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Entire positive p-k-convex radial solutions to p-k-Hessian equations and systems

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Abstract

In this paper, we analyze the existence of entire positive p-k-convex radial solution to a p-k-Hessian equation and system. Our methods are based on a new monotone iteration scheme.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Keller, J.: On solutions of \(\Delta u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)

    Article  MATH  Google Scholar 

  4. Osserman, R.: On the inequality \(\Delta u \ge f(u)\). Pac. J. Math. 7, 1641–1647 (1957)

    Article  MATH  Google Scholar 

  5. Naito, Y., Usami, H.: Entire solutions of the inequality \(div (A(|Du|) Du) \ge f (u)\). Math. Z. 225, 167–175 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138, 175–188 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lair, A., Mohammed, A.: Entire large solutions of semilinear elliptic equations of mixed type. Commun. Pure Appl. Anal. 8, 1607–1618 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cowan, C., Fazly, M.: On stable entire solutions of semi-linear elliptic equations with weights. Proc. Am. Math. Soc. 140, 2003–2012 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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  MATH  Google Scholar 

  10. Yang, Y., Zhang, X.: Necessary and sufficient conditions of entire subsolutions to Monge–Ampère type equations. Ann. Funct. Anal. 14, 4 (2023)

    Article  MATH  Google Scholar 

  11. Chrouda, M., Hassine, K.: Existence and asymptotic behaviour of entire large solutions for \(k\)-Hessian equations. J. Elliptic Parabol. Equ. 8, 469–481 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, Z., Zhou, S.: Existence of entire positive \(k\)-convex radial solutions to Hessian equations and systems with weights. Appl. Math. Lett. 50, 48–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Covei, D.: A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights. Acta Math. Sci. 37, 47–57 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

  17. Feng, M.: A class of singular coupled systems of superlinear Monge–Ampère equations. Acta Math. Appl. Sin. 38B, 1–18 (2022)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is sponsored by Beijing Natural Science Foundation under Grant No. 1212003.

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Correspondence to Xuemei Zhang.

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Kan, S., Zhang, X. Entire positive p-k-convex radial solutions to p-k-Hessian equations and systems. Lett Math Phys 113, 16 (2023). https://doi.org/10.1007/s11005-023-01642-6

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  • DOI: https://doi.org/10.1007/s11005-023-01642-6

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