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On the generalization of the Courant nodal domain theorem. (English) Zbl 1163.35449

This paper concerns with the nodal properties of the eigenfunctions (Fučik eigenfunctions) of -Δ p , where p>1, Δ p u:=·(|u| p-2 u) is the p-Laplacian. More precisely, the authors study the properties of the nonlinear eigenvalue problem

-Δ p u=λ|u| p-2 uinΩ,u=0onΩ,

and its more general version

-Δ p u=α|u| p-2 u + -β|u| p-2 u - inΩ,u=0onΩ,

where Ω is a bounded domain in N with smooth boundary Ω, and α,β,λ are real spectral parameters. The authors prove that, if u λ n is an eigenfunction associated with the nth variational eigenvalue, λ n , then u λ n has at most 2n-2 nodal domains. Moreover, if u λ n has n+k nodal domains then there is another eigenfunction with at most n-k nodal domains.

MSC:
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
47J10Nonlinear spectral theory, nonlinear eigenvalue problems
58E05Abstract critical point theory