Nonlinear elliptic equations in dimension two with potentials which can vanish at infinity.

• Santaria Leuyacc, Yony Raúl ^[1]
  1. [1] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 38, Nº. 2, 2019, págs. 325-351
• Idioma: inglés
• DOI: 10.4067/s0716-09172019000200325
• Enlaces
  • Texto completo
• Resumen

  • We will focus on the existence of nontrivial solutions to the following nonlinear elliptic equation −∆u + V (x)u = f(u), x ∈ R2, where V is a nonnegative function which can vanish at infinity or be unbounded from above, and f have exponential growth range. The proof involves a truncation argument combined with Mountain Pass Theorem and a Trudinger-Moser type inequality.

• Referencias bibliográficas
  • Adimurthi, P. N. Srikanth and S. L. Yadava, Phenomena of critical exponent in R2 Proc. Royal Soc. Edinb. 119A, pp. 19-25, (1991).
  • Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of R2 involving critical exponents....
  • F. S. B. Albuquerque, C. O. Alves and E.S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving...
  • D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2. Comm. Partial Differential Equations 1, pp. 407-...
  • D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial...
  • C. O. Alves, João Marcos do Ó, and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth,...
  • de Souza. Manassés and João Marcos do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth, J. Math. Anal. Appl.,...
  • Manasses de Souza On a singular elliptic problem involving critical growth in RN. Nonlinear Differ. Equ. Appl. 18, pp. 199-215, (2011).
  • João Marcos do Ó and Bernand Ruf, On a Schrödinger equation with periodic potential and critical growth in R2, Nonlinear differ. equ. appl....
  • J. M. do Ó, E. S. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl.,...
  • J. M. do Ó, E. S. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in RN, J. Differential Equations,...
  • João Marcos. do Ó, Marco A. S. Souto On a Class of Nonlinear Schrodinger Equations in R2 Involving Critical Growth, Journal of Differential...
  • J. M. do Ó, N-Laplacian equations in RN with critical growth, Abstr. Appl. Anal., 2, pp. 301-315, (1997).
  • O. Kavian, Introduction á la thèorie des points critiques et applications aux problèmes elliptiques, Springer-Verlag, Paris, (1993).
  • J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, pp. 1077-1092, (1970/71).
  • S. Pohožaev, The Sobolev embedding in the special case pl = n, Proceedings of the Tech. Sci conference on Adv. Sci. research Mathematics...
  • Soares, Sérgio H. Monari and Leuyacc, Yony R. Santaria Hamiltonian elliptic systems in dimension two with potentials which can vanish at infinity,...
  • J. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9,...
  • J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238,...
  • N. S. Trudinger, On embedding into Orlicz spaces and some applications,J. Math. Mech., 17, pp. 473-483, (1967).
  • M. Willem, Minimax Theorems, Boston: Birkhäuser, (1996).