Opuscula Math. 40, no. 1 (2020), 71-92
https://doi.org/10.7494/OpMath.2020.40.1.71

 
Opuscula Mathematica

Nonhomogeneous equations with critical exponential growth and lack of compactness

Giovany M. Figueiredo
Vicenţiu D. Rădulescu

Abstract. We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth.

Keywords: exponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iteration.

Mathematics Subject Classification: 35J62, 35A15, 35B30, 35B33, 58E05.

Full text (pdf)

  1. C.O. Alves, Existence and multiplicity of solutions for a class of quasilinear equations, Adv. Nonlinear Studies 5 (2005), 73-87.
  2. C.O. Alves, G.M. Figueiredo, Existence and multiplicity of positive solutions to a \(p\)-Laplacian equation in \(\mathbb{R}^N\), Differential and Integral Equations 19 (2006), 143-162.
  3. C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in \(\mathbb{R}^N\), J. Differential Equations 246 (2009), 1288-1311.
  4. C.O. Alves, G.M. Figueiredo, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Adv. Nonlinear Studies 11 (2011), 265-294.
  5. C.O. Alves, J.M. Bezerra do Ó, O.H. Miyagaki, On pertubations of a class of a periodic \(m\)-Laplacian equation with critical growth, Nonlinear Anal. 45 (2001), 849-863.
  6. C.O. Alves, M.A.S. Souto, On existence and concentration behavior of ground state solutions for a class of problems with critical growth, Comm. Pure Appl. Anal. 1 (2002), 417-431.
  7. J.M. Bezerra do Ó, \(N\)-Laplacian equations in \(\mathbb{R}^N\) with critical growth, Abstract and Applied Analysis 2 (1997), 301-315.
  8. H. Brezis, E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 8 (1983), 486-490.
  9. D.M. Cao, Nontrivial solutions of semilinear elliptic equation with critical exponent in \(\mathbb{R}^N\), Comm. Partial Diff. Equations 17 (1992), 407-435.
  10. L. Cherfils, Y. Il'yasov, On the stationary solutions of generalized reaction difusion equations with \(p\& q\)-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9-22.
  11. F. Gazzola, V.D. Rădulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in \(\mathbb{R}^n\), Differential Integral Equations 13 (2000), 47-60.
  12. Li Gongbao, Some properties of weak solutions of nonlinear scalar field equations, Annales Acad. Sci. Fennicae, Series A 14 (1989), 27-36.
  13. C. He, G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing \(p\&q\)-Laplacians, Annales Acad. Sci. Fennicae, Series A 33 (2008), 337-371.
  14. O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer, Heidelberg, 1983.
  15. P.L. Lions, The concentration-compacteness principle in the calculus of variation. The locally compact case, part II, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984), 223-283.
  16. J. Moser, A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-468.
  17. J.Y. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Partial Diff. Equations 131 (1990), 223-253.
  18. N.S. Papageorgiou, V.D. Rădulescu, Resonant \((p,2)\)-equations with asymmetric reaction, Analysis and Applications 13 (2015), 481-506.
  19. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, On a class of parametric \((p,2)\)-equations, Applied Mathematics and Optimization 75 (2017), 193-228.
  20. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, \((p,2)\)-equations symmetric at both zero and infinity, Advances in Nonlinear Analysis 7 (2018), 327-351.
  21. N.S. Papageorgiou, V.D. Rădulescu, D.D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Berlin, 2019.
  22. P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 27-42.
  23. N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math. XX (1967), 721-747.
  24. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150.
  25. M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
  • Vicenţiu D. Rădulescu (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-4615-5537
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
  • Department of Mathematics, University of Craiova, 200585 Craiova, Romania
  • "Simion Stoilow" Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei, 010702 Bucharest, Romania
  • Communicated by Marius Ghergu.
  • Received: 2019-02-13.
  • Accepted: 2019-02-27.
  • Published online: 2020-02-17.
Opuscula Mathematica - cover

Cite this article as:
Giovany M. Figueiredo, Vicenţiu D. Rădulescu, Nonhomogeneous equations with critical exponential growth and lack of compactness, Opuscula Math. 40, no. 1 (2020), 71-92, https://doi.org/10.7494/OpMath.2020.40.1.71

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.