Work

On sign-changing solutions for ( p , q )-Laplace equations with two parameters

Year: 2016

Type: article

Abstract: We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous ( p , q ) {(p,q)} -Laplace equations - Δ p ⁢ u - Δ q ⁢ u = α ⁢ | u | p - 2 ⁢ u + β ⁢ | u | q - 2 ⁢ u {-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p ≠ q {p\neq q} . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the ( α , β ) {(\alpha,\beta)} -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p - and q -Laplacians in one dimension.

Source: Advances in Nonlinear Analysis

DOI: 10.1515/anona-2016-0172

Authors: Vladimir Bobkov, Mieko Tanaka

Institutions: Russian Academy of Sciences, Institute of Mathematics with Computer Center, University of West Bohemia in Pilsen, Tokyo University of Science

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $1,469

Funders: Ministerstvo Školství, Mládeže a Tělovýchovy, Japan Society for the Promotion of Science