Existence of a nodal solution with minimal energy for a Kirchhoff equation

Abstract

We show the existence of a nodal solution (sign‐changing solution) for a Kirchhoff equation of the type −M0true∫Ω|∇u|2dxΔu=f(u)inΩ,u=0on∂Ω,where Ω is a bounded domain in R3, M is a general C1 class function and f is a superlinear C1 class function with subcritical growth. The proof is based on a minimization argument and a quantitative deformation lemma.

Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains

“…(c) Here we also get a general existence result of least energy nodal solutions for problem (P λ ) under more restricted assumptions on M and f , which covers and generalizes the result in [8], see Remark 5.2 in Section 5. Besides, the asymptotic behavior of the least energy nodal solutions of problem (P λ ) when λ converges to infinity and the energy doubling property are studied, which are not observed in [8], see Theorem 1.2 below.…”

“…Indeed, the truncation in the present paper is more refined and technical, see (2.2) and (2.3) in Section 2. In addition, we succeeded in proving that the nodal Nehari manifold M λ,θ is nonempty by using an elementary method and more analysis instead of Miranda's theorem which is adopted in [8], see Lemma 3.3 and its proof in Section 3. (b) Under certain assumptions on M and f , we get a constant sign solution u λ and a nodal solution w λ with the energy estimates Φ λ (w λ ) > 2Φ λ (u λ ) > 0 for λ large.…”

“…In [8], under certain assumptions on M and f , the authors show the existence of least energy nodal solution for (P λ ) on bounded smooth domains Ω ⊂ R 3 with λ = 1. This type problems include but are not restricted to the type (1.1), and much closer to problem (P λ ) we consider here.…”

“…Based on the variational method, an appropriated truncation argument and a priori estimates, the author shows the existence result of positive solution for the above problem and study the asymptotic behavior of that solution when λ converges to infinity. Motivated by the articles [6,8,21], we shall investigate the existence of constant sign and sign-changing solutions for problem (P λ ). The main differences are the following:…”

“…In the last ten years, by classical variational method, there are many interesting results about the existence and nonexistence of solutions, sign-changing solutions, ground state solutions, the existence of positive solutions and positive ground states, least energy nodal solutions, multiplicity of solutions, semiclassical limit and concentrations of solution to Kirchhoff type problems, see e.g. [1,4,[6][7][8][9][10][11]13,14,16,[19][20][21] and the references therein.…”