Abstract This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k {C}^{k} invariant center manifolds of the system with colored noise approximate that of the original system.
{"title":"Approximations of center manifolds for delay stochastic differential equations with additive noise","authors":"Longyu Wu, Jiaxin Gong, Juan Yang, J. Shu","doi":"10.1515/anona-2022-0301","DOIUrl":"https://doi.org/10.1515/anona-2022-0301","url":null,"abstract":"Abstract This article deals with approximations of center manifolds for delay stochastic differential equations with additive noise. We first prove the existence and smoothness of random center manifolds for these approximation equations. Then we show that the C k {C}^{k} invariant center manifolds of the system with colored noise approximate that of the original system.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43093064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e−z(−Δ)α2 {e}^{-z{left(-Delta )}^{frac{alpha }{2}}} for α>0,z∈C+ alpha gt 0,zin {{mathbb{C}}}^{+} . To this end, we establish the asymptotic estimates for P(z,x) Pleft(z,x) with z z satisfying 0<ω≤∣θ∣<π2 0lt omega le | theta | lt frac{pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H=(−Δ)α2+V H={left(-Delta )}^{tfrac{alpha }{2}}+V , where V V belongs to higher-order Kato class.
Abstract In this article, we study the following Kirchhoff equation: (0.1) − ( a + b ‖ ∇ u ‖ L 2 ( R 3 ) 2 ) Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 , -(a+bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2})Delta u+Vleft(| x| )u=fleft(u)hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}, where a , b > 0 a,bgt 0 , V V is a positive radial potential function, and f ( u ) fleft(u) is an asymptotically cubic term. The nonlocal term b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta u is 3-homogeneous in the sense that b ‖ ∇ t u ‖ L 2 ( R 3 ) 2 Δ ( t u ) = t 3 b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u bVert nabla tu{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta left(tu)={t}^{3}bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta u , so it competes complicatedly with the asymptotically cubic term f ( u ) fleft(u) , which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer k k , equation (0.1) has a radial nodal solution U k , 4 b {U}_{k,4}^{b} , which has exactly k + 1 k+1 nodal domains. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k , and for any sequence { b n } → 0 + , left{{b}_{n}right}to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges strongly to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R 3 ) {H}^{1}left({{mathbb{R}}}^{3}) , where U k , 4 0 {U}_{k,4}^{0} also has k + 1 k+1 nodal domains exactly and solves the classical Schrödinger equation: − a Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 . -aDelta u+Vleft(| x| )u=fleft(u)hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.
摘要本文研究了以下Kirchhoff方程:(0.1)−(a+b‖∇u‖l2 (R 3) 2) Δ u+V(∣x∣)u=f (u) In R 3， -(a+b Vertnabla{Vert ＿L}^{{2 }{}left ({{mathbb{R}}} ^{3})}^{2})Delta u+V left (| x|)u=f left (u) hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}} ^{3}，其中a,b > 0 a,b gt 0, V V是一个正径向势函数，f (u) f left (u)是一个渐近三次项。非局部项b‖∇u‖l2 (r2) 2 Δ u b Vertnabla{Vert ＿L}^{{2 }{}left ({{mathbb{R}}} ^{3})^}2 {}Delta u是3-齐次的，意思是b‖∇u‖l2 (r2) 2 Δ (r2) 2 Δ u b Vertnabla tu {Vert ＿L}^{{2 }{}left ({{mathbb{R}}} ^{3})}^{2 }Deltaleft (tu)={t}^{3b}Vertnabla u {Vert ＿L}^{{2}{}left ({{mathbb{R}}} ^3{)}^}2{}Delta u，所以它与渐近三次项f (u) f left (u)竞争很复杂，这与超三次情况完全不同。利用Miranda定理并对区域划分进行分类，通过粘接法和变分法证明了对于每一个正整数k k，方程(0.1)有一个径向节点解U k,4 b U k,{4^}b，它恰好有k+1个k+1个节点域。此外，我们证明了U k, 4b {U＿k},{4^}b的能量在k k中{是}严格递增的，并且对于任意序列b n{→0} +，{}{}left ｛{b＿n}{}right｝ to 0＿+，{直到}一{个子序列，U k, 4b n U＿k,4^}b＿n{在H 1 (R 3) H^1 }{}{{}{}}{}{}{}{}{}left ({{mathbb{R}}} ^3)中{强}收敛{于U k, 40 U＿k,4}^{b＿n在H 1 (R 3) H^1中也有k+1 k+1}节点{域，}并方程:−a Δ U + V(∣x∣)U = f (U)在R 3中。-a Delta u+V left (| x|)u=f left (u) hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}} ^3。我们的结果将Deng等人的结果从超立方情况扩展{到}渐近立方情况。
{"title":"Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term","authors":"Tao Wang, Yanling Yang, Hui Guo","doi":"10.1515/anona-2022-0323","DOIUrl":"https://doi.org/10.1515/anona-2022-0323","url":null,"abstract":"Abstract In this article, we study the following Kirchhoff equation: (0.1) − ( a + b ‖ ∇ u ‖ L 2 ( R 3 ) 2 ) Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 , -(a+bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2})Delta u+Vleft(| x| )u=fleft(u)hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}, where a , b > 0 a,bgt 0 , V V is a positive radial potential function, and f ( u ) fleft(u) is an asymptotically cubic term. The nonlocal term b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta u is 3-homogeneous in the sense that b ‖ ∇ t u ‖ L 2 ( R 3 ) 2 Δ ( t u ) = t 3 b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u bVert nabla tu{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta left(tu)={t}^{3}bVert nabla u{Vert }_{{L}^{2}left({{mathbb{R}}}^{3})}^{2}Delta u , so it competes complicatedly with the asymptotically cubic term f ( u ) fleft(u) , which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer k k , equation (0.1) has a radial nodal solution U k , 4 b {U}_{k,4}^{b} , which has exactly k + 1 k+1 nodal domains. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k , and for any sequence { b n } → 0 + , left{{b}_{n}right}to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges strongly to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R 3 ) {H}^{1}left({{mathbb{R}}}^{3}) , where U k , 4 0 {U}_{k,4}^{0} also has k + 1 k+1 nodal domains exactly and solves the classical Schrödinger equation: − a Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 . -aDelta u+Vleft(| x| )u=fleft(u)hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41354486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we develop a continuous periodic switching model depicting Wolbachia infection frequency dynamics in mosquito populations by releasing Wolbachia-infected mosquitoes, which is different from the discrete modeling efforts in the literature. We obtain sufficient conditions on the existence of a unique and exactly two periodic solutions and analyze the stability of each periodic solution, respectively. We also provide a brief discussion and several numerical examples to illustrate our theoretical results.
{"title":"Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model","authors":"Yantao Shi, Bo Zheng","doi":"10.1515/anona-2022-0297","DOIUrl":"https://doi.org/10.1515/anona-2022-0297","url":null,"abstract":"Abstract In this article, we develop a continuous periodic switching model depicting Wolbachia infection frequency dynamics in mosquito populations by releasing Wolbachia-infected mosquitoes, which is different from the discrete modeling efforts in the literature. We obtain sufficient conditions on the existence of a unique and exactly two periodic solutions and analyze the stability of each periodic solution, respectively. We also provide a brief discussion and several numerical examples to illustrate our theoretical results.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45611647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We deal with localized semiclassical states for singularly perturbed Kirchhoff-type equation as follows: − ε 2 a + ε b ∫ R 3 ∣ ∇ v ∣ 2 d x Δ v + V ( x ) v = P ( x ) f ( v ) , x ∈ R 3 , -left({varepsilon }^{2}a+varepsilon bmathop{int }limits_{{{mathbb{R}}}^{3}}| nabla v{| }^{2}{rm{d}}xright)Delta v+Vleft(x)v=Pleft(x)fleft(v),hspace{1em}xin {{mathbb{R}}}^{3}, where V , P ∈ C 1 ( R 3 , R ) V,Pin {C}^{1}left({{mathbb{R}}}^{3},{mathbb{R}}) and bounded away from zero. By applying the penalization approach together with the Nehari manifold approach in the studies of Szulkin and Weth, we obtain the existence of an infinite sequence of localized solutions of higher topological type. In addition, we also give a concrete set as the concentration position of these localized solutions. It is noted that, in our main results, f f only belongs to C ( R , R ) Cleft({mathbb{R}},{mathbb{R}}) and does not satisfy the Ambrosetti-Rabinowitz-type condition.
我们处理奇摄动kirchhoff型方程的局域半经典态:−ε 2a + ε b∫R 3∣∇v∣2d x Δ v + v (x) v = P (x) f (v)， x∈R 3， -left（{varepsilon }＾{2}a+varepsilon bmathop{int }limits＿{{{mathbb{R}}}＾{3.}}| nabla v{| }＾{2}{rm{d}}xright）Delta v+ vleft(x)v=Pleft(x)fleft(v);hspace{1em}xin {{mathbb{R}}}＾{3.}，其中V,P∈c1 (r3, R) V,Pin {c}＾{1}left（{{mathbb{R}}}＾{3.}，{mathbb{R}})，从零开始跳跃。将惩罚方法与Nehari流形方法一起应用于Szulkin和Weth的研究中，得到了一类高拓扑型局部解的无穷序列的存在性。此外，我们还给出了一个具体集合作为这些局部解的集中位置。值得注意的是，在我们的主要结果中，f只属于C (R, R) Cleft（{mathbb{R}}，{mathbb{R}})，不满足ambrosetti - rabinowitz型条件。
{"title":"Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation","authors":"Binhua Feng, Da-Bin Wang, Zhi-Guo Wu","doi":"10.1515/anona-2022-0296","DOIUrl":"https://doi.org/10.1515/anona-2022-0296","url":null,"abstract":"Abstract We deal with localized semiclassical states for singularly perturbed Kirchhoff-type equation as follows: − ε 2 a + ε b ∫ R 3 ∣ ∇ v ∣ 2 d x Δ v + V ( x ) v = P ( x ) f ( v ) , x ∈ R 3 , -left({varepsilon }^{2}a+varepsilon bmathop{int }limits_{{{mathbb{R}}}^{3}}| nabla v{| }^{2}{rm{d}}xright)Delta v+Vleft(x)v=Pleft(x)fleft(v),hspace{1em}xin {{mathbb{R}}}^{3}, where V , P ∈ C 1 ( R 3 , R ) V,Pin {C}^{1}left({{mathbb{R}}}^{3},{mathbb{R}}) and bounded away from zero. By applying the penalization approach together with the Nehari manifold approach in the studies of Szulkin and Weth, we obtain the existence of an infinite sequence of localized solutions of higher topological type. In addition, we also give a concrete set as the concentration position of these localized solutions. It is noted that, in our main results, f f only belongs to C ( R , R ) Cleft({mathbb{R}},{mathbb{R}}) and does not satisfy the Ambrosetti-Rabinowitz-type condition.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41607425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”","authors":"Vicentiu D. Rădulescu, Runzhang Xu","doi":"10.1515/anona-2022-0302","DOIUrl":"https://doi.org/10.1515/anona-2022-0302","url":null,"abstract":"","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47717068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we study the existence of ground state solutions for the Schrödinger-Poisson-Slater type equation with the Coulomb-Sobolev critical growth: − Δ u + 1 4 π ∣ x ∣ ∗ ∣ u ∣ 2 u = ∣ u ∣ u + μ ∣ u ∣ p − 2 u , in R 3 , -Delta u+left(frac{1}{4pi | x| }ast | u{| }^{2}right)u=| u| u+mu | u{| }^{p-2}u,hspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{3}, where μ > 0 mu gt 0 and 3 < p < 6 3lt plt 6 . With the help of the Nehari-Pohozaev method, we obtain a ground-state solution for the above equation by employing compactness arguments.
{"title":"Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent","authors":"Chun-Yu Lei, Jun Lei, H. Suo","doi":"10.1515/anona-2022-0299","DOIUrl":"https://doi.org/10.1515/anona-2022-0299","url":null,"abstract":"Abstract In this article, we study the existence of ground state solutions for the Schrödinger-Poisson-Slater type equation with the Coulomb-Sobolev critical growth: − Δ u + 1 4 π ∣ x ∣ ∗ ∣ u ∣ 2 u = ∣ u ∣ u + μ ∣ u ∣ p − 2 u , in R 3 , -Delta u+left(frac{1}{4pi | x| }ast | u{| }^{2}right)u=| u| u+mu | u{| }^{p-2}u,hspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{3}, where μ > 0 mu gt 0 and 3 < p < 6 3lt plt 6 . With the help of the Nehari-Pohozaev method, we obtain a ground-state solution for the above equation by employing compactness arguments.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45264599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {bf{A}} . More precisely, A {bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| xi | }^{qleft(x)-2}xi if ∣ ξ ∣ | xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| xi | }^{pleft(x)-2}xi if ∣ ξ ∣ | xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f ( x , u ) fleft(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.
摘要本文讨论了R N｛mathbb｛R｝｝^｛N｝中一个非线性加权拟线性方程的一对非平凡非负和非正解的存在性，该方程涉及一个双相广义变指数椭圆算子a｛bf｛a｝。更准确地说，A｛bf｛A｝｝具有类似于如果Şξ|nenenebc xi |很小则Şξ。用Cerami条件而不是经典的Palais-Smale条件证明了存在性，使得非线性项f（x，u）fleft（x，u）不一定满足Ambrosetti-Rabinowitz条件。
{"title":"Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition","authors":"Jingjing Liu, P. Pucci","doi":"10.1515/anona-2022-0292","DOIUrl":"https://doi.org/10.1515/anona-2022-0292","url":null,"abstract":"Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {bf{A}} . More precisely, A {bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| xi | }^{qleft(x)-2}xi if ∣ ξ ∣ | xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| xi | }^{pleft(x)-2}xi if ∣ ξ ∣ | xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f ( x , u ) fleft(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44334825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we study a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity ( − Δ ) s u = λ u + f ( x , u ) , in Ω , u = 0 , in R N Ω . left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{left(-Delta )}^{s}u=lambda u+fleft(x,u),hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega , u=0,hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}backslash Omega right.end{array}right. Multiplicity of nontrivial solutions is obtained when the parameter is near the eigenvalue of the fractional Laplace operator without Ambrosetti and Rabinowitz condition for the nonlinearity. Our methods are the combination of minimax method, bifurcation theory, and Morse theory.
{"title":"Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition","authors":"Leiga Zhao, Hongrui Cai, Yutong Chen","doi":"10.1515/anona-2022-0281","DOIUrl":"https://doi.org/10.1515/anona-2022-0281","url":null,"abstract":"Abstract In this article, we study a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity ( − Δ ) s u = λ u + f ( x , u ) , in Ω , u = 0 , in R N Ω . left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{left(-Delta )}^{s}u=lambda u+fleft(x,u),hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega , u=0,hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}backslash Omega right.end{array}right. Multiplicity of nontrivial solutions is obtained when the parameter is near the eigenvalue of the fractional Laplace operator without Ambrosetti and Rabinowitz condition for the nonlinearity. Our methods are the combination of minimax method, bifurcation theory, and Morse theory.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43801670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) Omega := {{mathbb{R}}}^{2}times left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) left({bf{v}},rho )in {H}^{s}left(Omega )times {{mathbb{H}}}^{s}left(Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}left(Omega )times {{mathbb{H}}}^{s+2}left(Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.
{"title":"Global Sobolev regular solution for Boussinesq system","authors":"Xiaofeng Zhao, Weijia Li, Weiping Yan","doi":"10.1515/anona-2022-0298","DOIUrl":"https://doi.org/10.1515/anona-2022-0298","url":null,"abstract":"Abstract This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω ≔ R 2 × ( 0 , R ) Omega := {{mathbb{R}}}^{2}times left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( Ω ) left({bf{v}},rho )in {H}^{s}left(Omega )times {{mathbb{H}}}^{s}left(Omega ) with the small initial data in the Sobolev space H s + 2 ( Ω ) × H s + 2 ( Ω ) {H}^{s+2}left(Omega )times {{mathbb{H}}}^{s+2}left(Omega ) . Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent s s can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44085152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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