Study of a class of generalized Schrödinger equations

Andrelino V. Santos, João R. Santos Júnior, Antonio Suárez

DOI: http://dx.doi.org/10.12775/TMNA.2020.008

Abstract


A class of generalized Schrödinger problems in a bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we combine monotony, bifurcation and variational methods.

Keywords


Generalized Schrödinger problems; existence of solutions; variational methods; sub-supersolution method; bifurcation method

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References


A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411–422.

A.V. Borovskiı̆ and A.L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys. 77 (1993), 562–573.

H.S. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B 5 (1993), 3539–3550.

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55–64.

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett. 70 (1993), 2082–2085.

D. C. Clark, A variant of the Lusternik–Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65–74.

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal. 56 (2004), 213–226.

A. De Bouard, N. Hayashi and J.C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys. 189 (1997), 73–105.

Y. Deng, S. Peng, and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations 258 (2015), 115–147.

J.M. do Ó, O.H. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations: The critical exponential case, Nonlinear Anal. 67 (2007), 3357–3372.

J.M. do Ó, O.H. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 249 (2010), 722–744.

G.M. Figueiredo, J.R. Santos Júnior and A. Suárez, Structure of the set of positive solutions of a nonlinear Schrödinger equation, Israel J. Math. 227 (2018), 485–505.

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901.

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. 37 (1980), 83–87.

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352–370.

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981), 3262–3267.

E.W. Laedke, K.H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), 2764–2769.

A.G. Litvak and A.M. Sergeev, One dimensional collapse of plasma waves, J. Exp. Theor. Phys. Lett. 27 (1978), 517–520.

J.Q. Liu, Y.Q. Wang and Z.Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations 187 (2003), 473–493.

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Research Notes in Mathematics, vol. 426, Chapman & Hall/CRC, Boca Raton, FL, 2001.

V.G. Makhankov and V.K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104 (1984), 1–86.

M. Poppenberg, K. Schmitt and Z.Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344.

G.R.W. Quispel and H.W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A 110 (1982), 41–80.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, Amer. Math. Soc., Providence, RI, 1986.

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. 80 (2013), 194–201.

Y. Shen and Y. Wang, A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal. 15 (2016), 853–870.


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