Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type

Abstract

In this paper, we study the following fourth-order elliptic equation with Kirchhoff-typewhere the constants a > 0, b ≥ 0. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution u b . Moreover, we also prove that the energy of u b is strictly larger than two times the ground state energy. Finally, we give a convergence property of u b when b as a parameter and b → 0.2000 Mathematics Subject Classification. 35J50, 35J60.

Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials

“…There have been many works about the existence of nontrivial solutions to (1.1) by using variational methods, see for example, [1,5,6,7,8,9,11,12,13,18,20,21,23,24,25,27,28,29,32,34,35] and the references therein. A typical way to deal with (1.1) is to use the mountain-pass theorem.…”

Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity

“…(1.3), see, for example, [4,29,36,44,46,[50][51][52]. However, except [21,57], there are very few papers considering sign-changing solutions. By combining constraint variation methods and deformation lemma, Zhang et al [57] studied sign-changing solution to problem (1.1) when K(x) ≡ 1.…”

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

“…Moreover, for any u , φ ∈ H , we have Clearly, the critical points of are weak solutions of . Furthermore, if u ∈ H is a solution of and u ± ≠0, then u is a sign‐changing solution of , where As far as we know, a variety of ways are used to obtain the sign‐changing solutions, such as by constructing invariant sets and descending flow , adopting the Ekeland's variational principle and the implicit function theorem , applying variational method together with the Brouwer degree theory , and using diagonal principle with non‐Nehari manifold method . Next, we give the following decomposition that plays an important role in seeking sign‐changing solutions for , for any u ∈ H , …”

“…As far as we know, a variety of ways are used to obtain the sign-changing solutions, such as by constructing invariant sets and descending flow [13,14], adopting the Ekeland's variational principle and the implicit function theorem [15], applying variational method together with the Brouwer degree theory [16], and using diagonal principle with non-Nehari manifold method [8,[17][18][19][20][21]. Next, we give the following decomposition that plays an important role in seeking sign-changing solutions for (1.3), for any u 2 H,…”