Spectrum of one dimensional p-Laplacian operator with indefinite weight

M. Moussa, University Ibn Tofail, Kenitra, Morocco
A. Anane, University Mohamed Ist, Oujda, Morocco
O. Chakrone, University Mohamed Ist, Oujda, Morocco

E. J. Qualitative Theory of Diff. Equ., No. 17. (2002), pp. 1-11.

Abstract: This paper is concerned with the nonlinear boundary eigenvalue problem
$$-(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u\qquad u \in I=]a,b[,\quad u(a)=u(b)=0,$$
where $p>1$, $\lambda$ is a real parameter, $m$ is an indefinite weight, and $a$, $b$ are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight $m$ and the domain $I$, the k-th eigenfunction, corresponding to the $k$-th eigenvalue, has exactly $k-1$ zeros in $(a,b)$. At the end, we give a simple variational formulation of eigenvalues.

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