Topological Methods in Nonlinear Analysis

Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media

Giovanni Molica Bisci and Vicenţiu D. Rădulescu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this work we obtain an existence result for a class of singular quasilinear elliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenberg type inequality, a critical point result for differentiable functionals is exploited, in order to prove the existence of a precise open interval of positive eigenvalues for which the treated problem admits at least one nontrivial weak solution. In the case of terms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 45, Number 2 (2015), 493-508.

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343993

Digital Object Identifier
doi:10.12775/TMNA.2015.024

Mathematical Reviews number (MathSciNet)
MR3408833

Zentralblatt MATH identifier
1367.35029

Citation

Bisci, Giovanni Molica; Rădulescu, Vicenţiu D. Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media. Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 493--508. doi:10.12775/TMNA.2015.024. https://projecteuclid.org/euclid.tmna/1459343993


Export citation

References

  • B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli–Kohn–Nirenberg inequalities, Calc. Var. Partial Differential Equations 23 (2005), 327–345.
  • B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian with a critical potential, Ann. Mat. Pura Appl. 182 (2003), 247–270.
  • M. Badiale and G. Tarantello, A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), 259–293.
  • G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 140 (2009), 1–16.
  • H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
  • L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259–275.
  • P. Caldiroli and R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. Partial Differential Equations 8 (1999), 365–387.
  • ––––, On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl. (NoDEA) 7 (2000), 187–199.
  • F. Catrina and Z.-Q. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence \rom(and nonexistence\rom) and symmetry of extremal function, Commun. Pure Appl. Math. 54 (2001), 229–258.
  • K.S. Chou and C.W. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. London Math. Soc. 48 (1993), 137–151.
  • R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer–Verlag, Berlin, 1985.
  • W. Fulks and J.S. Maybee, A singular nonlinear equation, Osaka J. Math. 12 (1960), 1–19.
  • M. Ghergu and V. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl. 311 (2005), 635–646.
  • ––––, Multiparameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 61–84.
  • ––––, Singular elliptic problems with lack of compactness, Ann. Mat. Pura Appl. 185 (2006), 63–79.
  • ––––, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, No. 37, Oxford University Press, New York, 2008.
  • N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703–5743.
  • Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), 487–512.
  • J. Hernández, F.J. Mancebo and J.M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 777–813.
  • L. Iturriaga, Existence and multiplicity results for some quasilinear elliptic equation with weights, J. Math. Anal. Appl. 339 (2008), 1084–1102.
  • A. Kristály, On singular elliptic equations involving oscillatory terms, Nonlinear Anal. 72 (2010), 1561–1569.
  • A. Kristály and Cs. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials, Proc. Amer. Math. Soc. 135 (2007), 2121–2126.
  • A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010.
  • E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math. 118 (1983), 349–374.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, Springer–Verlag, New York, 1989.
  • M. Mihăilescu, V. Rădulescu and D. Stancu, A Caffarelli–Kohn–Nirenberg type inequality with variable exponent and applications to \romPDE's, Complex Var. Elliptic Equ. 56 (2011), 659–669.
  • P. Pucci and R. Servadei, Existence, non-existence and regularity of radial ground states for $p$-Laplacian equations with singular weights, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 505–537.
  • B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401–410.
  • ––––, A bifurcation theory for some nonlinear elliptic equations, Colloq. Math. 95 (2003), 139–151.
  • D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations 18 (2003), 57–75.
  • Z.-Q. Wang and M. Willem, Singular minimization problems, J. Differential Equations 161 (2000), 307–320.
  • B. Xuan, The solvability of quasilinear Brezis–Nirenberg type problems with singular weights, Nonlinear Anal. 62 (2005), 703–725.