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

Authors: N. Ghoussoub and C. Yuan
Journal: Trans. Amer. Math. Soc. 352 (2000), 5703-5743
MSC (2000): Primary 35J20, 35J70, 47J30, 58E30
Published electronically: July 6, 2000
MathSciNet review: 1695021
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Abstract | References | Similar Articles | Additional Information


We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation:

$\left\{ \begin{matrix} {-\triangle_{p} u = \lambda \vert u\vert^{r-2}u + \mu ... ... }, {}} \ {\hphantom{-} u\vert _{\partial \Omega} = 0, } \end{matrix}\right.$

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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Additional Information

N. Ghoussoub
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

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
Article copyright: © Copyright 2000 American Mathematical Society