Mathematics Without Boundaries by Themistocles M. Rassias & Panos M. Pardalos

Author:Themistocles M. Rassias & Panos M. Pardalos
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

4 Asymptotically Homogeneous Case

The eigenvalue problem with Dirichlet boundary condition for the nonlinear operator means the existence of for which

has a nontrivial solution. The eigenvalue problem is hard because the operator , so , is not homogeneous. Even in the case of the p-Laplacian Δ p , which is homogeneous, the spectrum is not completely known (it is known what is called the beginning of the spectrum of as can be seen in [7], see also [25, 24)].

In order to overcome the lack of homogeneity in it was introduced in [23] an asymptotic -homogeneity at zero condition for the nonhomogeneous operator :

[(AH0)] There exist a positive function and a continuous function on such that

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