Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.29
Duan Wu, P. Niu
In this note, we prove the boundary and global C 1 , γ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations on a convex polyhedron by perturbation and iteration techniques.
本文用微扰和迭代技术证明了凸多面体上完全非线性均匀椭圆型方程黏性解的边界和全局c1， γ正则性。
{"title":" C 1 , γ ","authors":"Duan Wu, P. Niu","doi":"10.14232/ejqtde.2022.1.29","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.29","url":null,"abstract":"In this note, we prove the boundary and global C 1 , γ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations on a convex polyhedron by perturbation and iteration techniques.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.3
Fei Fang, Binlin Zhang
In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].
{"title":"Global existence and blow-up for semilinear parabolic equation with critical exponent in \u0000 \u0000 \u0000 R\u0000 \u0000 N","authors":"Fei Fang, Binlin Zhang","doi":"10.14232/ejqtde.2022.1.3","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.3","url":null,"abstract":"In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.28
B. Satco
In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.
{"title":"Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations","authors":"B. Satco","doi":"10.14232/ejqtde.2022.1.28","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.28","url":null,"abstract":"In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.9
A. Slavík
We study the asymptotic behavior of solutions to the multidimensional diffusion (heat) equation with continuous time and discrete space. We focus on initial-value problems with bounded initial data, and provide sufficient conditions for the existence of pointwise and uniform limits of solutions.
{"title":"Asymptotic behavior of solutions to the multidimensional semidiscrete diffusion equation","authors":"A. Slavík","doi":"10.14232/ejqtde.2022.1.9","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.9","url":null,"abstract":"We study the asymptotic behavior of solutions to the multidimensional diffusion (heat) equation with continuous time and discrete space. We focus on initial-value problems with bounded initial data, and provide sufficient conditions for the existence of pointwise and uniform limits of solutions.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.6
J. Cid, Rodrigo López Pouso, Jorge Rodríguez–López
In this paper, we present some uniqueness results for systems of ordinary differential equations. All of them are linked by a weak transversality condition at the initial condition, which generalizes those in the previous literature. Several examples are also provided to illustrate our results.
{"title":"Uniqueness criteria for ordinary differential equations with a generalized transversality condition at the initial condition","authors":"J. Cid, Rodrigo López Pouso, Jorge Rodríguez–López","doi":"10.14232/ejqtde.2022.1.6","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.6","url":null,"abstract":"In this paper, we present some uniqueness results for systems of ordinary differential equations. All of them are linked by a weak transversality condition at the initial condition, which generalizes those in the previous literature. Several examples are also provided to illustrate our results.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.43
Equations V. H. Linh, N. Nga, N. Tuan
This paper is concerned with the asymptotic behavior of solutions of linear differential-algebraic equations (DAEs) under small nonlinear perturbations. Some results on the asymptotic behavior of solutions which are well known for ordinary differential equations are extended to DAEs. The main tools are the projector-based decoupling and the contractive mapping principle. Under certain assumptions on the linear part and the nonlinear term, asymptotic behavior of solutions are characterized. As the main result, a Perron type theorem that establishes the exponential growth rate of solutions is formulated.
{"title":"Asymptotic behaviour of solutions of quasilinear\u0000 differential-algebraic equations","authors":"\t\tEquations\t\t\tV. H. Linh, N. Nga, N. Tuan","doi":"10.14232/ejqtde.2022.1.43","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.43","url":null,"abstract":"This paper is concerned with the asymptotic behavior of solutions of linear differential-algebraic equations (DAEs) under small nonlinear perturbations. Some results on the asymptotic behavior of solutions which are well known for ordinary differential equations are extended to DAEs. The main tools are the projector-based decoupling and the contractive mapping principle. Under certain assumptions on the linear part and the nonlinear term, asymptotic behavior of solutions are characterized. As the main result, a Perron type theorem that establishes the exponential growth rate of solutions is formulated.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88191435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
本文证明了齐次空间(rn， ρ， wdx)的有界域Ω∧R n, n≥1上的下列加权Sobolev不等式，在合适的相容条件下，正权函数(v, w， Ω 1， Ω 2，…，Ω n)和拟度量ρ，(∫Ω | f | q v w d z) 1 q≤C∑i = 1 N(∫Ω | f z i | p Ω i M S w d z) 1 p, f∈L i p 0 (Ω¯)，式中q≥p >1, ms为强极大算子。给出了非一致退化梯度不等式的若干推论。
{"title":"Sobolev inequality with non-uniformly degenerating gradient","authors":"F. Mamedov, S. Monsurrò","doi":"10.14232/ejqtde.2022.1.24","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.24","url":null,"abstract":"<jats:p>In this paper we prove the following weighted Sobolev inequality in a bounded domain <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:math>, of a homogeneous space <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math>, under suitable compatibility conditions on the positive weight functions <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> and on the quasi-metric <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>q</mml:mi> </mml:msup> <mml:mi>v</mml:mi> <mml:mspace width=\"thinmathspace\" /> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> ","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.30
Ling Ran, Shang-Jie Chen, Lin Li
<jats:p>In this article, we study the following degenerated Schrödinger equations: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>γ<!-- γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>λ<!-- λ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true" /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math>
在这篇文章中，我们研究了在薛定谔均等的追随者:{ − Δ γ u + λ V ( x ) u = f ( x ,在R N中，u ∈ E λ , 在λ> 0是一个参数,Δ γ是a degenerate elliptic接线员,潜在的V (x)具有潜在的潜力，而非线性f (x, u)在你的无限中都是超线性或次线性的。
{"title":"The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well","authors":"Ling Ran, Shang-Jie Chen, Lin Li","doi":"10.14232/ejqtde.2022.1.30","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.30","url":null,"abstract":"<jats:p>In this article, we study the following degenerated Schrödinger equations: <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>γ<!-- γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>λ<!-- λ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> ","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.14232/ejqtde.2022.1.36
Xiao-Feng Ke, Jia‐Feng Liao
In this paper, the existence of periodic solutions to the second order Hamiltonian systems is investigated. By introducing a new growth condition which generalizes the Ambrosetti–Rabinowitz condition, we prove a existence result of nontrivial $T$-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the potential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions.
{"title":"On the existence of periodic solutions to second order Hamiltonian systems","authors":"Xiao-Feng Ke, Jia‐Feng Liao","doi":"10.14232/ejqtde.2022.1.36","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.36","url":null,"abstract":"In this paper, the existence of periodic solutions to the second order Hamiltonian systems is investigated. By introducing a new growth condition which generalizes the Ambrosetti–Rabinowitz condition, we prove a existence result of nontrivial $T$-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the potential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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