Multiple solutions for a critical quasilinear equation with Hardy potential.

Gao, Fengshuang; Guo, Yuxia

In this paper, we investigate the following quasilinear equation involving a Hardy potential: ⎧ ⎪ ⎨ ⎪ ⎩ − N ∑ i , j = 1 D j (a i j (u) D i u) 1 2 N ∑ i , j = 1 a ′ i j (u) D i u D j u − μ | x | 2 u = a u | u | 2 ∗ − 2 u i n Ω , u = 0 o n ∂ Ω , (P) where 2 ∗ = 2 N N − 2 is the Sobolev critical exponent for the embedding of H 1 0 (Ω) into L p (Ω) , a > 0 is a constant and Ω ⊂ R N is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on a i j , when N ≥ 7 and μ ∈ [ 0 , μ ∗) for some constant μ ∗ , problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

Discrete & Continuous Dynamical Systems - Series S, 2019, Vol 12, Issue 7, p1977

1937-1632

Academic Journal

10.3934/dcdss.2019128