Advertisement

Multiple solutions to weakly coupled supercritical elliptic systems

  • Omar Cabrera
  • Mónica ClappEmail author
Article
  • 25 Downloads

Abstract

We study a weakly coupled supercritical elliptic system of the form
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |x_2|^\gamma \left( \mu _{1}|u|^{p-2}u+\lambda \alpha |u|^{\alpha -2}|v|^{\beta }u \right) &{}\quad \text {in }\Omega ,\\ -\Delta v = |x_2|^\gamma \left( \mu _{2}|v|^{p-2}v+\lambda \beta |u|^{\alpha }|v|^{\beta -2}v \right) &{}\quad \text {in }\Omega ,\\ u=v=0 &{}\quad \text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^{N}\), \(N\ge 3\), \(\gamma \ge 0\), \(\mu _{1},\mu _{2}>0\), \(\lambda \in {\mathbb {R}}\), \(\alpha , \beta >1\), \(\alpha +\beta = p\), and \(p\ge 2^{*}:=\frac{2N}{N-2}\). We assume that \(\Omega \) is invariant under the action of a group G of linear isometries, \({\mathbb {R}}^{N}\) is the sum \(F\oplus F^\perp \) of G-invariant linear subspaces, and \(x_2\) is the projection onto \(F^\perp \) of the point \(x\in \Omega \). Then, under some assumptions on \(\Omega \) and F, we establish the existence of infinitely many fully nontrivial G-invariant solutions to this system for \(p\ge 2^*\) up to some value which depends on the symmetries and on \(\gamma \). Our results apply, in particular, to the system with pure power nonlinearity (\(\gamma =0\)) and yield new existence and multiplicity results for the supercritical Hénon-type equation
$$\begin{aligned} -\Delta w = |x_2|^\gamma \,|w|^{p-2}w \quad \text {in }\Omega , \qquad w=0 \quad \text {on }\partial \Omega . \end{aligned}$$

Keywords

Weakly coupled elliptic system Bounded domain Supercritical nonlinearity Hénon-type equation Phase separation 

Mathematics Subject Classification

35J47 35B33 35B40 35J50 

Notes

References

  1. 1.
    Badiale, M., Serra, E.: Multiplicity results for the supercritical Hénon equation. Adv. Nonlinear Stud. 4(4), 453–467 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cao, D., Liu, Z., Peng, S.: Sign-changing bubble tower solutions for the supercritical Hénon-type equations. Ann. Mat. Pura Appl. (4) 197(4), 1227–1246 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Castro, A., Cossio, J., Neuberger, J.M.: A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt. J. Math. 27(4), 1041–1053 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205(2), 515–551 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case. Calc. Var. Partial Differ. Equ. 52(1–2), 423–467 (2015)CrossRefGoogle Scholar
  6. 6.
    Clapp, M., Faya, J.: Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains. Discrete Contin. Dyn. Syst. (to appear). Preprint arXiv:1805.10304
  7. 7.
    Clapp, M., Pacella, F.: Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size. Math. Z. 259(3), 575–589 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Clapp, M., Pacella, F.: Existence and asymptotic profile of nodal solutions to supercritical problems. Adv. Nonlinear Stud. 17(1), 87–97 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Clapp, M., Pistoia, A.: Existence and phase separation of entire solutions to a pure critical competitive elliptic system. Calc. Var. Partial Differ. Equ. 57, 23 (2018).  https://doi.org/10.1007/s00526-017-1283-9 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clapp, M., Rizzi, M.: Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents. Rend. Istit. Mat. Univ. Trieste 49, 53–71 (2017)MathSciNetGoogle Scholar
  11. 11.
    Conti, M., Terracini, S., Verzini, G.: Nehari’s problem and competing species systems. Ann. Inst. Henri Poincaré Anal. Nonlinéaire 19(6), 871–888 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dávila, J., Faya, J., Mahmoudi, F.: New type of solutions to a slightly subcritical Hénon type problem on general domains. J. Differ. Equ. 263(11), 7221–7249 (2017)CrossRefGoogle Scholar
  13. 13.
    dos Santos, E.M., Pacella, F.: Hénon-type equations and concentration on spheres. Indiana Univ. Math. J. 65(1), 273–306 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gladiali, F., Grossi, M., Neves, S.L.N.: Nonradial solutions for the Hénon equation in \(\mathbb{R}^N\). Adv. Math. 249, 1–36 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guo, Y., Li, B., Wei, J.: Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in \({\mathbb{R}}^3\). J. Differ. Equ. 256(10), 3463–3495 (2014)CrossRefGoogle Scholar
  16. 16.
    Hebey, E., Vaugon, M.: Sobolev spaces in the presence of symmetries. J. Math. Pures Appl. (9) 76(10), 859–881 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ivanov, S.V., Nazarov, A.I.: On weighted Sobolev embedding theorems for functions with symmetries. (Russian) Algebra i Analiz 18(1), 108–123 (2006) (translation in St. Petersburg Math. J. 18(1), 77–88 (2007))Google Scholar
  18. 18.
    Liu, J., Liu, X., Wang, Z.-Q.: Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth. J. Differ. Equ. 261(12), 7194–7236 (2016)CrossRefGoogle Scholar
  19. 19.
    Nachbin, L.: The Haar Integral. Translated from the Portuguese by Lulu Bechtolsheim. Robert E. Krieger Publishing Co., Huntington (1976)zbMATHGoogle Scholar
  20. 20.
    Ni, W.M.: A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31(6), 801–807 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Peng, S., Peng, Y., Wang, Z.-Q.: On elliptic systems with Sobolev critical growth. Calc. Var. Partial Differ. Equ. 55, 142 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pistoia, A., Soave, N.: On Coron’s problem for weakly coupled elliptic systems. Proc. Lond. Math. Soc. (3) 116(1), 33–67 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pistoia, A., Tavares, H.: Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions. J. Fixed Point Theory Appl. 19(1), 407–446 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35(3), 681–703 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Soave, N.: On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition. Calc. Var. Partial Differ. Equ. 53(3–4), 689–718 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Szulkin, A.: Ljusternik–Schnirelmann theory on \({\cal{C}}^1\)-manifolds. Ann. Inst. Henri Poincaré Anal. Nonlinéaire 5(2), 119–139 (1988)CrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoCoyoacánMexico

Personalised recommendations