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Proyecciones (Antofagasta)
Print version ISSN 0716-0917
Proyecciones (Antofagasta) vol.38 no.2 Antofagasta June 2019
http://dx.doi.org/10.4067/S0716-09172019000200325
Articles
Nonlinear elliptic equations in dimension two with potentials which can vanish at infinity
1Universidade de São Paulo Brasil e-mail : yonyraul@alumni.usp.br
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.
Keywords: Nonlinear elliptic equations; vanishing potentials; Trudinger-Moser inequality.
References
[1] Adimurthi, P. N. Srikanth and S. L. Yadava, Phenomena of critical exponent in R2 Proc. Royal Soc. Edinb. 119A, pp. 19-25, (1991). [ Links ]
[2] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of R2 involving critical exponents. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17, pp. 481-504. (1990). [ Links ]
[3] F. S. B. Albuquerque, C. O. Alves and E.S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R2. J. Math. Anal. Appl, 409, pp. 1021-1031, (2014). [ Links ]
[4] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2. Comm. Partial Differential Equations 1, pp. 407-435, (1992). [ Links ]
[5] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3, pp. 139-153, (1995). [ Links ]
[6] C. O. Alves, João Marcos do Ó. , and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth, Nonlinear Analysis, 56, pp. 781-791, (2004). [ Links ]
[7] de Souza. Manassés and João Marcos do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth, J. Math. Anal. Appl., 380, pp. 241-263, (2011). [ Links ]
[8] Manasses de Souza On a singular elliptic problem involving critical growth in RN . Nonlinear Differ. Equ. Appl. 18, pp. 199-215, (2011). [ Links ]
[9] João Marcos do Ó and Bernand Ruf , On a Schrödinger equation with periodic potential and critical growth in R2, Nonlinear differ. equ. appl. 13, pp. 167-192, (2006). [ Links ]
[10] J. M. do Ó, E. S. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345, pp. 286-304, (2008). [ Links ]
[11] J. M. do Ó, E. S. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in RN , J. Differential Equations, 246, pp. 1363-1386, (2009). [ Links ]
[12] João Marcos. do Ó, Marco A. S. Souto On a Class of Nonlinear Schrodinger Equations in R2 Involving Critical Growth, Journal of Differential Equations, 174, pp. 289-311, (2001) [ Links ]
[13] J. M. do Ó, N-Laplacian equations in RN with critical growth, Abstr. Appl. Anal., 2, pp. 301-315, (1997). [ Links ]
[14] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer-Verlag, Paris, (1993). [ Links ]
[15] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, pp. 1077-1092, (1970/71). [ Links ]
[16] S. Pohožaev, The Sobolev embedding in the special case pl = n, Proceedings of the Tech. Sci conference on Adv. Sci. research Mathematics sections 1964-1965 , Moscow. Energet. Inst., pp. 158-170, (1965). [ Links ]
[17] Soares, Sérgio H. Monari and Leuyacc, Yony R. Santaria Hamiltonian elliptic systems in dimension two with potentials which can vanish at infinity, Commun. Contemp. Math. 20 (2018), No. 8, 1750053, 37, pp. [ Links ]
[18] J. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9, pp. 571-583, (2007). [ Links ]
[19] J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238, pp. 201-219, (2007). [ Links ]
[20] N. S. Trudinger, On embedding into Orlicz spaces and some applications,J. Math. Mech., 17, pp. 473-483, (1967). [ Links ]
[21] M. Willem, Minimax Theorems, Boston: Birkhäuser, (1996) [ Links ]
Received: October 2017; Accepted: March 2019