On a class of Kirchhoff type problems

Ground state solutions and least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/comm

“…The proof of this lemma is almost the same as that of Lemma 2.5 in [12] (see also 30 in [9,5,11]). So we omit it here.…”

“…which implies that, {u n } is a bounded sequence of E. Therefore, by Lemma 1.1, there exists 9 u ∈ E such that u n ⇀ u and u ± n ⇀ u± in E as n → +∞, u ± n ⇀ u± in  E as n → +∞ 10 and u ± n → u ± in L s (R N ) as n → +∞ for 2 ≤ s < 2 * . Moreover, by Lemma 3.2 there 11 exists C > 0 such that ∥u ± n ∥ ≥ C and ∥u ± n ∥ p p ≥ C, which implies that u ± ̸ = 0.…”

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$ R N

“…Proof The proof of this lemma is almost the same as that of Lemma 2.5 in [5] (see also in [4,15,16]) . So we omit it here.…”

On a Kirchhoff type problem inRN

“…Our result on energy estimates for the sign-changing solutions to (P α,β ) with N = 1, 2, 3 and p ∈ (4, 2 * ) is much more precise than the corresponding result in [9]. Moreover, Theorem 1.3 seems to be the first result on the estimates of energy values for the sign-changing solutions to (P α,β ) with p ∈ (2, 4) ∩ (2, 2 * ).…”

“…The unique positive solution is radial symmetric and is also the unique least energy solution to (P 0,1 ), while the energy values of the sign-changing solutions go to infinity. On the other hand, to the best of our knowledge, for the Kirchhoff type problem (P α,β ), only the existence result of one positive solution with N = 1, 2, 3 and p ∈ (4, 2 * ) was established in [3] by Alves and Figueiredo. Simultaneously, in recent years, the Kirchhoff type problems in the whole space R N with N = 1, 2, 3 have been studied widely by the variational methods since then the nice work [8], and various existence results of the solutions to such problems were established, see for example [1,3,6,7,[9][10][11][12]14] and the references therein. Inspired by the above facts, the purpose of the current paper is to make a detailed description on the solutions of (P α,β ).…”