## Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

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- by N. Ghoussoub and C. Yuan
- Trans. Amer. Math. Soc.
**352**(2000), 5703-5743 - DOI: https://doi.org/10.1090/S0002-9947-00-02560-5
- Published electronically: July 6, 2000
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## Abstract:

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: \[ \left \{ \begin {matrix} {-\triangle _{p} u = \lambda |u|^{r-2}u + \mu \textstyle {\frac {|u|^{q-2}}{|x|^{s}}}u \quad \text {in $\Omega $}, {}} {\hphantom {-} u|_{\partial \Omega } = 0, }\hfill \end {matrix}\right . \] where $\lambda$ and $\mu$ are two positive parameters and $\Omega$ is a smooth bounded domain in $\mathbf {R}^n$ containing $0$ in its interior. The variational approach requires that $1 < p < n$, $p\leq q\leq p^{*}(s)\equiv \frac {n-s}{n-p}p$ and $p\leq r\leq p^*\equiv p^*(0)=\frac {np}{n-p}$, which we assume throughout. However, the situations differ widely with $q$ and $r$, and the interesting cases occur either at the critical Sobolev exponent ($r=p^*$) or in the Hardy-critical setting ($s=p=q$) or in the more general Hardy-Sobolev setting when $q=\frac {n-s}{n-p}p$. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen*dual sets*. Many of the results are new even in the case $p=2$, especially those corresponding to singularities (i.e., when $0<s\leq p)$.

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## Bibliographic Information

**N. Ghoussoub**- Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada
- MR Author ID: 73130
**C. Yuan**- Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada
- Received by editor(s): August 11, 1998
- Published electronically: July 6, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 5703-5743 - MSC (2000): Primary 35J20, 35J70, 47J30, 58E30
- DOI: https://doi.org/10.1090/S0002-9947-00-02560-5
- MathSciNet review: 1695021