The existence of nontrivial solutions to nonlinear elliptic equation of p
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- 作者：Gongbao Li ; Xiaoyan Liang
- 关键词： ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V0Y-4VDY7X8-D&_mathId=mml28&_user=10&_cdi=5659&_pii=S0362546X09000765&_rdoc=85&_issn=0362546X&_acct=C000029478&_version=1&_userid=578664&md
- 刊名：Nonlinear Analysis
- 出版年：2009
- 出版时间：1 September 2009-15 September
- 年：2009
- 卷：71
- 期：5-6
- 页码：2316-2334
- 全文大小：1017 K
文摘
In this paper, we study the following nonlinear elliptic equation of p–q-Laplacian type on RN: where 1<q≤p<N, and Δsu=div(us−2u) is the s-Laplacian of u. We prove that under suitable conditions on f(x,t), if g(x)≡0 and a(x)≡m>0, b(x)≡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 g*<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 RN, 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.