Ground state solutions for a quasilinear elliptic equation with general critical nonlinearity

References

• Bouard De., Hayashi A, Saut N. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun Math Phys. 1997;189:73–105. doi: 10.1007/s002200050191
   Web of Science ®Google Scholar
• Repovš D. Stationary waves of Schrödinger-type equations with variable exponent. Anal Appl. 2015;13:645–661. doi: 10.1142/S0219530514500420
   Web of Science ®Google Scholar
• Laedke E, Spatschek K. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys. 1983;24:2764–2769. doi: 10.1063/1.525675
   Web of Science ®Google Scholar
• Quispel GRW, Capel HW. Equation of motion for the Heisenberg spin chain. Phys A. 1982;110:41–80. doi: 10.1016/0378-4371(82)90104-2
   Web of Science ®Google Scholar
• Brandi H, Manus C, Mainfry G, et al. Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys Fluids. 1993;135:3539–3550. doi: 10.1063/1.860828
   Web of Science ®Google Scholar
• Jeanjean L, Tanaka K. A positive solution for a nonlinear Schrödinger equation on RN. Indiana Vniv Math J. 2005;54:443–464. doi: 10.1512/iumj.2005.54.2502
   Web of Science ®Google Scholar
• Poppenberg M, Schmitt K, Wang ZQ. On the existence of soliton solutions to quasilinear Schrödinger equations. Calculus Var Partial Differ Equa. 2002;14:329–344. doi: 10.1007/s005260100105
   Web of Science ®Google Scholar
• Kurihara S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Jpn. 1981;50:3262–3267. doi: 10.1143/JPSJ.50.3262
   Google Scholar
• Chen XL, Sudan RN. Necessary and sufficient conditions for self-focusing of short ultraintense laster pulse in underdense plasma. Phys Rev Lett. 1993;70:2082–2085. doi: 10.1103/PhysRevLett.70.2082
   PubMed Web of Science ®Google Scholar
• Che GF, Chen HB. Existence of multiple nontrivial solutions for a class of generalized quasilinear Schrödinger equations on RN. Bull Belg Math Soc Simon Sterin. 2018;25:39–53. doi: 10.36045/bbms/1523412051
   Google Scholar
• Chen JH, Tang XH, Cheng BT. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Comm Pure Appl Anal. 2019;18:493–517. doi: 10.3934/cpaa.2019025
   Web of Science ®Google Scholar
• Zhang J, Lin XY, Tang XH. Ground states solutions for a quasilinear Schrödinger equation. Mediterr J Math. 2017;14:84. DOI:10.1007/s00009-016-0816-3.
   Web of Science ®Google Scholar
• Xu LP, Chen HB. Ground state solutions for quasilinear Schrödinger equations via Pohozaev manifold in Orlicz space. J Diff Equa. 2018;265:4417–4441. doi: 10.1016/j.jde.2018.06.009
   Web of Science ®Google Scholar
• Chen ST, Tang XH. Existence of ground state solutions for quasilinear Schrödinger equations with variable potentials and almost necessary nonlinearities. Elec J. Diff Equa. 2018;2018:1–13. doi: 10.1186/s13662-017-1452-3
   Web of Science ®Google Scholar
• Fang XD. Positive solutions for quasilinear Schrödinger equations in RN. Comm Pure Appl Anal. 2017;16:1603–1615. doi: 10.3934/cpaa.2017077
   Google Scholar
• Liu XN, Chen HB. Positive solutions for a class of quasilinear Schrödinger equations with vanishing potentials. Bound Value Probl. 2017;35. DOI:10.1186/s13661-017-0769-x.
   Google Scholar
• Huang WT, Xiang JL. Soliton solutions for a quasilinear Schrödinger equations with critical exponent. Comm Pur Appl Anal. 2016;15:1309–1333. doi: 10.3934/cpaa.2016.15.1309
   Web of Science ®Google Scholar
• Liu XQ, Liu JQ, Wang ZQ. Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc. 2013;141:253–263. doi: 10.1090/S0002-9939-2012-11293-6
   Web of Science ®Google Scholar
• Chu CM, Liu HD. Existence of positive solutions for a quasilinear Schrödinger equation. Nonlinear Anal: Real World Appl. 2018;44:118–127. doi: 10.1016/j.nonrwa.2018.04.007
   Web of Science ®Google Scholar
• Liang ZP, Gao JF, Li AR. Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term. Appl Math Lett. 2019;89:22–27. doi: 10.1016/j.aml.2018.09.015
   Web of Science ®Google Scholar
• Kristály A, Repovš D. On the Schrödinger-Maxwell system involving sublinear terms. Nonliear Anal: Real World Appl. 2012;13:213–223.
   Web of Science ®Google Scholar
• Papageorgiou NS, Rǎdulescu VD, Repovš DD. Nonliear analysis-theory and methods, Springer Monographs in Mathematics, Springer Nature, Cham, 2019.
   Google Scholar
• Shen YT, Wang YJ. Standing waves for a class of quasilinear Schrödinger equations. Comp Var Ellip Equ. 2016;61:817–842. doi: 10.1080/17476933.2015.1119818
   Web of Science ®Google Scholar
• Shen YT, Wang YJ. Standing waves for a relativistic quasilinear asymptotically Schrödinger equation. Appl Anal. 2016;95:2553–2564. doi: 10.1080/00036811.2015.1100296
   Web of Science ®Google Scholar
• Shi HX, Chen HB. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Comm Pure Appl Anal. 2018;17:53–66. doi: 10.3934/cpaa.2018004
   Web of Science ®Google Scholar
• Shi HX, Chen HB. Positive solutions for generalized quasilinear Schrödinger equations with general vanishing at infinity. Appl Math Lett. 2016;61:137–142. doi: 10.1016/j.aml.2016.06.004
   Web of Science ®Google Scholar
• Chen JH, Tang XH, Cheng BT. Non-nehari manifold for a class of generalized quasilinear Schrödinger equations. Comm Pure Appl Anal. 2017;74:20–26.
   Google Scholar
• Chen JH, Tang XH, Cheng BT. Ground states for a class of generalized quasilinear Schrödinger equations in RN. Mediterr J Math. 2017;14:190. doi: 10.1007/s00009-017-0990-y
   Google Scholar
• Deng YB, Peng SJ, Yan SS. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Diff Equ. 2015;260:115–147. doi: 10.1016/j.jde.2014.09.006
   Web of Science ®Google Scholar
• Deng YB, Huang WT, Zhang S. Ground states solutions for quasilinear Schrödinger equations with critical growth and lower power subcritical perturbation. Adv Nonlinear Stud. 2019;19:219–237. doi: 10.1515/ans-2018-2029
   Web of Science ®Google Scholar
• Zhang J, Zou WM. The critical case for a Berestycki-Lions theorem. Sci China Math. 2014;57:541–554. doi: 10.1007/s11425-013-4687-9
   Web of Science ®Google Scholar
• Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on RN. Proc R Soc Edinburgh Sect A. 1999;129:787–809. doi: 10.1017/S0308210500013147
   Web of Science ®Google Scholar
• Liu JQ, Wang YQ, Wang ZQ. Soliton solutions for quasilinear Schrödinger equations. II. J Diff Equ. 2003;187:473–493. doi: 10.1016/S0022-0396(02)00064-5
   Web of Science ®Google Scholar
• Willem M. Minimax theorems. Nonlinear Differ Equ Appl Birkhäuser Berlin. 1996;24:1–159.
   Google Scholar
• Lions PL. The concentration-compactness principle in the calculus of variations: the locally case. Part II. Ann Inst H Poincare Anal Non-Lineaire. 1984;1:223–283. doi: 10.1016/S0294-1449(16)30422-X
   Google Scholar
• Zhu XP, Cao DM. The concentration-compactness principle in nonlinear elliptic equations. Acta Math Sci. 1989;9:307–328. doi: 10.1016/S0252-9602(18)30356-4
   Web of Science ®Google Scholar
• Jeanjean L, Tanaka K. A remark on least energy solutions in RN. Proc Amer Math Soc. 2003;131:2399–2408. doi: 10.1090/S0002-9939-02-06821-1
   Web of Science ®Google Scholar
• Garcia Azorero JP, Peral I. Alonso, Hardy inequalities and some critical elliptic and parabolic problems. J Differ Equ. 1998;144:441–476. doi: 10.1006/jdeq.1997.3375
   Web of Science ®Google Scholar