Existence of two positive solutions for indefinite Kirchhoff equations in R^3

Date

2016-01-25

Authors

Ding, Ling
Meng, Yi-Jie
Xiao, Shi-Wu
Zhang, Jin-Ling

Journal Title

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Volume Title

Publisher

Texas State University, Department of Mathematics

Abstract

In this article we study the Kirchhoff type equation -(1 + b ∫ℝ3 |∇u|2dx) ∆u + u = k(x) ƒ(u) + λh(x)u, x ∈ ℝ3, u ∈ H1(ℝ3), involving a linear part -∆u + u - λh(x)u which is coercive if 0 < λ < λ1(h) and is noncoercive if λ > λ1(h), a nonlocal nonlinear term -b ∫ℝ3 |∇u|2dx∆u and a sign-changing nonlinearity of the form k(x) ƒ(s), where b > 0, λ > 0 is a real parameter and λ1(h) is the first eigenvalue of -∆u + u = λh(x)u. Under suitable assumptions on ƒ and h, we obtain positives solution for λ ∈ (0, λ1(h)) and two positive solutions with a condition on k.

Description

Keywords

Indefinite Kirchhoff equation, Concentration compactness lemma, (PS) condition, Ekeland's variational principle

Citation

Ding, L., Meng, Y. J., Xiao, S. W., & Zhang, J. L. (2016). Existence of two positive solutions for indefinite Kirchhoff equations in R^3. <i>Electronic Journal of Differential Equations, 2016</i>(35), pp. 1-22.

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Attribution 4.0 International

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