The existence of radial solutions for a class of k-Hessian systems with the nonlinear gradient terms

• Published in: Journal of applied mathematics & computing Vol. 70; no. 3; pp. 2225 - 2240
• Main Authors: Yu, Zelong, Bai, Zhanbing
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2024
• Subjects: Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theory of Computation; Fixed point index; Jensen integral inequality; Monotone matrix; 34A34; Hessian system; 35J57; 34B18
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-024-02049-9

This paper mainly deals with the following k -Hessian system with the nonlinear gradients S k ( λ ( D 2 u i ) ) + | ∇ u i | k = φ i ( | v | , - u 1 , - u 2 , … , - u n ) , i n Ω , u i = 0 , i = 1 , 2 , … , n , o n ∂ Ω , where 1 ≤ k ≤ N , n ≥ 2 , Ω is the open unit ball in R N ( N ≥ 2 ) and S k ( λ ( D 2 u ) ) is the k -Hessian operator of u . Some results about the existence of radial solutions are obtained by making some appropriate assumptions about R + n -monotone matrices for φ i . Based on the Jensen integral inequality and fixed point theorem, we have overcome the computational difficulties of k -Hessian system with gradients, and obtained the conclusion that the system has at least one radial solution and at least two radial solutions.