Abstract
In this paper, we establish the existence of positive solutions for a class of quasilinear Schrödinger equations involving supercritical growth. By using a change of variables, the quasilinear equation is reduced to a semilinear equation. Then, variational method is used together with a truncation argument used in Del Pino and Felmer (Calc var Partial Differ Equ 4:121–137, 1996) and concentration compactness principle given in Zelati and Rabinowtiz (Commun Pure Appl Math 45:1217–1269, 1992.
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Alves C.O., Carrião P.C., Miyagaki O.H.: Nonlinear perturbations of a periodic elliptic problem with critical growth. J. Math. Anal. Appl. 260, 133–146 (2001)
Alves C.O., Figueiredo G.M.: Nonlinear perturbations of a periodic Kirchhoff equation in \({\mathbb{R}^N}\). Nonlinear Anal. 75, 2750–2759 (2012)
Alves C.O., Figueiredo G.M., Severo U.B.: Multiplicity of positive solutions for a class of quasilinear problems. Adv. Differ. Equ. 14, 911–942 (2009)
Alves C.O., Soares S.H.M., Souto M.A.S.: Schrödinger-Poisson equations with supercritical growth. Eletron. J. Differ. Equ. 1, 1–11 (2011)
Borovskii A., Galkin A.: Dynamical modulation of an ultrashort high-intensity laser pulse in matter. J. Exp. Theor. Phys. 77, 562–573 (1983)
Brandi H., Manus C., Mainfray G., Lehner T., Bonnaud G.: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B 5, 3539–3550 (1993)
Colin M.: Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2. Adv. Differ. Equ. 8, 1–28 (2003)
Colin M., Jeanjean L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56, 213–226 (2004)
Zelati V.C., Rabinowtiz P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}^N}\). Commun. Pure Appl. Math. 45, 1217–1269 (1992)
Del Pino M., Felmer P.L.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)
do Ó J.M., Miyagaki O.H., Soares S.H.M.: Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ. 248, 722–744 (2010)
do Ó J.M., Severo U.: Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun. Pure Appl. Anal. 8, 621–644 (2009)
do Ó J.M., Severo U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ. Equ. 38, 275–315 (2010)
Gilbard D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer, Berlin (1983)
Gloss E.: Existence and concentration of positive solutions for a quasilinear equation in \({\mathbb{R}^N}\). J. Math. Anal. Appl. 371, 465–484 (2010)
Kavian O.: Introduction á la théorie des points critiques et applications aux problems elliptiques. Springer, New York/ Berlin (1994)
Kurihura S.: Large-amplitude quasi-solitons in superfluids films. J. Phys. Soc. Japan 50, 3262–3267 (1981)
Li G., Wang C.: The existence of a nontrivial solution to p-Laplacian equations in \({\mathbb{R}^N}\) with supercritical growth. Math. Methods Appl. Sci. 36, 69–79 (2013)
Liu X., Liu J.-Q., Wang Z.-Q.: Ground states for quasilinear Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 46, 641–669 (2013)
Liu J., Wang Z.-Q.: Soliton solutions for quasilinear Schrödinger equations, I. Proc. Am. Math. Soc. 131, 441–448 (2003)
Liu J., Wang Y., Wang Z.-Q.: Soliton solutions for quasilinear Schrödinger equations, II. J. Differ. Equ. 187, 473–493 (2003)
Liu X.-Q., Liu J.-Q., Wang Z.-Q.: Quasilinear elliptic equations with critical growth via perturbation method. J. Differ. Equ. 248, 102–124 (2013)
Makhankov V.G., Fedyanin V.K.: Nonlinear effects in quasi-one-dimensional models of condensed matter theory. Phys. Rep. 104, 1–86 (1984)
Miyagaki O.H., Moreira S.I.: Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign. J. Math. Anal. Appl. 421, 643–655 (2015)
Moameni A.: Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in R N. J. Differ. Equ. 229, 570–587 (2006)
Moameni A.: Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in \({\mathbb{R}^N}\). Commun. Pure Appl. Anal. 7, 89–105 (2008)
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)
Silva E.A.B., Vieira G.F.: Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 39, 1–33 (2010)
Silva E.A.B., Vieira G.F.: Quasilinear asymptotically periodic Schrödinger equations with subcritical growth. Nonlinear Anal. 72, 2935–2949 (2010)
Wang Y.: Multiple solutions for quasilinear elliptic equations with critical growth. J. Korean Math. Soc. 48, 1269–1283 (2011)
Wang Y.J., Zhang Y.M., Shen Y.T.: Multiple solutions for quasilinear Schrödinger equations involving critical exponent. Appl. Math. Comput. 216, 849–856 (2010)
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Author was partially supported by INCTmat/Brazil, CNPq/Brazil and CAPES/Brazil (Proc. 2531/14-3).
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Figueiredo, G.M., Miyagaki, O.H. & Moreira, S.I. Nonlinear perturbations of a periodic Schrödinger equation with supercritical growth. Z. Angew. Math. Phys. 66, 2379–2394 (2015). https://doi.org/10.1007/s00033-015-0525-y
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DOI: https://doi.org/10.1007/s00033-015-0525-y