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Existence and uniqueness for a \(p\)-Laplacian nonlinear eigenvalue problem. (English) Zbl 1188.35125
Summary: We consider the Dirichlet eigenvalue problem
\[ -\text{div}(|\nabla u|^{p-2}\nabla u ) =\lambda \| u\|_q^{p-q}|u|^{q-2}u, \]
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 \(p=q\).

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35D30 Weak solutions to PDEs
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