# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022121
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## Soliton solutions for quasilinear modified Schrödinger equations in applied sciences

 Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

* Corresponding author: Anna Maria Candela

To Rosella Mininni, a beloved friend
Both the authors are members of the Research Group INdAM–GNAMPA

Received  February 2022 Revised  April 2022 Early access May 2022

Fund Project: The research that led to the present paper was partially supported by MIUR–PRIN Project "Qualitative and quantitative aspects of nonlinear PDEs" (2017JPCAPN 005) and by Fondi di Ricerca di Ateneo 2017/18 "Problemi differenziali non lineari"

In this paper, we prove the existence of nontrivial weak bounded solutions of the quasilinear modified Schrödinger problem
 $\left\{ \begin{array}{ll} -{\rm div}(g^2(u) \nabla u) + g(u) g^{\prime}(u) |\nabla u|^2 + V(x) u = f(x, u) &\hbox{in}\ \mathbb R^3 , \\ u > 0 &\hbox{in }\ \mathbb R^3 , \end{array} \right.$
where
 $V: \mathbb R^3\to \mathbb R$
,
 $f: \mathbb R^3\times \mathbb R\to \mathbb R$
are "good" functions and
 $g: \mathbb R\to \mathbb R$
is such that
 $g^2(u) = 1+\frac{[(l(u^2))^{\prime}]^2}{2}$
for a given
 $l\in\mathcal{C}^2( \mathbb R)$
.
By means of variational methods and an approximation argument, here we obtain an existence result for the superfluid film equation in Plasma Physics and for the equation which models the self–channelling of a high–power ultrashort laser, which derive from our model problem by taking
 $l(s) = s$
, respectively
 $l(s) = \sqrt{1+s}$
, in the previous definition of
 $g^2(u)$
.
Citation: Anna Maria Candela, Caterina Sportelli. Soliton solutions for quasilinear modified Schrödinger equations in applied sciences. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022121
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700.  doi: 10.1016/0022-247X(78)90013-6. [3] H. Berestycki and P.-L. Lions, Nonlinear scalar field equation Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [4] A. M. Candela and G. Palmieri, Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.  doi: 10.1515/ans-2006-0209. [5] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dynam. Syst., Suppl. 2009 (2009), 133-142. [6] A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$–linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 72, 39 pp. doi: 10.1007/s00526-017-1170-4. [7] A. M. Candela, G. Palmieri and A. Salvatore, Positive solutions of modified Schrödinger equations on unbounded domains, submitted. [8] A. M. Candela and A. Salvatore, Existence of radial bounded solutions for some quasilinear elliptic equations in $\mathbb R^N$, Nonlinear Anal., 191 (2020), 111625, 26 pp. doi: 10.1016/j.na.2019.111625. [9] A. M. Candela and A. Salvatore, Positive solutions for some generalized $p$–Laplacian type problems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1935-1945.  doi: 10.3934/dcdss.2020151. [10] A. M. Candela, A. Salvatore and C. Sportelli, Bounded solutions for quasilinear modified Schrödinger equation, submitted. [11] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008. [12] 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.  doi: 10.1007/s002200050191. [13] J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372.  doi: 10.1016/j.na.2006.10.018. [14] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508. [15] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. [16] 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.  doi: 10.1063/1.525675. [17] J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5. [18] J. Q. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2002), 441-448.  doi: 10.1090/S0002-9939-02-06783-7. [19] V. G. Makhan'kov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6. [20] 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.  doi: 10.1007/s005260100105. [21] Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005. [22] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700.  doi: 10.1016/0022-247X(78)90013-6. [3] H. Berestycki and P.-L. Lions, Nonlinear scalar field equation Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [4] A. M. Candela and G. Palmieri, Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.  doi: 10.1515/ans-2006-0209. [5] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dynam. Syst., Suppl. 2009 (2009), 133-142. [6] A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$–linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 72, 39 pp. doi: 10.1007/s00526-017-1170-4. [7] A. M. Candela, G. Palmieri and A. Salvatore, Positive solutions of modified Schrödinger equations on unbounded domains, submitted. [8] A. M. Candela and A. Salvatore, Existence of radial bounded solutions for some quasilinear elliptic equations in $\mathbb R^N$, Nonlinear Anal., 191 (2020), 111625, 26 pp. doi: 10.1016/j.na.2019.111625. [9] A. M. Candela and A. Salvatore, Positive solutions for some generalized $p$–Laplacian type problems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1935-1945.  doi: 10.3934/dcdss.2020151. [10] A. M. Candela, A. Salvatore and C. Sportelli, Bounded solutions for quasilinear modified Schrödinger equation, submitted. [11] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008. [12] 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.  doi: 10.1007/s002200050191. [13] J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372.  doi: 10.1016/j.na.2006.10.018. [14] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508. [15] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. [16] 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.  doi: 10.1063/1.525675. [17] J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5. [18] J. Q. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2002), 441-448.  doi: 10.1090/S0002-9939-02-06783-7. [19] V. G. Makhan'kov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6. [20] 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.  doi: 10.1007/s005260100105. [21] Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005. [22] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.
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