Infinitely many solutions for a Hénon-type system in hyperbolic space.

Abstract

This paper is devoted to studying the semilinear elliptic system of Hénon type ⎧⎩⎨⎪⎪−ΔBNu=K(d(x))Qu(u,v),−ΔBNv=K(d(x))Qv(u,v),u,v∈H1r(BN),N≥3,{−ΔBNu=K(d(x))Qu(u,v),−ΔBNv=K(d(x))Qv(u,v),u,v∈Hr1(BN),N≥3, in the hyperbolic space BNBN, where H1r(BN)={u∈H1(BN):u is radial}Hr1(BN)={u∈H1(BN):u is radial} and −ΔBN−ΔBN denotes the Laplace–Beltrami operator on BNBN, d(x)=dBN(0,x)d(x)=dBN(0,x), Q∈C1(R×R,R)Q∈C1(R×R,R) is p-homogeneous, and K≥0K≥0 is a continuous function. We prove a compactness result and, together with Clark’s theorem, we establish the existence of infinitely many solutions.