Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3

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Abstract

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities:. (0.1){-(a+b∫R3|Du|2)δu+V(x)u=|u|p-1u,x∈R3,u∈H1(R3),u>0,x∈R3, where a, b>. 0 are constants, 2. <. p<. 5 and V:R3→R. Under certain assumptions on V, we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.Our main results especially solve problem (0.1) in the case where p∈(2, 3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem{-(ε2a+εb∫R3|Du|2)δu+V(x)u=f(u),x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0 is a parameter, V(x) is a positive continuous potential and f(u)~|u|p-1u with 3<p<5 and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,33], which deal with Schrödinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem. © 2014 Elsevier Inc.

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Li, G., & Ye, H. (2014). Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. Journal of Differential Equations, 257(2), 566–600. https://doi.org/10.1016/j.jde.2014.04.011

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