Electron. J. Diff. Equ., Vol. 2010(2010), No. 26, pp. 1-10.

Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem

Giovanni Franzina, Pier Domenico Lamberti

We consider the Dirichlet eigenvalue problem
 -\hbox{div}(|\nabla u|^{p-2}\nabla u )
  =\lambda \| u\|_q^{p-q}|u|^{q-2}u,
where the unknowns $u\in W^{1,p}_0(\Omega )$ (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 $p\geq N$. 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 $p=q$.

Submitted January 14, 2010. Published February 16, 2010.
Math Subject Classifications: 35P30.
Key Words: p-laplacian; eigenvalues; existence; uniqueness results

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Giovanni Franzina
Dipartimento di Matematica, University of Trento
Trento, Italy
email: g.franzina@email.unitn.it
Pier Domenico Lamberti
Dipartimento di Matematica Pura e Applicata
University of Padova, Padova, Italy
email: lamberti@math.unipd.it

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