QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

doi:10.1007/s10473-022-0310-x

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 975-1002.doi: 10.1007/s10473-022-0310-x

QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

Edcarlos D. SILVA, Jefferson S. SILVA

Federal University of Goiás, Zip code 74001-970, Goiânia, Goiás-GO, Brazil

收稿日期:2020-09-09 修回日期:2020-10-14 出版日期:2022-06-26 发布日期:2022-06-24
通讯作者: Edcarlos D. SILVA,E-mail:edcarlos@ufg.br E-mail:edcarlos@ufg.br
基金资助:
Research was partially supported by CNPq with (429955/2018-9). The first author was partially suported by CNPq (309026/2020-2) and FAPDF with (16809.78.45403.25042017).

QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

Edcarlos D. SILVA, Jefferson S. SILVA

Federal University of Goiás, Zip code 74001-970, Goiânia, Goiás-GO, Brazil

Received:2020-09-09 Revised:2020-10-14 Online:2022-06-26 Published:2022-06-24
Contact: Edcarlos D. SILVA,E-mail:edcarlos@ufg.br E-mail:edcarlos@ufg.br
Supported by:
Research was partially supported by CNPq with (429955/2018-9). The first author was partially suported by CNPq (309026/2020-2) and FAPDF with (16809.78.45403.25042017).

摘要/Abstract

摘要： It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

关键词: Quasilinear Schrödinger equations, linking theorems, superlinear elliptic equations, critical nonlinearities, Bounded potentials

Abstract: It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

Key words: Quasilinear Schrödinger equations, linking theorems, superlinear elliptic equations, critical nonlinearities, Bounded potentials

中图分类号:

35J20

Edcarlos D. SILVA, Jefferson S. SILVA. QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES[J]. 数学物理学报, 2022, 42(3): 975-1002.

Edcarlos D. SILVA, Jefferson S. SILVA. QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES[J]. Acta mathematica scientia,Series A, 2022, 42(3): 975-1002.

参考文献 39

[1]	Alves C O, Wang Y, Shen Y. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differential Equations, 2015, 259:318-343
[2]	Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on RN. Comm Part Diff Eq, 1995, 20:1725-1741
[3]	de Bouard A, Hayashi N, Saut J C. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm Math Phys, 1997, 189:73-105
[4]	Brull L, Lange H. Solitary waves for quasilinear Schrödinger equations. Expo Math, 1986, 4:278-288
[5]	Chen X L, Sudan R N. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys Rev Lett, 1993, 70:2082-2085
[6]	Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation:a dual approach. Nonlinear Anal, 2004, 56:213-226
[7]	Del Pino M, Felmer P. Local Mountain Pass for semilinear elliptic problems in unbounded domains. Calc Var Partial Differential Equations, 1996, 4:121-137
[8]	Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differential Equations, 2015, 258:115-147
[9]	Furtado M F, Silva E D, Silva M L. Quasilinear Schrödinger equations with asymptotically linear nonlinearities. Adv Nonlinear Stud, 2014, 14:671-686
[10]	Furtado M F, Silva E D, Silva M L. Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin. Z Angew Math Phys, 2015, 66:277-291
[11]	Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys, 1980, 37:83-87
[12]	Kosevich A M, Ivanov B, Kovalev A S. Magnetic solitons. Phys Rep, 1990, 194:117-238
[13]	Kurihura S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan, 1981, 50:3263-3267
[14]	Landau L D, Lifschitz E M. Quantum Mechanics, Non-relativistic Theory. Institute of Physical Problems URSS, Academy of Sciences, 1958
[15]	Li Q, Wu X. Existence, multiplicity and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys, 2017, 58:041501
[16]	Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1:109-145; 223-283
[17]	Litvak A G, Sergeev A M. One dimensional collapse of plasma waves. JETP Lett, 1978, 27:517-520
[18]	Liu J Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc, 2002, 131:441-448
[19]	Liu J Q. Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations II. J Differential Equations, 2003, 187:473-493
[20]	Liu J Q, Wang Y Q, Wang Z Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29:879-901
[21]	Liu X, Liu J, Wang Z Q. Ground states for quasilinear Schrödinger equations with critical growth. Calc Var Partial Differential Equations, 2013, 46:641-669
[22]	Liu S, Zhou J. Standing waves for quasilinear Schrödinger equations with indefinite potentials. Journal of Differential Equations, 2018, 265:3970-3987
[23]	Makhankov V G, Fedyanin V K. Non-linear effects in quasi-one-dimensional models of condensed matter theory. Physics Reports, 1984, 104:1-86
[24]	Nakamura A. Damping and modification of exciton solitary waves. J Phys Soc Jpn, 1977, 42:1824-1835
[25]	do Ó J M, Severo U B. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun Pure Appl Anal, 2009, 8:621-644
[26]	do Ó J M, Miyagaki O H, Soares S H. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248:722-744
[27]	Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutions to quasilinear Schrödinger equations. Calc Var Partial Differential Equations, 2002, 14:329-344
[28]	Rabinowitz P. Minimax methods in critical point theory with applications to differential equations. CBMS Reg Conf Ser Math. Vol 65. Providence RI:Amer Math Soc, 1986
[29]	Rabinowitz P. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43:270-291
[30]	Schechter M, Tintarev K. Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems. Bull Soc Math Belg Ser B, 1992, 44:249-261
[31]	Schechter M. Linking methods in critical point theory. Boston, MA:Birkhäuser Boston, Inc, 1999
[32]	Schechter M. A variation of the mountain pass lemma and applications. J London Math Soc, 1991, 44(2):491-502
[33]	Silva E A B. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal, 1991, 16:455-477
[34]	Silva E A B, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with subcritical growth. Nonlinear Analysis, 2010, 72:2935-2949
[35]	Silva E A B, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Cal Var, 2010, 39:1-33
[36]	Silva E D, Silva J S. Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues. J Math Phys, 2019, 60:081504
[37]	Silva E D, Silva J S. Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure. Discrete Continuous Dynamical Systems-A, 2020, 40(9):5441-5470
[38]	Souto M A S, Soares S H M. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Commun Pure Appl Anal, 2013, 12(1):99-116
[39]	Willem M. Minimax Theorems. Basel, Berlin:Birkhäuser Boston, 1996

QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

摘要/Abstract

引用本文

使用本文

参考文献 39

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	李奇, 向长林. A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION[J]. 数学物理学报, 2022, 42(3): 1113-1124.
[2]	谭金岚, 李勇勇, 唐春雷. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL[J]. 数学物理学报, 2022, 42(3): 1125-1140.
[3]	徐嘉, 代国伟. 带p-Laplacian 扰动对称泛函无穷多个临界点的存在性[J]. 数学物理学报, 2014, 34(6): 1532-1541.
[4]	吴元泽, 黄毅生, 刘增. SIGN-CHANGING SOLUTIONS FOR SCHRÖDINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS[J]. 数学物理学报, 2014, 34(3): 691-702.
[5]	曾小雨, 万俊霞. 含有共振项的p-Laplace方程解的存在性[J]. 数学物理学报, 2014, 34(2): 227-233.
[6]	代国伟, 王文婷, 冯丽丽. 非光滑型对偶喷泉定理及其在一个微分包含问题中的应用[J]. 数学物理学报, 2012, 32(3): 530-539.
[7]	裘良华. 体积元保持变换下的一类特殊预给定数量曲率问题[J]. 数学物理学报, 2011, 31(5): 1317-1322.
[8]	陈淑红, 郭柏灵. 费米子气体附近BCS-BEC跨越的Ginzburg-Landau方程组整体解的存在性[J]. 数学物理学报, 2011, 31(5): 1359-1368.
[9]	葛斌, 薛小平, 周庆梅. 一类具有非光滑位势p(x)-Kirchhoff 型问题多解性[J]. 数学物理学报, 2011, 31(5): 1431-1438.
[10]	高红亚, 赵丽芳. 关于广义Beltrami方程组[J]. 数学物理学报, 2010, 30(4): 953-958.
[11]	刘祥清; 黄毅生; 邓志颖. 双调和方程特征值问题[J]. 数学物理学报, 2009, 29(1): 57-69.
[12]	姚庆六. 一类非线性 Dirichlet 边值问题的正径向解[J]. 数学物理学报, 2009, 29(1): 48-56.
[13]	章国庆; 刘三阳. R²中带不定权与临界位势的非线性椭圆问题[J]. 数学物理学报, 2008, 28(5): 929-936.
[14]	刘祥清; 黄毅生. 一类带权函数的拟线性椭圆方程[J]. 数学物理学报, 2007, 27(2): 277-287.
[15]	何吉欢. Schrodinger方程的变分迭代解法[J]. 数学物理学报, 2001, 21(zk): 577-583.