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

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 -Δp⁢u-Δq⁢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.

Keywords

• eigenvalue problem
• first eigenvalue
• second eigenvalue
• nodal solutions
• sign-changing solutions
• nehari manifold
• linking theorem
• descending flow
• 35j62
• 35j20
• 35p30