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The eigenvalue problem for the \(p\)-Laplacian-like equations. (English) Zbl 1023.35036

Summary: We consider the eigenvalue problem for the following \(p\)-Laplacian-like equation: \[ -\text{div} (a(|Du|^p)|Du|^{p-2}Du)=\lambda f(x,u) \quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial \Omega, \] where \(\Omega \subset \mathbb{R}^n\) is a bounded smooth domain. When \(\lambda\) is small enough, a multiplicity result for eigenfunctions are obtained. Two examples from nonlinear quantized mechanics and capillary phenomena, respectively, are given for applications of the theorems.

MSC:

35J60 Nonlinear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

multiplicity
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