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# Eigenvalues of $-\Delta_p -\Delta_q$ Under Neumann Boundary Condition

Published:2016-06-02
Printed: Sep 2016
• Mihai Mihăilescu,
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
• Gheorghe Moroşanu,
Department of Mathematics and its Applications, Central European University, 1051 Budapest, Hungary
 Format: LaTeX MathJax PDF

## Abstract

The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable.
 Keywords: eigenvalue problem, Sobolev space, Nehari manifold, variational methods
 MSC Classifications: 35J60 - Nonlinear elliptic equations 35J92 - Quasilinear elliptic equations with $p$-Laplacian 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 49R05 - Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)

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