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

Advances in Nonlinear Analysis

article

Vladimir Bobkov¹  Mieko Tanaka³ 

[1] Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences;Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia;Department of Mathematics, Tokyo University of Science

关键词: eigenvalue problem;    first eigenvalue;    second eigenvalue;    nodal solutions;    sign-changing solutions;    Nehari manifold;    linking theorem;    descending flow;   

DOI  :  10.1515/anona-2016-0172

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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.

【 授权许可】

CC BY