Skip to main content
SpringerLink
Log in

Nodal Solutions for Supercritical Laplace Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we study radial solutions for the following equation

$$\Delta u(x)+f (u(x), |x|) = 0,$$

where \({x \in {\mathbb{R}^{n}}}\), n > 2, f is subcritical for r small and u large and supercritical for r large and u small, with respect to the Sobolev critical exponent \({2^{*} = \frac{2n}{n-2}}\). The solutions are classified and characterized by their asymptotic behaviour and nodal properties. In an appropriate super-linear setting, we give an asymptotic condition sufficient to guarantee the existence of at least one ground state with fast decay with exactly j zeroes for any j ≥ 0. Under the same assumptions, we also find uncountably many ground states with slow decay, singular ground states with fast decay and singular ground states with slow decay, all of them with exactly j zeroes. Our approach, based on Fowler transformation and invariant manifold theory, enables us to deal with a wide family of potentials allowing spatial inhomogeneity and a quite general dependence on u. In particular, for the Matukuma-type potential, we show a kind of structural stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bidaut-Véron M.F.: Local and global behavior of solutions of quasilinear equations of Emden-Fowler type. Arch. Rational Mech. Anal. 107, 293–324 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bamon R., Flores I., Del Pino M.: Ground states of semilinear elliptic equations: a geometric approach. Ann. Inst. Henry Poincaré 17, 551–581 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Battelli F., Johnson R.: Singular ground states of the scalar curvature equation in \({{\mathbb{R}^{n}}}\). Diff. Int. Equ. 14, 123–139 (2000)

    MathSciNet  Google Scholar 

  4. Capietto A., Dambrosio W., Zanolin F.: Infinitely many radial solutions to a boundary value problem in a ball. Ann. Mat. Pura Appl. 179, 159–188 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng K.S., Chern J.L.: Existence of positive entire solutions of some semilinear elliptic equations. J. Diff. Equ. 98, 169–180 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Chern J.L., Chen Z.Y., Tang Y.L.: Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations. Trans. Am. Math. Soc. 363(6), 3211–3231 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chern J.L., Yanagida E.: Structure of the sets of regular and singular radial solutions for a semilinear elliptic equation. J. Differ. Equ. 224, 440–463 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Coffman C.V., Ullrich D.F.: On the continuation of solutions of a certain non-linear differential equation. Monatsh. Math. 71, 385–392 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Damascelli L., Pacella F., Ramaswamy M.: Symmetry of ground states of p-Laplace equations via the Moving Plane Method. Arch. Rat. Mech. Anal. 148, 291–308 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Erbe L., Tang M.: Structure of positive radial solutions of semilinear elliptic equations. J. Differ. Equ. 133, 179–202 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Felmer P., Quaas A., Tang M.: On the complex structure of positive solutions to Matukuma-type equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 869–887 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Fowler R.H.: Further studies of Emden’s and similar differential equations. Q. J. Math. 2, 259–288 (1931)

    Article  ADS  MATH  Google Scholar 

  13. Franca M.: Classification of positive solution of p-Laplace equation with a growth term. Arch. Math. (Brno) 40(4), 415–434 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Franca M.: Fowler transformation and radial solutions for quasilinear elliptic equations. Part 1: the subcritical and supercritical case. Can. Math. Appl. Quart. 16, 123–159 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Franca M.: Structure theorems for positive radial solutions of the generalized scalar curvature equation. Funkcialaj Ekvacioj 52, 343–369 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Franca M.: Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type. Ann. Mat. Pura Appl. 189, 67–94 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Franca M.: Positive solutions for semilinear elliptic equations: two simple models with several bifurcations. J. Dyn. Differ. Equ. 23, 573–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Franca M.: Positive solutions of semilinear elliptic equations: a dynamical approach. Differ. Int. Equ. 26, 505–554 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Franca M., Johnson R.: Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case. Adv. Nonlinear Stud. 4, 93–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. García-Huidobro M., Manasevich R., Yarur C.: On the structure of positive radial solutions to an equation containing p-Laplacian with weights. J. Differ. Equ. 223, 51–95 (2006)

    Article  ADS  MATH  Google Scholar 

  21. Gidas B., Ni W.M., Nirenberg L.: Symmetry of positive solutions of nonlinear elliptic equations in \({{\mathbb{R}^{n}}}\). Adv. Math. Suppl. Stud. 7, 369–403 (1981)

    MathSciNet  MATH  Google Scholar 

  22. Hale, J.: Ordinary Differential Equation. Pure Appl. Math. 21, (1980)

  23. Hirsch M., Pugh C., Shub M.: Invariant Manifolds, Lecture Notes in Math., vol. 583. Springer-Verlag, New York (1977)

    Google Scholar 

  24. Johnson R.: Concerning a theorem of Sell. J. Differ. Equ. 30, 324–339 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Johnson R., Pan X.B., Yi Y.F.: Singular ground states of semilinear elliptic equations via invariant manifold theory. Nonlinear Anal. Th. Meth. Appl. 20, 1279–1302 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Johnson R., Pan X.B., Yi Y.F.: The Melnikov method and elliptic equation with critical exponent. Indiana Math. J. 43, 1045–1077 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jones C., Küpper T.: On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 17(4), 803–835 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kabeya Y., Yanagida E., Yotsutani S.: Existence of nodal fast-decay solutions to \({{\rm div}(|\nabla u|^{m-2}\nabla u)+K(|x|)|u|^{q-1}u=0}\) in \({{\mathbb{R}^{n}}}\). Differ. Integral Equ. 9, 981–1004 (1996)

    MathSciNet  MATH  Google Scholar 

  29. Kawano N., Yanagida E., Yotsutani S.: Structure theorems for positive radial solutions to \({\Delta u+K(|x|)u^{q}=0}\) in \({{\mathbb{R}^{n}}}\). Funkcialaj Ekvacioj 36, 557–579 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Morishita H., Yanagida E., Yotsutani S.: Structure of positive radial solutions including singular solutions to Matukuma’s equation. Commun. Pure Appl. Anal. 4(4), 871–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Naito Y.: Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation. Hiroshima Math. J. 24, 177–220 (1994)

    MathSciNet  MATH  Google Scholar 

  32. Ni W.M.: On the elliptic equation \({\Delta u + K (x)u^{\frac{n+2}{n-2}} = 0}\), its generalization, and applications in geometry. Indiana Univ. Math. J. 31, 493–529 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ni W.M., Serrin J.: Nonexistence theorems for quasilinear partial differential equations. Rend. Circolo Mat. Palermo (Centenary supplement), Ser. II 8, 171–185 (1985)

    MathSciNet  MATH  Google Scholar 

  34. Ni W.M., Yotsutani S.: On Matukuma’s equation and related topics. Proc. Jpn. Acad. Ser. A 62, 260–263 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ni W.M., Yotsutani S.: Semilinear elliptic equations of Matukuma-type and related topics. Jpn. J. Appl. Math. 5, 1–32 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Papini, D.: Boundary value problems for second order differential equations with superlinear terms: a topological approach. Ph.D. Thesis, S.I.S.S.A, Trieste (2000)

  37. Pucci P., García-Huidobro M., Manasevich R., Serrin J.: Qualitative properties of ground states for singular elliptic equations with weights. Ann. Mat. Pura Appl., 185, S205–S243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Serrin J., Zou H.: Symmetry of ground states of quasilinear elliptic equations. Arch. Rational Mech. Anal. 148, 265–290 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Yanagida E.: Structure of radial solutions to \({\Delta u +K(|x|) |u|^{p-1}u = 0}\) in \({{\mathbb{R}^{n}}}\). SIAM J. Math. Anal. 27(3), 997–1014 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yanagida E., Yotsutani S.: Classification of the structure of positive radial solutions to \({\Delta u +K (|x|) u^{p} = 0}\) in \({{\mathbb{R}^{n}}}\). Arch. Rational Mech. Anal. 124, 239–259 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Yanagida E., Yotsutani S.: Existence of nodal fast-decay solutions to \({\Delta u +K (|x|) |u|^{p-1}u = 0}\) in \({{\mathbb{R}^{n}}}\). Nonlinear Anal. 22, 1005–1015 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yanagida E., Yotsutani S.: Existence of positive radial solutions to \({\Delta u +K (|x|) u^{p}=0}\) in \({{\mathbb{R}^{n}}}\). J. Differ. Equ. 115, 477–502 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Francesca Dalbono

  2. Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 1, 60131, Ancona, Italy

    Matteo Franca

Authors
  1. Francesca Dalbono
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matteo Franca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Franca.

Additional information

Communicated by W. Schlag

F. Dalbono supported by G.N.A.M.P.A.

M. Franca supported by G.N.A.M.P.A.—INdAM (Italy) and MURST (Italy).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dalbono, F., Franca, M. Nodal Solutions for Supercritical Laplace Equations. Commun. Math. Phys. 347, 875–901 (2016). https://doi.org/10.1007/s00220-015-2546-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2546-y

Keywords

Search

Navigation