Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the typ e-Delta(BN)u-lambda u=a(x)|u|(p-1)u+finB(N),u is an element of H-1(B-N), where B-N denotes the hyperbolic space, 1<p<2(& lowast;)-1:=N+2/N-2,lambda<(N-1)(2)/4,and f is an element of H-1(B-N)(f(sic)0) is a non-negative functional. The potentiala is an element of L-infinity(B-N)isassumed to be strictly positive, such that lim(d(x,0)->infinity)a(x)-> 1,whered(x,0)denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that a(x)<= 1. Then the casea(x)>= 1 is considered,and the existence of two positive solutions is proved. In both cases, it is assumedthat mu({x:a(x)not equal 1})>0.Subsequently, we establish the existence of two positivesolutions fora(x)equivalent to 1 and prove asymptotic estimates for positive solutions usingbarrier-type arguments. The proofs for existence combine variational arguments, keyenergy estimates involvinghyperbolic bubbles.