Geodesic
Eigenvalue problems for the \(p\)-Laplacian with indefinite weights
Electronic journal of differential equations, Tome 2001 (2001)
We consider the eigenvalue problem $-\Delta_p u=\lambda V(x) |u|^{p-2} u, u\in W_0^{1,p} (\Omega)$ where $p greater than 1, \Delta_p$ is the p-Laplacian operator, $\lambda greater than 0, \Omega$ is a bounded domain in $\mathbb{R}^N$ and $V$ is a given function in $L^s (\Omega) ( s$ depending on $p$ and $N$). The weight function $V$ may change sign and has nontrivial positive part. We prove that the least positive eigenvalue is simple, isolated in the spectrum and it is the unique eigenvalue associated to a nonnegative eigenfunction. Furthermore, we prove the strict monotonicity of the least positive eigenvalue with respect to the domain and the weight.
Classification : 35J20, 35J70, 35P05, 35P30
Keywords: nonlinear eigenvalue problem, p-Laplacian, indefinite weight 
@article{EJDE_2001__2001__a148,
     author = {Cuesta,  Mabel},
     title = {Eigenvalue problems for the {\(p\)-Laplacian} with indefinite weights},
     journal = {Electronic journal of differential equations},
     year = {2001},
     volume = {2001},
     zbl = {0964.35110},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a148/}
}
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JO  - Electronic journal of differential equations
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%A Cuesta,  Mabel
%T Eigenvalue problems for the \(p\)-Laplacian with indefinite weights
%J Electronic journal of differential equations
%D 2001
%V 2001
%U http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a148/
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%F EJDE_2001__2001__a148
Cuesta,  Mabel. Eigenvalue problems for the \(p\)-Laplacian with indefinite weights. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a148/