Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem
We consider the Dirichlet eigenvalue problem $$ -mathop{ m div}(| abla u|^{p-2} abla u ) =lambda | u|_q^{p-q}|u|^{q-2}u, $$ where the unknowns $uin W^{1,p}_0(Omega )$ (the eigenfunction) and $lambda >0$ (the eigenvalue), $Omega $ is an arbitrary domain in $mathbb{R}^N$ with finite measure,...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2010-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2010/26/abstr.html |
| Summary: | We consider the Dirichlet eigenvalue problem $$ -mathop{ m div}(| abla u|^{p-2} abla u ) =lambda | u|_q^{p-q}|u|^{q-2}u, $$ where the unknowns $uin W^{1,p}_0(Omega )$ (the eigenfunction) and $lambda >0$ (the eigenvalue), $Omega $ is an arbitrary domain in $mathbb{R}^N$ with finite measure, $1<p<infty $, $1<q< p^*$, $p^*=Np/(N-p)$ if $1<p<N$ and $p^*=infty $ if $pgeq N$. We study several existence and uniqueness results as well as some properties of the solutions. Moreover, we indicate how to extend to the general case some proofs known in the classical case $p=q$. |
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| ISSN: | 1072-6691 |