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Multiple peak solutions for polyharmonic equation with critical growth
Author(s) -
Guo Yuxia,
Liu Ting
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900428
Subject(s) - mathematics , eigenfunction , minimax , sobolev space , conjecture , mathematical analysis , unit sphere , embedding , eigenvalues and eigenvectors , pure mathematics , elliptic curve , domain (mathematical analysis) , physics , quantum mechanics , artificial intelligence , computer science , mathematical optimization
This paper is concerned with the following elliptic problem: P( − Δ ) m u = u + m ∗ − 1 + λ u − s 1 φ 1 , inB 1 ,u ∈ D 0 m , 2 ( B 1 ) ,( P )where( − Δ ) m is the polyharmonic operator,m ∗ = 2 N N − 2 mis the critical Sobolev embedding exponent. B 1 is the unit ball in R N , s 1 and λ > 0 are parameters,φ 1 > 0 is the eigenfunction of( − Δ ) m , D 0 m , 2 ( B 1 ) corresponding to the first eigenvalue λ 1 withmax y ∈ B 1φ 1 ( 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.