Ding, LingMeng, Yi-JieXiao, Shi-WuZhang, Jin-Ling2023-06-062023-06-062016-01-25Ding, L., Meng, Y. J., Xiao, S. W., & Zhang, J. L. (2016). Existence of two positive solutions for indefinite Kirchhoff equations in R^3. Electronic Journal of Differential Equations, 2016(35), pp. 1-22.1072-6691https://hdl.handle.net/10877/16909In 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.Text22 pages1 file (.pdf)enAttribution 4.0 InternationalIndefinite Kirchhoff equationConcentration compactness lemma(PS) conditionEkeland's variational principleArticle