Abstract
This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of these solutions with respect to two parameters \(\kappa \) and \(\lambda \) that appear in the equation. The proof of our main results relies on bifurcation techniques, the sub- and supersolution method and a construction of an appropriate large solution.
Similar content being viewed by others
References
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)
Ambrosetti, A.: On the existence of multiple solutions for a class of nonlinear boundary value problems. Rc. Semin. Mat. Univ. Padova 49, 195–204 (1973)
Ambrosetti, A., Lupo, D.: On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal. 8, 1145–1150 (1984)
Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal. 3, 635–645 (1979)
Adachi, S., Watanabe, T.: \(G\)-invariant positive solutions for a quasilinear Schrödinger equation. Adv. Differ. Equ. 16, 289–324 (2011)
Adachi, S., Watanabe, T.: Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity. Nonlinear Anal. 108, 275–290 (2014)
Adachi, S., Watanabe, T.: Uniqueness of the ground state solutions of quasilinear Schrödinger equations. Nonlinear Anal. 75, 819–833 (2012)
Arrieta, J.M., Pardo, R., Rodríguez-Bernal, A.: Asymptotic behavior of degenerate logistic equations. J. Differ. Equ. 259, 6368–6398 (2015)
Berestycki, H., Lions, P.-L.: Some applications of the method of super and sub-solutions. Bifurcation and Nonlinear Eigenvalue Problems (Proceedings. Session, University Paris XIII, Villetaneuse, 1978), Volume 782 of Lecture Notes in Mathematics, pp. 16–41. Springer, Berlin (1980)
Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)
Brizhik, L., Eremko, A., Piette, B., Zakrzewski, W.J.: Static solutions of a \(D\)-dimensional modified nonlinear Schrödinger equation. Nonlinearity 16, 1481–1497 (2003)
Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003)
Cîrstea, F.C., Radulescu, V.: Existence and uniqueness of blow-up solutions for a class of logistic equations. Commun. Contemp. Math. 4, 559–586 (2002)
Colin, M., Jeanjean, L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56, 213–226 (2004)
Delgado, M., López-Gómez, L., Suárez, A.: Non-linear versus linear diffusion from classical solutions to metasolutions. Adv. Differ. Equ. 7, 1101–1124 (2004)
do Ó, J.M., Severo, U.B.: Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun. Pure Appl. Anal. 8, 621–644 (2009)
do Ó, J.M., Miyagaki, H., Moreira, S.I.: On a quasilinear Schrödinger problem at resonance. Adv. Nonlinear Stud. 16, 569–580 (2016)
Du, Y., Huang, Q.: Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J. Math. Anal. 31, 1–18 (1999)
Fraile, J.M., Koch, P., López-Gómez, J., Merino, S.: Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation. J. Differ. Equ. 127, 295–319 (1996)
Figueiredo, G.M., Santos Júnior, J.R., Suárez, A.: Structure of the set of positive solutions of a non-linear Schrödinger equation. Israel J. Math. 227, 485–505 (2018)
Gámez, J.L.: Sub- and super-solutions in bifurcation problems. Nonlinear Anal. 28, 625–632 (1997)
García-Melián, J., Gómez-Reñasco, R., López-Gómez, J., Sabina de Lis, J.C.: Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs. Arch. Ration. Mech. Anal. 145, 261–289 (1998)
García-Melián, J., Letelier-Albornoz, R., Sabina de Lis, J.: Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up. Proc. Am. Math. Soc. 129, 3593–3602 (2001)
Liu, J.Q., Wang, Y.Q., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations II. J. Differ. Equ. 187, 473–493 (2003)
López-Gómez, J.: Approaching metasolutions by classical solutions. Differ. Integral Equ. 14, 739–750 (2001)
López-Gómez, J.: Linear Second Order Elliptic Operators. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013)
López-Gómez, J.: Metasolutions of Parabolic Equations in Population Dynamics. CRC Press, Boca Raton, FL (2016)
López-Gómez, J., Molina-Meyer, M.: The maximum principle for cooperative weakly coupled elliptic systems and some applications. Differ. Integral Equ. 7, 383–398 (1994)
López-Gómez, J.: Spectral Theory and Nonlinear Function Analysis. Chapman & Hall, Boca Raton (2001)
Hartmann, B., Zakrzewski, W.J.: Electrons on hexagonal lattices and applications to nanotubes. Phys. Rev. B 68, 184302 (2003)
Okubo, A.: Diffusion and Ecological Problems: Mathematical Models, Biomathematics, vol. 10. Springer, Berlin (1980)
Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, vol. 14, 2nd edn. Springer, New York (2001)
Poppenberg, M., Schmitt, K., Wang, Z.-Q.: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 14, 329–344 (2002)
Struwe, M.: A note on a result of Ambrosetti and Mancini. Ann. Mat. Pura Appl. 131, 107–115 (1982)
Acknowledgements
Research partially supported by CAPES and CNPq Grants 308735/2016-1 and 307770/2015-0. The authors thank to the referee for her/his comments and suggestions which improve notably this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cintra, W., Medeiros, E. & Severo, U. On positive solutions for a class of quasilinear elliptic equations. Z. Angew. Math. Phys. 70, 79 (2019). https://doi.org/10.1007/s00033-019-1121-3
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-019-1121-3