Abstract view
Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition


Published:20160602
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
Abstract
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
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 wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
MSC Classifications: 
35J60, 35J92, 46E30, 49R05 show english descriptions
Nonlinear elliptic equations Quasilinear elliptic equations with $p$Laplacian Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)
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)
