### Existence theory for quasilinear elliptic equations via a regularization approach

DOI: http://dx.doi.org/10.12775/TMNA.2017.008

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#### References

A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on R, Discrete Contin. Dyn. Syst. 9 (2003), 55–68.

D. Arcoya and L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, Nonlinear Differential Equations Appl. 6 (1999), 79–100.

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

H. 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.

Y. Brihaye and B. Hartmann, Solitons on nanotubes and fullerenes as solutions of a modified nonlinear Schrödinger equation, Advances in Soliton Research, 135–151, Nova Sci. Publ., Hauppauge, 2006.

L. Brizhik, A. Eremko, B. Piette and W.J. Zakrzewski, Static solutions of a Ddimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481–1497.

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Expos. Math. 4 (1986), 278–288.

A. Canino and M. Degiovanni, Nonsmooth critical point theory and quasilinear elliptic equations, Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472, 1995, 1–50.

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, Courant Institute of Mathematical Sciences, Providence, 2005.

T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561.

J. Chen and B. Guo, Blow-up and strong instability result for a quasilinear Schrödinger equation, Appl. Math. Model 33 (2009), 4192–4200.

J. Chen, Y. Li and Z.-Q. Wang, Stability of standing waves for a class of quasilinear Schrödinger equations, European J. Appl. Math. 23 (2012), 611–633.

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 Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213–226.

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasilinear Schrödinger equations, Nonlinearity 23 (2010), 1353–1358.

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.

Y. Deng, S. Peng and S. Yan, Solitary wave solutions to a quasilinear Schrödinger equation modeling the self-channeling of a high-power ultrashort laser in matter, preprint.

B. Hartmann and W.J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B 68 (2003), 184–302.

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

I.D. Iliev and K.P. Kirchev, Stability and instability of solitary waves for onedimensional singular Schrödinger equations, Differential Integral Equations 6 (1993), 685–703.

C.E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasilinear Schrödinger equations, Invent. Math. 158 (2004), 343–388.

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons, Phys. Rep. 194 (1990), 117–238.

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

H. Lange, M. Poppenperg and H. Teismann, Nash–Moser methods for the solutions of quasilinear Schrödinger equations, Comm. Partial Differential Equations 24 (1999), 1399–1418.

J. Liu, X. Liu and Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations 39 (2014), 2216–2239.

J. Liu, Y. Wang and Z.-Q. Wang, Soliton Solutions for Quasilinear Schrödinger Equations II, J. Differential Equations, 187 (2003), 473–493.

J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901.

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2003), 441–448.

J. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equation 257 (2014), 2874–2899.

X. Liu, J. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.

X. Liu, J. Liu and Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013), 102–124.

M. Poppenberg, On the local well posedness of quasilinear Schrödinger equations in arbitrary space dimension, J. Differential Equations 172 (2001), 83–115.

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calculus of Variations and PDEs 14 (2002), 329–344.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, 1986.

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E 50 (1994), 687–689.

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

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