含参数拟线性非齐次椭圆型方程的多重解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 286-296.

含参数拟线性非齐次椭圆型方程的多重解

宋洪雪^1,^2,*(),魏云峰^2,³

¹ 南京邮电大学理学院南京 210023
² 河海大学理学院南京 210098
³ 南京审计大学统计与数学学院南京 211815

收稿日期:2018-01-15 出版日期:2019-04-26 发布日期:2019-05-05
通讯作者: 宋洪雪 E-mail:songhx@njupt.edu.cn
基金资助:
国家自然科学基金(61503198);中国博士后科学基金面上项目(2017M611664);南京邮电大学校级科研基金(NY217092);南京邮电大学校级科研基金(NY218076)

Multiple Solutions for Quasilinear Nonhomogeneous Elliptic Equations with a Parameter

Hongxue Song^1,^2,*(),Yunfeng Wei^2,³

¹ School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023
² College of Science, Hohai University, Nanjing 210098
³ School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815

Received:2018-01-15 Online:2019-04-26 Published:2019-05-05
Contact: Hongxue Song E-mail:songhx@njupt.edu.cn
Supported by:
the NSFC(61503198);the China Postdoctoral Science Foundations(2017M611664);the NUPTSF(NY217092);the NUPTSF(NY218076)

摘要/Abstract

摘要：

该文研究如下形式的拟线性非齐次椭圆型方程

$ \begin{eqnarray*}\label{eq:0a1} -\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N}, \end{eqnarray*} $

其中$1 <p\leq N$ ($N\geq 3$), $\frac{1}{2} <\alpha\leq 1 $, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $h \in C(\mathbb{R}, \mathbb{R})$,而且扰动项$g\in L^{p'}(\mathbb{R}^{N})$,这里$p'=\frac{p}{p-1}$.利用变量代换结合极小极大方法可以证明该问题存在多重解.

关键词: 拟线性椭圆型方程, Ekeland变分原理, P-S序列

Abstract:

In this paper, we study the following quasilinear nonhomogeneous elliptic equations of the form

$-\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N}, $

where $1 <p\leq N$ ($N\geq 3$), $\frac{1}{2} <\alpha\leq 1 $, $V\in C(\mathbb{R}^{N}, \mathbb{R})$, $h \in C(\mathbb{R}, \mathbb{R})$ and $g\in L^{p'}(\mathbb{R}^{N})$, where $p'=\frac{p}{p-1}$, is a disturbance term. Using a variable replacement and minimax method, we show the existence and multiplicity of solutions to this problem.

Key words: Quasilinear elliptic equations, Ekeland's variational principle, Palais-Smale sequences

中图分类号:

O175.25

宋洪雪,魏云峰. 含参数拟线性非齐次椭圆型方程的多重解[J]. 数学物理学报, 2019, 39(2): 286-296.

Hongxue Song,Yunfeng Wei. Multiple Solutions for Quasilinear Nonhomogeneous Elliptic Equations with a Parameter[J]. Acta mathematica scientia,Series A, 2019, 39(2): 286-296.

参考文献

1	Adachi S , Watanabe T . Uniqueness of the ground state solutions of quasilinear Schr?dinger equations. Nonlinear Anal, 2012, 75: 819- 833
2	Bass F , Nasanov N N . Nonlinear electromagnetic spin waves. Phys Reports, 1990, 189: 165- 223
3	Berestycki H , Lions P L . Nonlinear scalar field equations I. Arch Ration Mech Anal, 1983, 82: 313- 346
4	Chabrowski J. Variational Methods for Potential Operator Equations. Berlin: Walter de Gruyter, 1997
5	Colin M , Jeanjean L . Solutions for a quasilinear Schr?dinger equation:a dual approach. Nonlinear Anal, 2004, 56: 213- 226
6	Ekeland I . Nonconvex minimization problems. Bull Amer Math Soc, 1979, 1: 443- 474
7	Figueiredo G M . Quasilinear equations with dependence on the gradient via Mountain Pass techniques in $\mathbb{R}^N$. Applied Mathematics and Computation, 2008, 203: 14- 18
8	Hasse R W . A general method for the solution of nonlinear soliton and kink Schr?dinger equation. Z Phys B, 1980, 37: 83- 87
9	Kurihura S . Large-amplitude quasi-solitons in superfluids. J Phys Soc Japan, 1981, 50: 3262- 3267
10	Laedke E , Spatschek K . Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1963, 24: 2764- 2769
11	李静, 陈才生. $\mathbb{R}^N$上一类拟线性Schr?dinger方程的Nehari流形解. 数学物理学报, 2016, 36A (3): 507- 520
11	Li J , Chen C . Nehari manifold method for solutions to a class of quasilinear Schr?dinger equations in $\mathbb{R}^N$. Acta Math Sci, 2016, 36A (3): 507- 520
12	Li Q , Yang Z D . Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic systems. J Partial Differ Equa, 2015, 28: 370- 382
13	Li Z X , Shen R T . Nonsmooth critical point theorems and its applications to quasilinear Schr?dinger equations. Acta Math Sci, 2016, 36B (1): 73- 86
14	Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case, part 1-2. Ann Inst H Poincaré, 1984, 1: 109-145, 223-283
15	Liu J Q , Wang Z Q . Soliton solutions for quasilinear Schr?dinger equations. Proc Amer Math Soc, 2003, 131: 441- 448
16	Liu J Q , Wang Y , Wang Z Q . Solutions for quasilinear Schr?dinger equations, Ⅱ. J Differential Equations, 2003, 187: 473- 493
17	do ó J M B , Severo U . Solitary waves for a class of quasilinear Schr?dinger equations in dimension two. Calc Var Partial Differential Equations, 2010, 38: 275- 315
18	do ó J M B . Semilinear Dirichlet problems for the N-Laplacian in $\mathbb{R}^N$ with nonlinearities in critical growth range. Differential Integral Equations, 1996, 9: 967- 979
19	do ó J M B , Miyagaki O H , Soares S H M . Soliton solutions for quasilinear Schr?dinger equations with critical growth. J Differential Equations, 2010, 248: 722- 744
20	Poppenberg M , Schmitt K , Wang Z Q . On the existence of soliton solutions to quasilinear Schr?dinger equations. Calc Var Partial Differential Equations, 2002, 14: 329- 344
21	Ritchie B . Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E, 1994, 50: 687- 689
22	Schechter M. Linking Methods in Critical Point Theory. Boston: Birkh?user, 1999
23	Severo U . Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian. Elec J Diff Equa, 2008, 56: 1- 16
24	Silva E B , Vieira G F . Quasilinear asymptotically periodic Schr?dinger equations with subcritical growth. Nonlinear Anal, 2010, 72: 2935- 2949
25	Tang X H . New super-quadratic conditions on ground state solutions for superlinear Schr?dinger equation. Adv Nonlinear Stud, 2014, 14: 349- 361
26	Wu X . Multiple solutions for quasilinear Schr?dinger equations with a parameter. J Differential Equations, 2014, 256: 2619- 2632
27	Xiang C L . Gradient estimates for solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential. Acta Math Sci, 2017, 37B (1): 58- 68
28	Yang H W , Zhu B , Chen J F . Hamiltonian analysis and dual vector spectral elements for 2D Maxwell eigenproblems. Commun Comput Phys, 2017, 21: 515- 525
29	Yang H W , Xu Z H , Yang D Z , et al. ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect. Adv Differ Equa, 2016, 167: 1- 22

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