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Existence and asymptotic behavior of nontrivial p-k-convex radial solutions for p-k-Hessian equations

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Abstract

We study, via the eigenvalue theory of completely continuous operators, the existence and asymptotic behavior of nontrivial p-k-convex radial solutions for a p-k-Hessian equation. This is probably the first time that p-k-Hessian equations have been studied by employing this technique. Several new nonexistence conclusions are also derived in this paper.

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References

  1. Trudinger, N., Wang, X.: Hessian measures. II. Ann. Math. 150, 579–604 (1999)

    Article  MathSciNet  Google Scholar 

  2. Pogorelov, A.: The Minkowski Multi-dimensional Problem. Wiley, New York (1978)

    Google Scholar 

  3. Figalli, A.: The Monge-Ampère equation and its applications. Zurich Lectures in Advanced Mathematics. European Mathematical Society (2017)

  4. Wang, G., Yang, Z., Zhang, L., Baleanu, D.: Radial solutions of a nonlinear \(k\)-Hessian system involving a nonlinear operator. Commun. Nonlinear Sci. Numer. Simul. 91, 105396 (2020)

    Article  MathSciNet  Google Scholar 

  5. Mohammed, A., Porru, G.: On Monge-Ampère equations with nonlinear gradient terms-infinite boundary value problems. J. Differ. Equ. 300, 426–457 (2021)

    Article  ADS  Google Scholar 

  6. Jiang, F., Trudinger, N.S., Yang, X.: On the Dirichlet problem for Monge–Ampère type equations. Calc. Var. 49, 1223–1236 (2014)

    Article  MathSciNet  Google Scholar 

  7. Zhang, X., Liu, L., Wu, Y., Cui, Y.: A sufficient and necessary condition of existence of blow-up radial solutions for a \(k\)-Hessian equation with a nonlinear operator. Nonlinear Anal. Model. Control 25, 126–143 (2020)

    MathSciNet  Google Scholar 

  8. Zhang, X., Xu, P., Wu, Y.: The eigenvalue problem of a singular \(k\)-Hessian equation. Appl. Math. Lett. 124, 107666 (2022)

    Article  MathSciNet  Google Scholar 

  9. Dai, L., Li, H.: Entire subsolutions of Monge–Ampère type equations. Commun. Pure Appl. Anal. 19, 19–30 (2020)

    Article  MathSciNet  Google Scholar 

  10. Dai, L.: Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete Contin. Dyn. Syst. Ser. A. 40, 579–596 (2020)

    Article  MathSciNet  Google Scholar 

  11. Covei, D.P.: A remark on the existence of entire large and bounded solutions to a \((k_1, k_2)\)-Hessian system with gradient term. Acta Math. Sin 33B, 761–774 (2017)

    Article  Google Scholar 

  12. Zhang, Z.: Large solutions to the Monge–Ampère equations with nonlinear gradient terms: existence and boundary behavior. J. Differ. Equ. 264, 263–296 (2018)

    Article  ADS  Google Scholar 

  13. Zhang, X., Feng, M.: Blow-up solutions to the Monge–Ampère equation with a gradient term: sharp conditions for the existence and asymptotic estimates. Calc. Var. 61, 208 (2022)

    Article  Google Scholar 

  14. Cui, J.: Existence and nonexistence of entire \(k\)-convex radial solutions to Hessian type system. Adv. Differ. Equ. 2021, 462 (2021)

    Article  MathSciNet  Google Scholar 

  15. Zhang, X., Feng, M.: The existence and asymptotic behavior of boundary blow-up solutions to the \(k\)-Hessian equation. J. Differ. Equ. 267, 4626–4672 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  16. Zhang, X., Feng, M.: Boundary blow-up solutions to singular \(k\)-Hessian equations with gradient terms: sufficient and necessary conditions and asymptotic behavior. J. Differ. Equ. 375, 475–513 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  17. Wan, H., Shi, Y., Qiao, X.: Entire large solutions to the \(k\)-Hessian equations with weights: existence, uniqueness and asymptotic behavior. J. Math. Anal. Appl. 503, 125301 (2021)

    Article  MathSciNet  Google Scholar 

  18. Gao, C., He, X., Ran, M.: On a power-type coupled system of \(k\)-Hessian equations. Quaest. Math. 44, 1593–1612 (2021)

    Article  MathSciNet  Google Scholar 

  19. Feng, M.: A class of singular \(k_i\)-Hessian systems. Topol. Methods Nonlinear Anal. 62, 341–365 (2023)

    MathSciNet  Google Scholar 

  20. Bao, J., Feng, Q.: Necessary and sufficient conditions on global solvability for the \(p\)-\(k\)-Hessian inequalities. Canad. Math. Bull. 65, 1004–1019 (2022)

    Article  MathSciNet  Google Scholar 

  21. Fan, W., Dai, L., Wang, B.: Positive radially symmetric entire solutions of \(p\)-\(k\)-Hessian equations and systems. Mathematics 10, 3258 (2022)

    Article  Google Scholar 

  22. Feng, M., Zhang, X.: The existence of infinitely many boundary blow-up solutions to the \(p\)-\(k\)-Hessian equation. Adv. Nonlinear Stud. 23, 20220074 (2023)

    Article  MathSciNet  Google Scholar 

  23. Kan, S., Zhang, X.: Entire positive\(p\)-\(k\)-convex radial solutions to \(p\)-\(k\)-Hessian equations and systems. Lett. Math. Phys. 113, 16 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  24. Zhang, X., Yang, Y.: Necessary and sufficient conditions for the existence of entire subsolutions to \(p\)-\(k\)-Hessian equations. Nonlinear Anal. 233, 113299 (2023)

    Article  MathSciNet  Google Scholar 

  25. Feng, M.: Eigenvalue problems for singular \(p\)-Monge-Ampère equations. J. Math. Anal. Appl. 528, 127538 (2023)

    Article  Google Scholar 

  26. Feng, M.: Nontrivial \(p\)-convex solutions to singular \(p\)-Monge–Ampere problems: existence multiplicity and nonexistence. Commun. Anal. Mech. 16, 71–93 (2024)

    Article  MathSciNet  Google Scholar 

  27. Gao, Z.: Uniqueness theorem for negative solutions of fully nonlinear elliptic equations in a ball. Nonlinear Anal. 242, 113495 (2024)

    Article  MathSciNet  Google Scholar 

  28. Guo, D.: Eigenvalue and eigenvectors of nonlinear operators. Chin. Ann. Math. 2B, 65–80 (1981)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12371112) and the Beijing Natural Science Foundation (1212003). The authors want to express their gratitude to the reviewers for careful reading and valuable suggestions, which improve the value of the article.

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  1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Meiqiang Feng & Yichen Lu

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  1. Meiqiang Feng
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  2. Yichen Lu
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Correspondence to Meiqiang Feng.

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Feng, M., Lu, Y. Existence and asymptotic behavior of nontrivial p-k-convex radial solutions for p-k-Hessian equations. Lett Math Phys 114, 107 (2024). https://doi.org/10.1007/s11005-024-01858-0

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