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2000 Solutions for a quasilinear elliptic equation with critical Sobolev exponent and perturbations on ${\bf R}^N$
Huan-Song Zhou
Differential Integral Equations 13(4-6): 595-612 (2000).

Abstract

We consider the following quasilinear elliptic problem \begin{equation} \begin{cases} -\mbox{div } |\nabla u|^{p-2}\nabla u)+c|u|^{p-2}u=|u|^ {p^*-2}u+f(x,u)+h(x)\\ u\in W^{1, p}(R^N), \ \ \ N>p\geq 2, \end{cases} \tag*{*} \end{equation} where $c>0, $ $ p^*=\frac{Np}{N-p},$ $ h(x)\in W^{-1, \frac{p}{p-1}}(R^N)$ (i.e., the dual space of $W^{1, p}(R^N)$), $f(x,0)=0 $ and $ f(x,u)$ is a lower-order perturbation of $|u|^ {p^*-2}u$ in the sense that $\lim_{u\rightarrow \infty}\frac{f(x,u)}{|u|^ {p^*-2}u}=0$. It is well known that (*) has only a trivial solution if $f(x,u)\equiv h(x)\equiv 0$ by a Pohozaev type identity, but (*) has a nontrivial solution if there is a subcritical perturbation, e.g., $h(x){\equiv} 0$ and $f(x,u){\not\equiv} 0$. In this paper, we prove that (*) has at least two distinct solutions if there are two perturbations, i.e., $f(x,u) {\not\equiv} 0 $ and $ h(x){\not\equiv} 0$ (inhomogeneous term) with $\| h\|$ small enough.

Citation

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Huan-Song Zhou. "Solutions for a quasilinear elliptic equation with critical Sobolev exponent and perturbations on ${\bf R}^N$." Differential Integral Equations 13 (4-6) 595 - 612, 2000.

Information

Published: 2000
First available in Project Euclid: 21 December 2012

zbMATH: 0970.35055
MathSciNet: MR1750041

Subjects:
Primary: 35J65
Secondary: 35B33, 58E05

Rights: Copyright © 2000 Khayyam Publishing, Inc.

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Vol.13 • No. 4-6 • 2000
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