Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{varepsilon }^{2}Delta u+Vleft(x)u+{varepsilon }^{-alpha }left({I}_{alpha }ast | u{| }^{2})u=lambda | u{| }^{p-1}uhspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where ε , λ > 0 varepsilon ,lambda gt 0 are parameters, N ⩾ 2 Ngeqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 left(alpha +6)hspace{0.1em}text{/}hspace{0.1em}left(alpha +2)lt plt {2}^{ast }-1 , I α {I}_{alpha } is the Riesz potential with 0 < α < N 0lt alpha lt N , and V ∈ C ( R N , R ) Vin {mathcal{C}}left({{mathbb{R}}}^{N},{mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.
{"title":"Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations","authors":"Yiqing Li, Binlin Zhang, Xiumei Han","doi":"10.1515/anona-2022-0293","DOIUrl":"https://doi.org/10.1515/anona-2022-0293","url":null,"abstract":"Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{varepsilon }^{2}Delta u+Vleft(x)u+{varepsilon }^{-alpha }left({I}_{alpha }ast | u{| }^{2})u=lambda | u{| }^{p-1}uhspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where ε , λ > 0 varepsilon ,lambda gt 0 are parameters, N ⩾ 2 Ngeqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 left(alpha +6)hspace{0.1em}text{/}hspace{0.1em}left(alpha +2)lt plt {2}^{ast }-1 , I α {I}_{alpha } is the Riesz potential with 0 < α < N 0lt alpha lt N , and V ∈ C ( R N , R ) Vin {mathcal{C}}left({{mathbb{R}}}^{N},{mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42968190","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}
F. Farroni, L. Greco, G. Moscariello, Gabriella Zecca
Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.
{"title":"Noncoercive parabolic obstacle problems","authors":"F. Farroni, L. Greco, G. Moscariello, Gabriella Zecca","doi":"10.1515/anona-2022-0322","DOIUrl":"https://doi.org/10.1515/anona-2022-0322","url":null,"abstract":"Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48861225","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 periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.
{"title":"Periodic solutions to a class of distributed delay differential equations via variational methods","authors":"Huafeng Xiao, Zhiming Guo","doi":"10.1515/anona-2022-0305","DOIUrl":"https://doi.org/10.1515/anona-2022-0305","url":null,"abstract":"Abstract In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42369970","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 consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.
{"title":"Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices","authors":"Zaitao Liang, Shengjun Li, Xin Li","doi":"10.1515/anona-2022-0287","DOIUrl":"https://doi.org/10.1515/anona-2022-0287","url":null,"abstract":"Abstract In this article, we consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47145885","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 presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.
{"title":"Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications","authors":"A. Taheri, V. Vahidifar","doi":"10.1515/anona-2022-0288","DOIUrl":"https://doi.org/10.1515/anona-2022-0288","url":null,"abstract":"Abstract This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41408547","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 investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ G xin G , where G G is a domain in R N {{mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ R N xin {{mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.
{"title":"Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition","authors":"Cuiling Liu, Xingyong Zhang","doi":"10.1515/anona-2022-0289","DOIUrl":"https://doi.org/10.1515/anona-2022-0289","url":null,"abstract":"Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ G xin G , where G G is a domain in R N {{mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ R N xin {{mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41664225","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 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":null,"pages":null},"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 We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -left(a+bmathop{int }limits_{{{mathbb{R}}}^{2}}| nabla u{| }^{2}{rm{d}}xright)Delta u+left(hleft(x)+mu Vleft(x))u=Kleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{2}, where a , b , μ > 0 a,b,mu gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-gamma } with γ ∈ ( 0 , 2 ) gamma in left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 mu gt 0 large and the concentration behavior of solutions as μ → + ∞ mu to +infty .
摘要:我们关注以下具有指数临界非线性的kirchhoff型方程- a+b∫r2∣∇u∣2d x Δ u+ (h (x)+ μ V (x))u=K (x)f (u)在r2中，- left (a+b mathop{int }limits ＿ {{{mathbb{R}}} ^{2}}| nabla u{|} ^{2x}{rm{d}}right) Delta u+ left (h left (x)+ mu V left (x))u=K left (x)f left (u)^hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}2{，}其中a,b， μ > 0 a,b，mugt 0，势V V有一个有界的零点集合，在无穷远处衰减为∣x∣−γ | x| ^{-}{gamma，其中γ}∈(0,2)gammainleft(0,2)，权K K有有限奇点，在无穷远处可以呈指数增长。利用截断技术，在一些权重Sobolev空间中，我们得到了μ > 0 mugt 0大的山口解的存在性和解的集中行为为μ→+∞muto + infty。
{"title":"Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities","authors":"Jian Zhang, Xue Bao, Jianjun Zhang","doi":"10.1515/anona-2022-0317","DOIUrl":"https://doi.org/10.1515/anona-2022-0317","url":null,"abstract":"Abstract We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -left(a+bmathop{int }limits_{{{mathbb{R}}}^{2}}| nabla u{| }^{2}{rm{d}}xright)Delta u+left(hleft(x)+mu Vleft(x))u=Kleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{2}, where a , b , μ > 0 a,b,mu gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-gamma } with γ ∈ ( 0 , 2 ) gamma in left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 mu gt 0 large and the concentration behavior of solutions as μ → + ∞ mu to +infty .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67260729","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.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：