Revista Matemática Complutense

, Volume 32, Issue 1, pp 1–18 | Cite as

Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method

  • Giovany M. FigueiredoEmail author
  • Fernando Bruno M. Nunes


In this paper we will be concerned with the problem
$$\begin{aligned} -\,\text{ div }(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u)= f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$
where \(\Omega \subset \mathbb {R}^{N}\) is bounded, \(1{<}p{<}N\), \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a superlinear continuous function with exponential subcritical or exponential critical growth and the function a is \(C^{1}\). We use as a main tool the Nehari manifold method and our results include a large class of problems.


p-N Laplacian Critical exponential growth Trudinger-Moser inequality 

Mathematics Subject Classification

Primary 35J60 Secondary 35C20 35B33 49J45 


  1. 1.
    Alves, C.O., Figueiredo, G.M.: On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in \(\mathbb{R}^{N}\). J. Differ. Equ. 246, 1288–1311 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Souto, M.A.S.: Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z. Angew. Math. Phys. 65, 1153 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alves, C.O., Freitas, L.R., Soares, S.H.M.: Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth. Differ. Integral Equ. 24(11–12), 1047–1062 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alves, M.J., Assunção, R.B., Miyagaki, O.H.: Existence result for a class of quasilinear elliptic equations with \((pq)\)-Laplacian and vanishing potentials. Ill. J. Math. 59(3), 545575 (2015)Google Scholar
  5. 5.
    Aouaoui, S.: Three nontrivial solutions for some elliptic equation involving the \(N\)-Laplacian. Electron. J. Qual. Theory Differ. Equ. 2, 12 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Averna, D., Montreanu, D., Tornatore, E.: Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence. Appl. Math. Lett. 61, 102–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barile, S., Figueiredo, G.M.: Existence of a least energy nodal solution for a class of p&q-quasilinear elliptic equations. Adv. Nonlinear Stud. 14, 511–530 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Barile, S., Figueiredo, G.M.: Existence of least energy positive, negative and nodal solutions for a class of p&q-problems with potentials vanishing at infinity. J. Math. Appl. Anal. 427, 1205–1233 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barile, S., Figueiredo, G.M.: Some classes of eigenvalues problems for generalized p&q-Laplacian type operators on bounded domains. Nonlinear Anal. 119, 457–468 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bartolo, R., Candela, A.M., Salvatore, A.: On a class of superlinear \((p, q )\)-Laplacian type equations on \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 438, 29–41 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)CrossRefGoogle Scholar
  13. 13.
    Chaves, M.F., Ercole, G., Miyagaki, O.H.: Existence of a nontrivial solution for the \((p, q)\)-Laplacian with \(p\)-critical exponent. Bound. Value Probl. 236, 15 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chaves, M.F., Ercole, G., Miyagaki, O.H.: Existence of a nontrivial solution for the \((p, q)\)-Laplacian in \(\mathbb{R}^{N}\) without the Ambrosetti–Rabinowitz condition. Nonlinear Anal. 114, 133–141 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, L., Chen, C., Xiu, Z.: Positive solution for a quasilinear equation with critical growth in \({\mathbb{R}}^{N}\). Ann. Polon. Math. 116(3), 251262 (2016)MathSciNetGoogle Scholar
  16. 16.
    Cherfils, L.L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian. CPAA 1, 4 (2004)zbMATHGoogle Scholar
  17. 17.
    de Araujo, A.L.A., Montenegro, M.: Existence of solution for a general class of elliptic equations with exponential growth. Ann. Mat. 195, 1737–1748 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \(\mathbb{R}^{2}\) with nonlinearities in the critical growth range. Calc. Var. 3, 139–153 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    de Souza, M.: Existence of solutions to equations of \(N\)-Laplacian type with Trudinger–Moser nonlinearities. Appl. Anal. 93(10), 2111–2125 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Faria, L.F.O., Miyagaki, O.H., Motreanu, D.: Comparison and positive solutions for problems with \((p-q)\)-Laplacian and convection term. Proc. Edinb. Math. Soc. 57(2), 687–698 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Faria, L.F.O., Miyagaki, O.H., Tanaka, M.: Existence of a positive solution for problems with \((p, q)\)-Laplacian and convection term in \(\mathbb{R}^{N}\). Bound. Value Prob. 2016, 158 (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Figueiredo, G.M., Ramos, H.: Quoirin, ground states of elliptic problems involving nonhomogeneous operators. Indiana Univ. Math. J. 65(3), 779–795 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Freitas, L.R.: Mutiplicity of solutions for a class of quasilinear equations with exponential critical growth. Nonlinear Anal. 95, 607–624 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Goyal, S., Sreenadh, K.: The Nehari manifold approach for N-Laplace equation with singular and exponential nonlinearities in \(\mathbb{R}^{N}\). Commun. Contemp. Math. 17, 1450011 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    He, C., Li, G.: The existence of a nontrivial solution to the p & q-Laplacian problem with nonlinearity asymptotic to \(u^{p-1}\) at infinity in \(\mathbb{R}^{N}\). Nonlinear Anal. (2007). Google Scholar
  26. 26.
    He, C., Li, G.: The regularity of weak solutions to nonlinear scalar field elliptic equations containing p&q-Laplacians. Ann. Acad. Sci. Fenn. Math. 33, 337–371 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    He, C., Li, G.: The regularity of weak solutions to nonlinear scalar field elliptic equations containing p&q-Laplacians. Ann. Acad. Sci. Fenn. 33, 337–371 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kavian, O.: Introduction à la théorie des Points Critiques. Springer, Berlin (1991)zbMATHGoogle Scholar
  29. 29.
    Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (1997)Google Scholar
  30. 30.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 185–201 (1985)MathSciNetGoogle Scholar
  31. 31.
    Mugnai, D., Papageorgiou, N.: Wang’s multiplicity result superlinear \((p-q)\)-equations without Ambrosetti–Rabinowitz condition. Trans. Am. Math. Soc. 366, 4919–4937 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nguyen, L., Guozhen, L.: N-Laplacian equations in \(\mathbb{R}^{N}\) with subcritical and critical growth without the Ambrosetti–Rabinowitz condition. Adv. Nonlinear Stud. 13(2), 289–308 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Pei, R., Zhang, J.: Nontrivial solutions for a class of quasilinear elliptic equations. J. Math. Res. Appl. 36, 8796 (2016)MathSciNetGoogle Scholar
  34. 34.
    Pereira, D.S.: Existence of infinitely many sign-changing solutions for elliptic problems with critical exponential growth. Electron. J. Differ. Equ. 119, 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Tiwari, S.: N-Laplacian critical problem with discontinuous nonlinearities. Adv. Nonlinear Anal. 4(2), 109–121 (2015)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Trudinger, N.S.: On imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Willem, M.: Minimax methods, Handbook of nonconvex analysis and applications, pp. 597–632. Int. Press, Somerville (2010)Google Scholar
  38. 38.
    Wu, M., Yang, Z.: A class of \(p\)\(q\)-Laplacian type equation with potentials eigenvalue problem in \(\mathbb{R}^{N}\). Bound. Value Prob. (2009). (ID 185319) Google Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  • Giovany M. Figueiredo
    • 1
    Email author
  • Fernando Bruno M. Nunes
    • 2
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de Engenharia AmbientalUniversidade Estadual do Amapá - UEAPMacapáBrazil

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