Positive, negative and least energy nodal solutions for Kirchhoff equations in ℝN. Issue 10 (3rd October 2021)

• Record Type: Journal Article
• Authors: Xu, Liping; Chen, Haibo
• Is Part Of: Complex variables and elliptic equations. Volume 66:Issue 10(2021)
• Page Start: 1676
• Page End: 1698
• Publication Date: 2021-10-03
• Subjects: Kirchhoff equations -- variational approach -- positive -- negative and least energy sign-changing solutions; 35J20 -- 35J65 -- 35J60; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2020.1779234
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : The paper deals with the following Kirchhoff equations: 1 − a + b ∫ R N | ∇ u | 2 d x △ u + V ( x ) u = K ( x ) f ( u ) i n R N, u ∈ H 1 ( R N ), where N ≥ 3, f ( u ) is C 1 real function satisfying quasicritical growth at infinity, and V ( x ), K ( x ) are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence of at least a positive and a negative solution. Moreover, using a quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u 0 with two nodal domains. Finally, we show that the energy of u 0 is strictly larger than the ground state energy.