Differential and Integral Equations

Solutions for a quasilinear elliptic equation with critical Sobolev exponent and perturbations on ${\bf R}^N$

Huan-Song Zhou

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Differential Integral Equations Volume 13, Number 4-6 (2000), 595-612.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356061240

Mathematical Reviews number (MathSciNet)
MR1750041

Zentralblatt MATH identifier
0970.35055

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B33: Critical exponents 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Citation

Zhou, Huan-Song. Solutions for a quasilinear elliptic equation with critical Sobolev exponent and perturbations on ${\bf R}^N$. Differential Integral Equations 13 (2000), no. 4-6, 595--612. https://projecteuclid.org/euclid.die/1356061240.


Export citation