Differential and Integral Equations

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

Huan-Song Zhou

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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


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