Ground State Solutions of Nehari-Pohozaev Type for a Class of Reaction-Diffusion System

Chen Peng,

College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Hubei Yichang 443002

 基金资助: 湖北省教育厅重点项目.  D20161206

 Fund supported: the Key Projects of Hubei Provincial Department of Education.  D20161206

Abstract

In this paper, we consider a class of nonlinear reaction diffusion systems, by using the non Nehari manifold method in strongly indefinite functional theory, a more direct and simple method to prove the ground state solution is given when the nonlinear term is superlinear. The existence of Nehari pankov type ground state solution is proved without strict monotone condition, and some new results are obtained.

Keywords： Reaction-diffusion system ; Ground states ; Strongly indefinite functional

Chen Peng. Ground State Solutions of Nehari-Pohozaev Type for a Class of Reaction-Diffusion System. Acta Mathematica Scientia[J], 2021, 41(5): 1347-1356 doi:

1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} \partial_t u-\Delta_x u+b(t, x)\cdot \nabla_x u+V(x)u = H_v(t, x, u, v), \\ - \partial_t v-\Delta_x v-b(t, x)\cdot \nabla_x v+V(x)v = H_u(t, x, u, v), \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} \Phi(z) = \frac{1}{2}(\|z^+\|^2-\|z^-\|^2)-\Psi(z), \end{eqnarray}$

$\begin{eqnarray} \langle \Phi'(z), \zeta\rangle = (z^+, \zeta^+)-(z^-, \zeta^-)-\int_{{{\Bbb R}}\times{{\Bbb R}}^N}H_z(t, x, z)\cdot \zeta, \end{eqnarray}$

$\begin{eqnarray} \langle \Phi'(z), z\rangle = \|z^+\|^2-\|z^-\|^2-\langle \Psi'(z), z\rangle. \end{eqnarray}$

$z_0 = (u_0, v_0) \in E$是系统(1.1) 的解, 则$z_0 \in {\cal N}^-$, 其中

$\rm(1)$   $\sigma(A) = \sigma_e(A)$, 即$A$有本质谱;

$\rm(2)$   $\sigma(A)\subset {{\Bbb R}} \setminus {(-a, a)} $$\sigma (A) 关于原点对称; \rm(3) 设 0 \leq \mu_1\leq \cdots \leq \mu_l 是算子 L^2$$ \inf {\sigma_{ess}} L^2$下方的所有特征值, 则$\{\pm \mu_i^{1/2}, i = 1, $$\cdots , l\}$$ L$的所有特征值.

$\rm(i)$  $\psi\in C^1(X, {{\Bbb R}} )$下方有界且是弱序列下半连续;

$\rm(ii)$  $\psi'$弱序列连续;

$\rm(iii)$  存在$r>\rho>0$, $e\in X^+ $$\|e\| = 1 使得 其中 则存在常数 c\in [\kappa, \sup \varphi(Q)] 和序列 \{u_n\}\subset X 满足 以下, 为叙述方便, 记 E 上定义如下泛函 这里 \Psi(z) = \int_{{{\Bbb R}}\times {{\Bbb R}}^N} H(t, x, z) . 定理的假设保证了 \Phi\in C^1(E, {{\Bbb R}})$$ \Phi$是弱序列下半连续的. 进一步, $\Phi$的临界点对应于系统(1.1) 的解.

3 定理的证明

注意到, 由条件(H1) 知$H(t, x, z)\geq 0$. 结合Fatou引理, $\Psi$是非负的且弱下半连续的. 通过标准的方法易证$\Psi'$是弱序列连续的.

$\begin{eqnarray} \Phi(z)\geq \Phi(\theta z+\zeta)+\frac{1}{2}\|\zeta\|^2+\frac{1-\theta^2}{2} \langle \Phi'(z), z\rangle-\theta \langle \Phi'(z), \zeta\rangle. \end{eqnarray}$

由(1.2)–(1.4) 式与(H4) 可得

$\begin{eqnarray} \Phi(z)\geq \Phi(\theta z+\zeta). \end{eqnarray}$

$\begin{eqnarray} \Phi(z)\geq \frac{\theta^2}{2}\|z\|^2+\frac{1-\theta^2}{2}\langle \Phi'(z), z\rangle +\theta^2\langle \Phi'(z), z^-\rangle-\int_{{{\Bbb R}}\times {{\Bbb R}}^N}H(t, x, \theta z^+). \end{eqnarray}$

$\rm(i)$  存在$\rho>0$使得

$\rm(ii)$  对所有的$z\in {\cal N}^-, \|z^+\|\geq \max\{\|z^-\|, \sqrt{2m}\}$.

当非线性项$H(t, x, z)$满足超线性增长时, 这个证明是标准的, 参见文献[17].

容易看到, 引理3.5可由引理3.1和引理3.3直接推得.

$\begin{eqnarray} \Phi(z_n)\rightarrow c_*, \ \ \|\Phi'(z_n)\|(1+\|z_n\|)\rightarrow 0. \end{eqnarray}$

$\begin{eqnarray} \Phi(z_n)\rightarrow c\geq 0, \ \ \langle \Phi'(z_n), z_n^+\rangle\rightarrow 0, \ \ \langle \Phi'(z_n), z_n^-\rangle\rightarrow 0 \end{eqnarray}$

为了证明序列$\{z_n\}$的有界性, 采用反证法. 假设$\|z_n\|\rightarrow \infty$.$w_n = z_n/{\|z_n\|}$, 则$\|w_n\| = 1$. 由Sobolev嵌入定理可知, 存在常数$C_1 > 0$使得$\|w_n\|_2\leq C_1$. 如果

$\begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\sup\int_{{{\Bbb R}}\times{{\Bbb R}}^N}H(t, x, Rz_n^+/{\|z_n\|})\\ & = & \lim\limits_{n\rightarrow \infty}\sup\int_{{{\Bbb R}}\times{{\Bbb R}}^N}H(t, x, Rw_n^+)\\ &\leq & \lim\limits_{n\rightarrow \infty}\sup R^2\varepsilon\int_{{{\Bbb R}}\times{{\Bbb R}}^N}|w_n^+|^2+ \lim\limits_{n\rightarrow \infty}\sup R^pC_\varepsilon\int_{{{\Bbb R}}\times{{\Bbb R}}^N}|w_n^+|^2\\ &\leq &\varepsilon (RC_1)^2 = \frac{1}{4}. \end{eqnarray}$

$\theta_n = R/{\|z_n\|}$, 则由(3.3), (3.5)和(3.6) 式可得

$\zeta_n(x) = \xi_n(x + k_n)$, 则

$\begin{eqnarray} \int_{B_{1+\sqrt{N+1}}(0)}|\zeta_n^+|^2>\frac{\delta}{2}. \end{eqnarray}$

参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Bartsch T , Ding Y .

Homoclinic solutions of an infinite-dimensional Hamiltonian system

Math Z, 2002, 240, 289- 310

Clment P , Felmer P , Mitidieri E .

Homoclinic orbits for a class of infinite dimensional Hamiltonian systems

Ann Sc Norm Super Pisa, 1997, 24, 367- 393

De Figueiredo D , Ding Y .

Strongly indefinite functions and multiple solutions of elliptic systems

Trans Amer Math Soc, 2003, 355, 2973- 2989

De Figueiredo D , Felmer P L .

Trans Amer Math Soc, 1994, 343, 97- 116

Ding Y , Luan S , Willem M .

Solutions of a system of diffusion equations

J Fixed Point Theory Appl, 2007, 2, 117- 139

Ding Y . Variational Methods for Strongly Indefinite Problems. Singapore: World Scientific Press, 2008

Ding Y , Xu T .

Effect of external potentials in a coupled system of multi-component incongruent diffusion

Topol Method Nonl Anal, 2019, 54, 715- 750

Ding Y , Xu T .

Concentrating patterns of reaction-diffusion systems: a variational approach

Trans Amer Math Soc, 2007, 369, 97- 138

Gu L , Zhou H .

An improved fountain Theorem and its application

Adv Nonlinear Stud, 2016, 17 (4): 727- 738

Guo Y , Zeng X , Zhou H .

Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

Ann I H Poincare-AN, 2016, 33 (3): 809- 828

Itô S . Diffusion Equations. Providence, RI: American Mathematical Society, 1992

Vazquez J L. The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion//Bonforte M, Grillo G. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Berlin: Springer, 2017: 205-278

Kryszewski W , Szulkin A .

An infinite dimensional morse theorem with applications

Trans Amer Math Soc, 1997, 349, 3184- 3234

Li G , Szulkin A .

An asymptotically periodic Schrödinger equation with indefinite linear part

Commun Contemp Math, 2002, 4, 763- 776

Li G , Yang J .

Asymptotically linear elliptic systems

Comm Partial Differential Equations, 2004, 29, 925- 954

Nagasawa M . Schrödinger Equations and Diffusion Theory. Boston: Birkhäuser, 1993

Saad M , Gomez J .

Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel

Physica A, 2018, 509, 703- 716

Szulkin A , Weth T .

Ground state solutions for some indefinite problems

J Funct Anal, 2009, 257, 3802- 3822

Tang X , Chen S , Lin X , Yu J .

Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions

J Differ Equa, 2020, 268, 4663- 4690

Tang X .

Non-Nehari manifold method for superlinear Schrödinger equation

Taiwanese J Math, 2014, 18, 1957- 1979

Tang X, Chen S. Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differential Equations, 2017, 55, Article nomber: 110

Wang Z , Zhou H .

Radial sign-changing solution for fractional Schrodinger equation

Discrete Cont Dyn-A, 2016, 36 (1): 499- 508

Wei Y , Yang M .

Existence of solutions for a system of diffusion equations with spectrum point zero

Z Angew Math Phys, 2014, 65, 325- 337

Yang M , Shen Z , Ding Y .

On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities

Chinese Ann Math, 2011, 32B (1): 45- 58

Yang M .

Nonstationary homoclinic orbits for an infinite-dimensional Hamiltonian system

J Math Phys, 2010, 51, 102701

Zeng X , Zhang Y , Zhou H .

Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent

Commun Contemp Math, 2014, 16 (6): 1450034

Zhang J , Tang X , Zhang W .

Nonlinear Anal, 2014, 95, 1- 10

Zhao F , Ding Y .

On a diffusion system with bounded potential

Discrete Contin Dyn Syst, 2009, 23, 1073- 1086

Zhao L , Zhao F .

On ground state solutions for superlinear Hamiltonian elliptic systems

Z Angew Math Phys, 2013, 64, 403- 418

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