## Differential and Integral Equations

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

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

#### 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