Spectrum of the A_p-Laplacian Operator - doi: 10.5269/bspm.v23i1-2.7466

Resumen

This work deals with the nonlinear boundary eigenvalue problem:

\begin{equation*}(V.P_{A,\rho,I})

\left\{
\begin{aligned}

& A_p u = \lambda \rho(x) |u|^{p-2}u in I =]a; b[;\\
& u(a) = u(b) = 0;

\end{aligned}

\right.

\end{equation*}
where A_p is called the A_p-Laplacian operator and defined by A_pu =(\Gamma(x) |u'|^{p-2}u')', p > 1, \gamma is a real parameter, \rho is an indefinite weight, a, b are real numbers and \Gamma\in C^1(I) \cap C^0 (\overline{I}) and it is nonnegative on \overline{I}.
We prove in this paper that the spectrum of the A_p-Laplacian operator is given by a sequence of eigenvalues. Moreover, each eigenvalue is simple, isolated and verifies the strict monotonicity property with respect to the weight \rho and the domain I. The k¡th eigenfunction corresponding to the k-th eigenvalue has exactly k-1 zeros in (a; b). Finally, we give a simple variational formulation of eigenvalues.