Advertisement

Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in \({{\mathbb {R}}}^N\)

  • Wedad Albalawi
  • Carlo MercuriEmail author
  • Vitaly Moroz
Article
  • 12 Downloads

Abstract

We study the asymptotic behaviour of positive groundstate solutions to the quasilinear elliptic equation
where \(1<p<N \), \(p<q<l<+\infty \) and \(\varepsilon > 0 \) is a small parameter. For \({\varepsilon }\rightarrow 0\), we give a characterization of asymptotic regimes as a function of the parameters q, l and N. In particular, we show that the behaviour of the groundstates is sensitive to whether q is less than, equal to, or greater than the critical Sobolev exponent \(p^{*} :=\frac{pN}{N-p}\).

Keywords

Groundstates Liouville-type theorems Quasilinear equations Singular perturbation 

Mathematics Subject Classification

35J92 (35B33 · 35B53 · 35B38) 

Notes

References

  1. 1.
    Allegretto, W., Huang, Y.X.: A Picone’s identity for the \(p\)-Laplacian and applications. Nonlinear Anal. 32(7), 819–830 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \(\mathbb{R}^{n}\), Progress in Mathematics, vol. 240. Birkhäuser Verlag, Basel (2006)zbMATHGoogle Scholar
  3. 3.
    Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. ure Appl. Math. 42(3), 271–297 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. III. Nucleation in a 2-component incompressible fluid. J. Chem. Phys. 31(3), 688–699 (1959)Google Scholar
  7. 7.
    Coleman, S.: Fate of the false vacuum: semiclassical theory. Phys. Rev. D 15(10), 2929 (1977)Google Scholar
  8. 8.
    Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65(3), 851–1112 (1993)zbMATHGoogle Scholar
  9. 9.
    Degiovanni, M., Musesti, A., Squassina, M.: On the regularity of solutions in the Pucci–Serrin identity. Calc. Var. Partial Differ. Equ. 18(3), 317–334 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Díaz, J.I., Saá, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305(12), 521–524 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    DiBenedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Farina, A., Mercuri, C., Willem, M.: A Liouville theorem for the \(p\)-Laplacian and related questions. arXiv:1711.11552v2 (2017)
  13. 13.
    Fraas, M., Pinchover, Y.: Isolated singularities of positive solutions of \(p\)-Laplacian type equations in \(\mathbb{R}^d\). J. Differ. Equ. 254(3), 1097–1119 (2013)zbMATHGoogle Scholar
  14. 14.
    Gazzola, F., Serrin, J.: Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(4), 477–504 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gazzola, F., Serrin, J., Tang, M.: Existence of ground states and free boundary problems for quasilinear elliptic operators. Adv. Differ. Equ. 5(1–3), 1–30 (2000)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guedda, M., Véron, L.: Local and global properties of solutions of quasilinear elliptic equations. J. Differ. Equ. 76(1), 159–189 (1988)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13(1), 879–902 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case I. Rev. Mat. Iberoam. 1(1), 145–201 (1985)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liskevich, V., Lyakhova, S., Moroz, V.: Positive solutions to nonlinear \(p\)-Laplace equations with Hardy potential in exterior domains. J. Differ. Equ. 232(1), 212–252 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Mercuri, C., Riey, G., Sciunzi, B.: A regularity result for the \(p\)-Laplacian near uniform ellipticity. SIAM J. Math. Anal. 48(3), 2059–2075 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mercuri, C., Squassina, M.: Global compactness for a class of quasi-linear elliptic problems. Manuscr. Math. 140(1), 119–144 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mercuri, C., Willem, M.: A global compactness result for the \(p\)-Laplacian involving critical nonlinearities. Discrete Contin. Dyn. Syst. 28(2), 469–493 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Merle, F., Peletier, L.A.: Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The nonradial case. J. Funct. Anal. 105(1), 1–41 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Moroz, V., Muratov, C.B.: Asymptotic properties of ground states of scalar field equations with a vanishing parameter. J. Eur. Math. Soc. (JEMS) 16(5), 1081–1109 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Muratov, C., Vanden-Eijnden, E.: Breakup of universality in the generalized spinodal nucleation theory. J. Stat. Phys. 114(3–4), 605–623 (2004)zbMATHGoogle Scholar
  26. 26.
    Pohožaev, S.: On the eigenfunctions of the equation \(\Delta u + \lambda f/(u) = 0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965)MathSciNetGoogle Scholar
  27. 27.
    Poliakovsky, A., Shafrir, I.: A Comparison Principle for the \(p\)-Laplacian, Elliptic and Parabolic Problems (Rolduc/Gaeta, 2001), pp. 243–252 (2002)Google Scholar
  28. 28.
    Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35(3), 681–703 (1986)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Pucci, P., Serrin, J.: Uniqueness of ground states for quasilinear elliptic operators. Indiana Univ. Math. J. 47(2), 501–528 (1998)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Pucci, P., Serrin, J.: The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhäuser Verlag, Basel (2007)zbMATHGoogle Scholar
  31. 31.
    Pucci, P., Servadei, R.: Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J. 57(7), 3329–3363 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sciunzi, B.: Classification of positive \(D^{1, p}(\mathbb{R}^{N})\)-solutions to the critical \(p\)-Laplace equation in \(\mathbb{R}^{N}\). Adv. Math. 291, 12–23 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Serrin, J., Tang, M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49(3), 897–923 (2000)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shafrir, I.: Asymptotic behaviour of minimizing sequences for Hardy’s inequality. Commun. Contemp. Math. 2(2), 151–189 (2000)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Struwe, M.: Variational Methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (1990)zbMATHGoogle Scholar
  36. 36.
    Su, J., Wang, Z.-Q., Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 238(1), 201–219 (2007)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tang, M.: Existence and uniqueness of fast decay entire solutions of quasilinear elliptic equations. J. Differ. Equ. 164(1), 155–179 (2000)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)MathSciNetzbMATHGoogle Scholar
  40. 40.
    van Saarloos, W., Hohenberg, P.C.: Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations. Phys. D 56(4), 303–367 (1992)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Vétois, J.: A priori estimates and application to the symmetry of solutions for critical \(p\)-Laplace equations. J. Differ. Equ. 260, 149–161 (2016)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Willem, M.: Functional Analysis: Fundamentals and Applications. Cornerstones. Springer, New York (2013)zbMATHGoogle Scholar
  44. 44.
    Unger, C., Klein, W.: Nucleation theory near the classical spinodal. Phys. Rev. B 29(5), 2698–2708 (1984)Google 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.College of SciencesPrincess Nourah Bint Abdulrahman UniversityRiyadhKingdom of Saudi Arabia
  2. 2.Department of Mathematics, Computational FoundrySwansea UniversitySwanseaUK

Personalised recommendations