The existence of nontrivial solutions to nonlinear elliptic equation of p-q-Laplacian type on RN

In this paper, we study the following nonlinear elliptic equation of p-q-Laplacian type on R-N: {-Delta(p)u + a(x)|u|(p-2)u- Delta(q)u + b(x)|u|(q-2)u = f(x, u) + g(x), x is an element of R-N u is an element of W equivalent to W-1,W-p(R-N) boolean AND W-1,W-q(R-N) where 1 < q <= p < N, and Delta(s)u = div(|del u|(s-2)del u) is the s-Laplacian of u. We prove that under suitable conditions on f (x, t), if g(x) equivalent to 0 and a(x) equivalent to m > 0, b(x) equivalent to n > 0 for some constants m and n, then the problem (*) has at least one nontrivial weak solution (see Theorem 1.12), generalizing a similar result for p-Laplacian type equation in [J.F. Yang. X.P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded Domains(I)Positive mass case, Acta Math. Sci. 7 (1987) 341-359]. Also, we prove that under essentially the same assumptions on f (x, t) as that in Theorem 1.12. there exists a constant C > 0. such that if parallel to g parallel to(*) < C, then the problem (*) possesses at least two nontrivial weak solutions (see Theorem 1.15), generalizing a similar result in [D.M. Cao. G.B. Li, Huansong Zhou, The existence of two solutions to quasilinear elliptic equations on R-N Chinese J. Contemp. Math. 17 (3) (1996) 277-285] for p-Laplacian type equation. Since our assumptions on f (x, t) are weaker than that in the above-mentioned reference, Theorem 1.15 is better than the main result in the same even if p = q. (C) 2009 Elsevier Ltd. All rights reserved.