Solutions for critical quasilinear elliptic equations in RN

In this paper, we consider the following critical quasilinear problems in R-N : - Sigma(N)(i,j=1) D-j(b(ij)(v)D(i)v)+1/2 Sigma(N)(i,j=1) b(ij)'(v)D(i)vD(j)v+a(x)v = mu vertical bar v vertical bar(qs-2)v + vertical bar v vertical bar(2)*(s-2)v, in R-N, v(x) -> 0 as vertical bar x vertical bar -> infinity, where N >= 4, b(ij)(v) similar to 1 + vertical bar v vertical bar(2s-2), s >= 1, mu > 0 is a constant, q < 2*, 2* = 2N/N-2 is the Sobolev critical exponent for the embedding of H-1(R-N) into L-p(R-N), 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 b(ij). Finally a regularization approach combined with a compact argument helps us to prove the existence of infinitely many solutions. Published under license by AIP Publishing.