Existence of sign-changing solutions for 𝑝(𝑥)-Laplacian Kirchhoff type problem in ℝ^𝑁

Abstract

The 𝑝(𝑥)-Laplacian Kirchhoff type equation involving the nonlocal term 𝑏 ∫_ℝ^𝑁 (1/𝑝(𝑥)) |∇𝑢|^𝑝(𝑥)𝑑𝑥 is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution 𝑢_𝑏 which has precisely two nodal domains. Moreover, the convergence property of 𝑢_𝑏 as the parameter 𝑏 ↘ 0 is also obtained.