Nodal Solutions of a Perturbed Elliptic Problem

Document Type

Article

Publication Date

2008

Publication Title

Topological Methods in Nonlinear Analysis

DOI

10.12775/TMNA.2008.035

ISSN

1230-3429

Abstract

Multiple nodal solutions are obtained for the elliptic problem

−Δu=f(x,u) + εg (x,u) in Ω,

u= 0 on ∂Ω,

where ε is a parameter, Ω is a smooth bounded domain in RN, f∈C(Ω¯×R), and g∈C(Ω¯×R). For a superlinear C1 function f which is odd in u and for any C1 function g, we prove that for any j∈N there exists εj>0 such that if |ε|≤εj then the above problem possesses at least j distinct nodal solutions. Except C1 continuity no further condition is needed for g. We also prove a similar result for a continuous sublinear function f and for any continuous function g. Results obtained here refine earlier results of S. J. Li and Z. L. Liu in which the nodal property of the solutions was not considered.

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