Opuscula Math. 39, no. 2 (2019), 175-194

Opuscula Mathematica

Infinitely many solutions for some nonlinear supercritical problems with break of symmetry

Anna Maria Candela
Addolorata Salvatore

Abstract. In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem \[\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}\] where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N\geq 3\), and \(A(x,t,\xi)\), \(g(x,t)\), \(h(x)\) are given functions, with \(A_t = \frac{\partial A}{\partial t}\), \(a = \nabla_{\xi} A\), such that \(A(x,\cdot,\cdot)\) is even and \(g(x,\cdot)\) is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if \(A(x,t,\xi)\) grows fast enough with respect to \(t\), then the nonlinear term related to \(g(x,t)\) may have also a supercritical growth.

Keywords: quasilinear elliptic equation, weak Cerami-Palais-Smale condition, Ambrosetti-Rabinowitz condition, break of symmetry, perturbation method, supercritical growth.

Mathematics Subject Classification: 35J20, 35J62, 35J66, 58E05.

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  1. V. Ambrosio, J. Mawhin, G. Molica Bisci, (Super)Critical nonlocal equations with periodic boundary conditions, Sel. Math. New Ser. 24 (2018), 3723-3751.
  2. D. Arcoya, L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal. 134 (1996), 249-274.
  3. D. Arcoya, L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, Nonlinear Differential Equations Appl. 6 (1999), 79-100.
  4. A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1-32.
  5. P. Bolle, N. Ghoussoub, H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math. 101 (2000), 325-350.
  6. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext XIV, Springer, New York, 2011.
  7. A.M. Candela, G. Palmieri, Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud. 6 (2006), 269-286.
  8. A.M. Candela, G. Palmieri, Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations 34 (2009), 495-530.
  9. A.M. Candela, G. Palmieri, Some abstract critical point theorems and applications, [in:] Dynamical Systems, Differential Equations and Applications, X. Hou, X. Lu, A. Miranville, J. Su and J. Zhu (eds), Discrete Contin. Dynam. Syst. Suppl. 2009 (2009), 133-142.
  10. A.M. Candela, G. Palmieri, Multiplicity results for some quasilinear equations in lack of symmetry, Adv. Nonlinear Anal. 1 (2012), 121-157.
  11. A.M. Candela, G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically \(p\)-linear terms, Calc. Var. Partial Differential Equations 56:72 (2017).
  12. A.M. Candela, A. Salvatore, Multiplicity results of an elliptic equation with non-homogeneous boundary conditions, Topol. Methods Nonlinear Anal. 11 (1998), 1-18.
  13. A.M. Candela, A. Salvatore, Some applications of a perturbative method to elliptic equations with non-homogeneous boundary conditions, Nonlinear Anal. 53 (2003), 299-317.
  14. A.M. Candela, G. Palmieri, K. Perera, Multiple solutions for \(p\)-Laplacian type problems with asymptotically \(p\)-linear terms via a cohomological index theory, J. Differential Equations 259 (2015), 235-263.
  15. A.M. Candela, G. Palmieri, A. Salvatore, Some results on supercritical quasilinear elliptic problems, preprint (2017).
  16. A.M. Candela, G. Palmieri, A. Salvatore, Infinitely many solutions for quasilinear elliptic equations with lack of symmetry, Nonlinear Anal. 172 (2018), 141-162.
  17. A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal. 6 (1995), 357-370.
  18. G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), 332-336.
  19. O.A. Ladyzhenskaya, N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
  20. R.S. Palais, Critical point theory and the minimax principle. In "Global Analysis", Proc. Sympos. Pure Math. 15, Amer. Math. Soc., Providence R.I. (1970), 185-202.
  21. B. Pellacci, M. Squassina, Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations 201 (2004), 25-62.
  22. P.H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), 753-769.
  23. P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, 1986.
  24. M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), 335-364.
  25. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed., Ergeb. Math. Grenzgeb. (4) 34, Springer-Verlag, Berlin, 2008.
  26. K. Tanaka, Morse indices at critical points related to the Symmetric Mountain Pass Theorem and applications, Comm. Partial Differential Equations 14 (1989), 99-128.
  • Communicated by Giovanni Molica Bisci.
  • Received: 2018-05-04.
  • Revised: 2018-11-11.
  • Accepted: 2018-11-13.
  • Published online: 2018-12-07.
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Cite this article as:
Anna Maria Candela, Addolorata Salvatore, Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math. 39, no. 2 (2019), 175-194, https://doi.org/10.7494/OpMath.2019.39.2.175

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