Existence theorems for a class of systems involving two quasilinear operators

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

In this paper, we study the existence of positive radialsolutions for a class of quasilinear systems of the form$$\begin{cases}\Delta_{\phi_1}u=a_1(|x|)f_1(v),\Delta_{\phi_2}v=a_2(|x|)f_2(u),\end{cases}\quad x\in \mathbb{R}^N, \quad N\ge 3,$$where $\Delta_{\phi}w:=\operatorname{div}(\phi(|\nabla w|)\nabla w)$,under appropriate conditions on the functions $\phi_1$, $\phi_2$,the weights $a_1$, $a_2$ and the non-linearities $f_1$, $f_2$.The conditions imposed for the existence of such solutions are different from those in previous results.

About the authors

Dragos-Patru Covei

The Bucharest Uviversity of Economic Studies

Doctor of physico-mathematical sciences, Associate professor

References

  1. D.-P. Covei, “On the radial solutions of a system with weights under the Keller–Osserman condition”, J. Math. Anal. Appl., 447:1 (2017), 167–180
  2. D. P. Covei, “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. (Engl. Ser.), 33:6 (2017), 761–774
  3. B. Franchi, E. Lanconelli, J. Serrin, “Existence and uniqueness of nonnegative solutions of quasilinear equations in $R^N$”, Adv. Math., 118:2 (1996), 177–243
  4. G. Diaz, “A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition”, Math. Ann., 353:1 (2012), 145–159
  5. H. Grosse, A. Martin, Particle physics and the Schrödinger equation, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol., 6, Cambridge Univ. Press, Cambridge, 1997, xii+167 pp.
  6. A. Hamydy, M. Massar, N. Tsouli, “Existence of blow-up solutions for a non-linear equation with gradient term in $mathbb R^N$”, J. Math. Anal. Appl., 377:1 (2011), 161–169
  7. J. Jaroš, K. Takaŝi, “On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations”, Proc. Roy. Soc. Edinburgh Sect. A, 145:5 (2015), 1007–1028
  8. J. B. Keller, “On solution of $Delta u=f(u)$”, Comm. Pure Appl. Math., 10:4 (1957), 503–510
  9. A. A. Kon'kov, “On properties of solutions of quasilinear second-order elliptic inequalities”, Nonlinear Anal., 123/124 (2015), 89–114
  10. A. V. Lair, “A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems”, J. Math. Anal. Appl., 365:1 (2010), 103–108
  11. A. V. Lair, “Entire large solutions to semilinear elliptic systems”, J. Math. Anal. Appl., 382:1 (2011), 324–333
  12. A. V. Lair, A. Mohammed, “Entire large solutions of semilinear elliptic equations of mixed type”, Commun. Pure Appl. Anal., 8:5 (2009), 1607–1618
  13. Hong Li, Pei Zhang, Zhijun Zhang, “A remark on the existence of entire positive solutions for a class of semilinear elliptic systems”, J. Math. Anal. Appl., 365:1 (2010), 338–341
  14. Z. A. Luthey, Piecewise analytical solutions method for the radial Schroedinger equation, Ph.D. thesis, Harvard Univ., Cambridge, 1975 (no paging)
  15. А. Г. Лосев, Е. А. Мазепа, “Об асимптотическом поведении положительных решений некоторых квазилинейных неравенств на модельных римановых многообразиях”, Уфимск. матем. журн., 5:1 (2013), 83–89
  16. G. M. Lieberman, “Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations”, J. Anal. Math., 115:1 (2011), 213–249
  17. Е. А. Мазепа, “Положительные решения квазилинейных эллиптических неравенств на модельных римановых многообразиях”, Изв. вузов. Матем., 2015, № 9, 22–30
  18. A. Mohammed, “Boundary behavior of blow-up solutions to some weighted non-linear differential equations”, Electron. J. Differential Equations, 2002 (2002), 78, 15 pp.
  19. Y. Naito, H. Usami, “Entire solutions of the inequality $operatorname{div}(A(|Du|)Du)ge f(u)$”, Math. Z., 225:1 (1997), 167–175
  20. Y. Naito, H. Usami, “Nonexistence results of positive entire solutions for quasilinear elliptic inequalities”, Canad. Math. Bull., 40:2 (1997), 244–253
  21. R. Osserman, “On the inequality $Delta uge f(u)$”, Pacific J. Math., 7:4 (1957), 1641–1647
  22. C. L. Pripoae, “Non-holonomic economical systems”, Conference “Applied differential geometry: general relativity”–Workshop “Global analysis, differential geometry, Lie algebras”, BSG Proc., 10, Geom. Balkan Press, Bucharest, 2004, 142–149
  23. M. D. Smooke, “Error estimates for piecewise perturbation series solutions of the radial Schrödinger equation”, SIAM J. Numer. Anal., 20:2 (1983), 279–295
  24. Haitao Yang, “On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^N$”, Commun. Pure Appl. Anal., 4:1 (2005), 187–198
  25. Zhijun Zhang, Song Zhou, “Existence of entire positive $k$-convex radial solutions to Hessian equations and systems with weights”, Appl. Math. Lett., 50 (2015), 48–55
  26. Xinguang Zhang, “A necessary and sufficient condition for the existence of large solutions to ‘mixed’ type elliptic systems”, Appl. Math. Lett., 25:12 (2012), 2359–2364
  27. Zhijun Zhang, “Existence of positive radial solutions for quasilinear elliptic equations and systems”, Electron. J. Differential Equations, 2016 (2016), 50, 9 pp.
  28. Song Zhou, “Existence of entire radial solutions to a class of quasilinear elliptic equations and systems”, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 38, 10 pp.
  29. N. Fukagai, K. Narukawa, “On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems”, Ann. Mat. Pura Appl. (4), 186:3 (2007), 539–564
  30. М. А. Красносельский, Я. Б. Рутицкий, Выпуклые функции и пространства Орлича, Физматгиз, М., 1958, 271 с.
  31. J. Soria, Tent spaces based on weighted Lorentz spaces. Carleson measures, Ph.D. thesis, Graduate School of Arts and Sciences, Washington Univ., St. Louis, 1990, 121 pp.

Copyright (c) 2019 Covei D.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies