argument, we construct the solutions to (P) with many peaks near the boundary but not on the boundary of the domain. Moreover, we prove that the number of solutions for the problem (P) is unbounded as the parameter tends to infinity, therefore proving the Lazer–McKenna conjecture for the higher order case with critical growth.

Date: 2021
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https://doi.org/10.1002/mana.201900428

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