Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

By Fubao Zhang, Miao Du. Journal of Differential Equations, 2020

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Abstract In this paper, we consider the following Kirchhoff type problem { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + λ V ( x ) u = | u | p − 2 u in R 3 , u ∈ H 1 ( R 3 ) , where a > 0 is a constant, b and λ are positive parameters, and 2 p 6 . Suppose that the nonnegative continuous potential V represents a potential well with the bottom V − 1 ( 0 ) , the equation has been extensively studied in the case 4 ≤ p 6 . In contrast, no existence result of solutions is available for the case 2 p 4 due to the presence of the term ( ∫ R 3 | ∇ u | 2 d x ) Δ u . By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2 p 4 . Moreover, we also explore the decay rate of the positive solutions as | x | → ∞ as well as their asymptotic behavior as b → 0 and λ → ∞ .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

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