Equation with positive coefficient in the quasilinear term and vanishing potential

Jose Aires, Marco A. S. Souto

DOI: http://dx.doi.org/10.12775/TMNA.2015.069

Abstract


In this paper we study the existence of nontrivial classical solution for
the quasilinear Schr\"odinger equation:
$$ - \Delta u +V(x)u+\frac{\kappa}{2}\Delta
(u^{2})u= f(u),
$$%
in $\mathbb{R}^N$, where $N\geq 3$, $f$ has
subcritical growth and $V$ is a nonnegative potential. For this purpose, we use variational methods combined with perturbation arguments, penalization technics of Del Pino and Felmer and Moser iteration. As a main novelty with respect to some previous results, in our work we are able to deal with the case $\kappa > 0$ and the potential can vanish at infinity.


Keywords


Quasilinear Schrodinger equation; subcritical growth; Vanishing potentials

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References


J. F. L. Aires and M. A. S. Souto, it Existence of solutions for a quasilinear Schrodinger equation with vanishing potentials, J. Math. Anal. Appl. 416 (2014), 924-946.

C. O. Alves, Y. Wang and Y. Shen, Soliton solutions for for class of quasilinear Schrodinger equations with a parameter, J. Differential Equations 259 (2015), 318-343.

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of elliptic equations in RN with vanishing potentials, J. Differential Equations 252 (2012), 5555-5568.

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrodinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), 1977-1991.

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005), 117-144.

A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrodinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006), 317-348.

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrodinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005), 1321-1332.

W. D. Bastos, O. H. Miyagaki and R. S. Vieira, Existence of solutions for a class of degenerate quasilinear elliptic equation in RN with vanishing potentials, Boundary Value Problems 92 (2013), doi:10.1186/1687-2770-2013-92.

V. Benci and G. Cerami, Existence of positive solutions of the equation -Delta u+a(x)u = u^{(N+2)=(N-2)}, J. Funct. Anal. 88 (1) (1990), 90-117.

V. Benci, C. R. Grisanti and A. M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrodinger equations with V (1) = 0, Topol. Methods in Nonlinear Anal. 26 (2005), 203-219.

V. Benci, C. R. Grisanti and A. M. Micheletti, Existence of solutions for the nonlinear Schrodinger equation with V (1) = 0, Progr. Nonlinear Differential Equations Appl. 66 (2005), 53-65.

H. Berestycki and P.L. Lions, Nonlinear scalar field equations I, Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-346.

C. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys. 77 (1983), 562-573.

D. Bonheure and J. Van Schaftingen, Groundstates for nonlinear Schrodinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. 189 (2010), 273-301.

L. Brull, H. Lange and Koln, Stationary, oscillatory and solitary waves type solutions of singular nonlinear Schrodinger equations, Math. Mech. Appl. sci. 8 (1986), 559-575.

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.

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrodinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213-226.

A. De Bouard and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrodinger equation, Comm. Math. Phys. 189 (1997), 73-105.

M. Del Pino and P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996), 121-137.

M. Ghimenti and A.M. Micheletti, Existence of minimal nodal solutions for the nonlinear Schrodinger equations with V (1) = 0, Adv. Differential Equations 11 (2006), no. 12, 1375-1396.

S. Kurihara, Large-amplitude quasi-solitons in super uids films, J. Phys. Soc. Japan 50 (1981), 3262-3267.

H. Lange, M. Poppenberg and H. Teisniann, Nash-More methods for the solution of quasilinear Schrodinger equations, Commun. Partial Differential Equations 24 (7-8) (1999), 1399-1418.

P.L. Lions, The concentration compactness principle in the calculus of variations, The locally compact case, Parts I and II, Ann. Inst. H. Poincare Anal. Non. Lineaire (1984), 109-145, 223-283.

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

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

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrodinger equations, J. Math. Phys. 54 (2013), doi: 10.1063/1.4811394.

M. Willem, Minimax Theorems, Birkhauser, (1986).


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