Positive and sign-changing solutions for Kirchhoff equations with indefinite potential

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Abstract

We deal with the nonlinear Kirchhoff problem −(a + b ∫_R³ |∇u|²dx) Δu + V(x)u = f(u), x ∈ R³, (P) where a is a positive constant, b > 0 is a parameter, the potential function V is allowed to change its sign, and the nonlinearity f ∈ C(R, R) exhibits subcritical growth. Under some suitable conditions on V, we first prove that the problem has a positive ground state solution for all b > 0. Then, by using a more general global compactness lemma and a sign-changing Nehari manifold, combined with the method of constructing a sign-changing (PS)_c sequence, we show the existence of a least energy sign-changing solution for b > 0 that is sufficiently small. Moreover, the asymptotic behavior b ↘ 0 is established.

Keywords

Kirchhoff problem / positive solutions / sign-changing solutions / variational methods / indefinite potential

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