Infinitely many sign-changing solutions for Kirchhoff type problems in R3

By Jijiang Sun, Lin Li, Matija Cencelj, Boštjan Gabrovšek. Nonlinear Analysis, 2019

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Abstract In this paper, we consider the following nonlinear Kirchhoff type problem: − a + b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) , in R 3 , u ∈ H 1 ( R 3 ) , where a , b > 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik–Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity | u | p − 2 u with p ∈ ( 2 , 4 ] .

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/67d69b7d2724138e73cef025763b9f171db197f8