摘要：
In this paper, we study the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. The method of upper and lower solutions pair is applied to establish the existence of traveling waves. We prove that the nature of nonlinear selection is to find a lower solution pair with the first species decaying in a faster rate. By novel constructions of various upper and lower solutions, we obtain a number of new results on the minimal speed determinacy. Numerical simulations are carried out to illustrate all of our discoveries. (C) 2020 Elsevier Inc. All rights reserved.
摘要：
In this paper, Lie symmetry analysis is performed for the equation derived from $(2+1)$-dimensional higher order Broer-Kaup equation. Meanwhile, the optimal system and similarity reductions based on the Lie group method are obtained. Furthermore, the conservation law is studied via the Ibragimov’s method.
作者机构：
[Ouyang, Zigen; Liu, Zhisu] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.;[Zhang, Jianjun] Chongqing Jiaotong Univ, Coll Math & Stat, Chongqing 400074, Peoples R China.;[Zhang, Jianjun] Univ Insubria, Dipartimento Sci & Alta Tecnol, Via GB Vico 46, I-21100 Varese, Italy.
通讯机构：
[Zhang, Jianjun] C;[Zhang, Jianjun] U;Chongqing Jiaotong Univ, Coll Math & Stat, Chongqing 400074, Peoples R China.;Univ Insubria, Dipartimento Sci & Alta Tecnol, Via GB Vico 46, I-21100 Varese, Italy.
关键词：
gauged Schrodinger equations;sign-changing solutions;invariant sets of descending flow
摘要：
We are concerned with sign-changing solutions of the following gauged nonlinear Schrodinger equation in dimension two including the so-called Chern-Simons term {-Delta u + omega u + (h2(vertical bar x vertical bar)/vertical bar x vertical bar(2) + integral(+infinity)(vertical bar x vertical bar) h(s)/s u(2)(s) ds) u = u = lambda vertical bar u vertical bar(p-2) u in R-2, u(x) = u(vertical bar x vertical bar) is an element of H-1(R-2), where omega, lambda > 0, p is an element of (4, 6) and h(s) = 1/2 integral(s)(0) tau u(2)(tau)d tau. Via a novel perturbation approach and the method of invariant sets of descending flow, we investigate the existence and multiplicity of sign-changing solutions. Moreover, energy doubling is established, i.e. the energy of sign-changing solution w(lambda) is strictly larger than twice that of the ground state energy for lambda > 0 large. Finally, for any sequence lambda(n) -> infinity as n -> infinity, up to a subsequence, lambda(1/p-2)(n) w(lambda n) -> w strongly in H-rad(1)(R-2) as n -> infinity, where w is a sign-changing solution of -Delta u + omega u = vertical bar u vertical bar(p-2)u, u is an element of H-rad(1) (R-2).
期刊：
Advances in Difference Equations,2018年2018(1):1-18 ISSN：1687-1847
通讯作者：
Liu, Hongliang
作者机构：
[Xiao, Qizhen; Liu, Hongliang; Ouyang, Zigen] Univ South China, Sch Math & Phys, Hengyang, Peoples R China.
通讯机构：
[Liu, Hongliang] U;Univ South China, Sch Math & Phys, Hengyang, Peoples R China.
关键词：
Biharmonic equation;Variational methods;High energy solutions;Concentration of solutions
摘要：
We consider the following nonlinear biharmonic equations:
$$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$
where
$V_{\lambda }(x)$
is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.
期刊：
TAIWANESE JOURNAL OF MATHEMATICS,2017年21(2):403-428 ISSN：1027-5487
通讯作者：
Chen, Huiwen
作者机构：
[Chen, Huiwen; Ouyang, Zigen] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.;[He, Zhimin] Cent S Univ, Sch Math & Stat, Changsha 410083, Hunan, Peoples R China.;[Li, Jianli] Hunan Normal Univ, Dept Math, Changsha 410081, Hunan, Peoples R China.
通讯机构：
[Chen, Huiwen] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
Homoclinic solutions;Discrete Hamiltonian systems;Asymptotically quadratic;Supquadratic;Critical point theory;Variational methods
摘要：
In this paper, we deal with the second order discrete Hamiltonian system Delta[p(n) Delta u (n - 1)]- L(n)u(n) + Delta W(n; u ( n)) = 0, where L : Z -> R (NxN) is positive de fi nite for su ffi ciently large vertical bar n vertical bar epsilon Z and W(n; x) is inde fi nite sign. By using critical point theory, we establish some new criteria to guarantee that the above system has in fi nitely many nontrivial homoclinic solutions under the assumption that W(n; x) is asymptotically quadratic and supquadratic, respectively. Our results generalize and improve some existing results in the literature.
作者机构：
[Yang, Liu] Hengyang Normal Univ, Dept Math & Comp Sci, Hengyang 421008, Hunan, Peoples R China.;[Ouyang, Zigen; Liu, Zhisu] Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
通讯机构：
[Liu, Zhisu] U;Univ South China, Sch Math & Phys, Hengyang 421001, Hunan, Peoples R China.
关键词：
Multiplicity;Kirchhoff type equation;Critical growth
摘要：
In this paper, we study the following Kirchhoff type equation with critical growth {-(a + b integral(Omega) vertical bar del u vertical bar(2)dx) del u = lambda u + mu vertical bar u vertical bar(2)u + vertical bar u vertical bar(4)u in Omega, u = 0 on partial derivative Omega, where a > 0, b >= 0 and Omega is a smooth bounded domain in R-3. When the real parameter mu is larger than some positive constant, we investigate the multiplicity of nontrivial solutions for the above problem with parameter lambda belonging to a left neighborhood of the Dirichlet eigenvalue of the Laplacian operator -Delta. (C) 2016 Elsevier Ltd. All rights reserved.
