Solutions for critical quasilinear elliptic equations in RN

Abstract

In this paper, we consider the following critical quasilinear problems in RN: −∑i,j=1NDj(bij(v)Div)+12∑i,j=1Nbij′(v)DivDjv+a(x)v=μ|v|qs−2v+|v|2*s−2v,inRN,v(x)→0 as|x| →∞, where N ≥ 4, bij(v) ∼ 1 + |v|2s−2, s ≥ 1, μ > 0 is a constant, q < 2*, 2*=2NN−2 is the Sobolev critical exponent for the embedding of H1(RN) into Lp(RN), and a(x) is a potential function which is finite and positive and satisfies some decay assumptions. We first use the truncation to overcome the difficulties caused by both the unbounded domain and the critical exponents. Then we perform a change of variables to overcome the difficulty caused by the unboundness of bij. Finally a regularization approach combined with a compact argument helps us to prove the existence of infinitely many solutions.