Abstract

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system

⎧⎩⎨−Δλu=|v|p−1v−Δλv=|u|q−1uu=v=0in Ω,in Ω,on ∂Ω,

in a bounded domain Ω⊂RN with smooth boundary ∂Ω. Here p,q>1, and Δλ is the strongly degenerate operator of the form

Δλu=∑j=1N∂∂xj(λ2j(x)∂u∂xj),

where λ(x)=(λ1(x),…,λN(x)) satisfies certain conditions.