Abstract This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , left{begin{array}{ll}-M(Vert nabla u{Vert }_{2}^{2})Delta u=fleft(x,u),& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{B}_{1}left(0), u=0,& hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial {B}_{1}left(0),end{array}right. where B 1 ( 0 ) {B}_{1}left(0) is the unit ball in R 2 {{mathbb{R}}}^{2} , M : R + → R + M:{{mathbb{R}}}^{+}to {{mathbb{R}}}^{+} is a Kirchhoff function, and f ( x , t ) fleft(x,t) has supercritical exponential growth on t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] exp {[}({beta }_{0}+| xhspace{-0.25em}{| }^{alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) exp ({beta }_{0}{t}^{2+| x{| }^{alpha }}) with β 0 {beta }_{0} , α > 0 alpha gt 0 . Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ] {mathrm{liminf}}_{tto infty }frac{tfleft(x,t)}{exp {[}({beta }_{0}+| hspace{-0.25em}xhspace{-0.25em}{| }^{alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ x ∣ α ) {mathrm{liminf}}_{tto infty }frac{tfleft(x,t)}{exp ({beta }_{0}{t}^{2+| x{| }^{alpha }})} , respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo (Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen (Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M ( t ) = 1 M(t)=1 . In particular, if the weighted term ∣ x ∣ α | xhspace{-0.25em}{| }^{alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) (Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.
摘要本文讨论了以下非线性超临界椭圆问题：−M（‖õu‖2 2）Δu＝f（x，u），在B1（0）中，u＝0，在B.1（0）上， left ｛begin｛array｝{ll}-M（Vertnabla u｛Vert｝_｛2｝^｛2}）Delta u=fleft（x，u），&&hspace｛0.1em｝text｛in｝space｛0.1em｝sspace｛0.33em｝{B}_{1} left（0），u=0，&&hspace｛0.1em｝text｛on｝space｛0.1em｝sace｛0.33em｝partial{B}_{1} left（0），end｛array｝right。其中B1（0）{B}_{1} left（0）是R 2中的单位球｛mathbb｛R｝｝^｛2｝，M:R+→ R+M:｛｛mathbb｛R｝｝^｛+｝to｛math bb｛R｝}^｛+｝是基尔霍夫函数，f（x，t）fleft（x，t）在t上具有超临界指数增长，表现为exp[（β0+ŞxŞα）t2]exp｛[｝β｛0｝｛t｝^｛2+|x｛|｝^｝alpha｝｝）与β0｛beta｝_｛0}，α＞0alphagt 0。在深入分析和详细估计的基础上，我们用变分法得到了上述问题的Nehari型基态解。此外，我们可以在liminf t上的较弱假设下确定极小极大水平的精细上界→ ∞ t f（x，t）exp[（β0+ŞxŞα）t2]｛mathrm｛liminf｝｝_{-0.25em}xhspace｛-0.25em｝｛|｝^｛alpha｝）｛t｝^｝2｝]｝和liminf t→ ∞ 分别为t f（x，t）exp（β0 t 2+ŞxŞα）｛mathrm｛liminf｝｝_｛ttoinfty｝frac｛tfleft（x，t）｝｛exp（β_｛0｝｛t｝^｛2+|x｛|｝^｝alpha｝）｝。我们的结果推广和改进了G.M.Figueiredo和U.B.Severo（具有指数临界增长的基尔霍夫问题的基态解，Milan J.Math.84（2016），第1，23–39页）和Q.a.Ngó和V.H.Nguyen（超临界Moser-Trudinger不等式和相关椭圆问题，Calc.Var.P偏微分Equ.59（2020），第2页，论文69，30.）对于M（t）＝1M（t）=1。特别地，如果加权项ŞxŞα|xhspace｛-0.25em｝｛|}^｛alpha｝正在消失，我们可以立即获得S.T.Chen、x.H.Tang和J.Y.Wei（2021）（关于具有临界指数增长的平面Kirchhoff型椭圆问题的改进结果，Z.Angew.Math.Phys.72（2021），no.1，论文no.38，定理1.3和定理1.4）中的结果。
{"title":"On the planar Kirchhoff-type problem involving supercritical exponential growth","authors":"Limin Zhang, Xianhua Tang, Peng Chen","doi":"10.1515/anona-2022-0250","DOIUrl":"https://doi.org/10.1515/anona-2022-0250","url":null,"abstract":"Abstract This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , left{begin{array}{ll}-M(Vert nabla u{Vert }_{2}^{2})Delta u=fleft(x,u),& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{B}_{1}left(0), u=0,& hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial {B}_{1}left(0),end{array}right. where B 1 ( 0 ) {B}_{1}left(0) is the unit ball in R 2 {{mathbb{R}}}^{2} , M : R + → R + M:{{mathbb{R}}}^{+}to {{mathbb{R}}}^{+} is a Kirchhoff function, and f ( x , t ) fleft(x,t) has supercritical exponential growth on t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] exp {[}({beta }_{0}+| xhspace{-0.25em}{| }^{alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) exp ({beta }_{0}{t}^{2+| x{| }^{alpha }}) with β 0 {beta }_{0} , α > 0 alpha gt 0 . Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ] {mathrm{liminf}}_{tto infty }frac{tfleft(x,t)}{exp {[}({beta }_{0}+| hspace{-0.25em}xhspace{-0.25em}{| }^{alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ x ∣ α ) {mathrm{liminf}}_{tto infty }frac{tfleft(x,t)}{exp ({beta }_{0}{t}^{2+| x{| }^{alpha }})} , respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo (Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen (Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M ( t ) = 1 M(t)=1 . In particular, if the weighted term ∣ x ∣ α | xhspace{-0.25em}{| }^{alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) (Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1412 - 1446"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43009193","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 study distributional solutions of semilinear biharmonic equations of the type Δ2u+f(u)=0 onℝN, {Delta ^2}u + f(u) = 0quad on;{{mathbb R} ^N}, where f is a continuous function satisfying f (t)t ≥ c |t|q+1 for all t ∈ ℝ with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimension N and on the sign of the solutions.
摘要我们研究了Δ2u+f（u）=0型双线性双调和方程的分布解 在…上ℝN、 ｛Delta^2｝u+f（u）=0quad on ；｛mathbb R｝^N｝，其中f是满足f（t）t≥c|t|q+1的连续函数，对于所有t∈ℝ 其中c>0和q>1。利用一种主要基于谨慎选择合适的加权检验函数的新方法和Hardy-Rellich不等式的新版本，我们证明了几个独立于维数N和解的符号的Liouville定理。
{"title":"Entire solutions of certain fourth order elliptic problems and related inequalities","authors":"L. D’Ambrosio, E. Mitidieri","doi":"10.1515/anona-2021-0217","DOIUrl":"https://doi.org/10.1515/anona-2021-0217","url":null,"abstract":"Abstract We study distributional solutions of semilinear biharmonic equations of the type Δ2u+f(u)=0 onℝN, {Delta ^2}u + f(u) = 0quad on;{{mathbb R} ^N}, where f is a continuous function satisfying f (t)t ≥ c |t|q+1 for all t ∈ ℝ with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimension N and on the sign of the solutions.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"785 - 829"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49655468","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 consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.
{"title":"Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues","authors":"A. Vitolo","doi":"10.1515/anona-2022-0241","DOIUrl":"https://doi.org/10.1515/anona-2022-0241","url":null,"abstract":"Abstract We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1182 - 1200"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67260573","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 Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=Nleft(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {mathcal{ {mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.
{"title":"On the nonlinear perturbations of self-adjoint operators","authors":"Michal Beldzinski, M. Galewski, Witold Majdak","doi":"10.1515/anona-2022-0235","DOIUrl":"https://doi.org/10.1515/anona-2022-0235","url":null,"abstract":"Abstract Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=Nleft(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {mathcal{ {mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1117 - 1133"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46654667","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 study positive solutions to the one-dimensional generalized double phase problems of the form: − ( a ( t ) φ p ( u ′ ) + b ( t ) φ q ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , left{begin{array}{l}-(aleft(t){varphi }_{p}left(u^{prime} )+bleft(t){varphi }_{q}left(u^{prime} ))^{prime} =lambda hleft(t)fleft(u),hspace{1em}tin left(0,1), uleft(0)=0=uleft(1),end{array}right. where 1 < p < q < ∞ 1lt plt qlt infty , φ m ( s ) ≔ ∣ s ∣ m − 2 s {varphi }_{m}left(s):= | s{| }^{m-2}s , a , b ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) a,bin Cleft(left[0,1],{[}0,infty )) , h ∈ L 1 ( ( 0 , 1 ) , ( 0 , ∞ ) ) ∩ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) , hin {L}^{1}left(left(0,1),left(0,infty ))cap Cleft(left(0,1),left(0,infty )), and f ∈ C ( [ 0 , ∞ ) , R ) fin Cleft({[}0,infty ),{mathbb{R}}) is nondecreasing. More precisely, we show various existence results including the existence of at least two or three positive solutions according to the behaviors of f ( s ) fleft(s) near zero and infinity. Both positone (i.e., f ( 0 ) ≥ 0 fleft(0)ge 0 ) and semipositone (i.e., f ( 0 ) < 0 fleft(0)lt 0 ) problems are considered, and the results are obtained through the Krasnoselskii-type fixed point theorem. We also apply these results to show the existence of positive radial solutions for high-dimensional generalized double phase problems on the exterior of a ball.
{"title":"Analysis of positive solutions to one-dimensional generalized double phase problems","authors":"B. Son, Inbo Sim","doi":"10.1515/anona-2022-0240","DOIUrl":"https://doi.org/10.1515/anona-2022-0240","url":null,"abstract":"Abstract We study positive solutions to the one-dimensional generalized double phase problems of the form: − ( a ( t ) φ p ( u ′ ) + b ( t ) φ q ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , left{begin{array}{l}-(aleft(t){varphi }_{p}left(u^{prime} )+bleft(t){varphi }_{q}left(u^{prime} ))^{prime} =lambda hleft(t)fleft(u),hspace{1em}tin left(0,1), uleft(0)=0=uleft(1),end{array}right. where 1 < p < q < ∞ 1lt plt qlt infty , φ m ( s ) ≔ ∣ s ∣ m − 2 s {varphi }_{m}left(s):= | s{| }^{m-2}s , a , b ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) a,bin Cleft(left[0,1],{[}0,infty )) , h ∈ L 1 ( ( 0 , 1 ) , ( 0 , ∞ ) ) ∩ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) , hin {L}^{1}left(left(0,1),left(0,infty ))cap Cleft(left(0,1),left(0,infty )), and f ∈ C ( [ 0 , ∞ ) , R ) fin Cleft({[}0,infty ),{mathbb{R}}) is nondecreasing. More precisely, we show various existence results including the existence of at least two or three positive solutions according to the behaviors of f ( s ) fleft(s) near zero and infinity. Both positone (i.e., f ( 0 ) ≥ 0 fleft(0)ge 0 ) and semipositone (i.e., f ( 0 ) < 0 fleft(0)lt 0 ) problems are considered, and the results are obtained through the Krasnoselskii-type fixed point theorem. We also apply these results to show the existence of positive radial solutions for high-dimensional generalized double phase problems on the exterior of a ball.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1365 - 1382"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47680649","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 the following Schrödinger-Poisson system: − Δ u + u + k ( x ) ϕ ( x ) u = f ( x ) ∣ u ∣ p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = k ( x ) u 2 , x ∈ R 3 , left{begin{array}{ll}-Delta u+u+kleft(x)phi left(x)u=fleft(x)| u{| }^{p-1}u+gleft(x),& xin {{mathbb{R}}}^{3}, -Delta phi =kleft(x){u}^{2},& xin {{mathbb{R}}}^{3},end{array}right. with p ∈ ( 3 , 5 ) pin left(3,5) . Under suitable assumptions on potentials f ( x ) fleft(x) , g ( x ) gleft(x) and k ( x ) kleft(x) , then at least four positive solutions for the above system can be obtained for sufficiently small ‖ g ‖ H − 1 ( R 3 ) Vert g{Vert }_{{H}^{-1}left({{mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.
{"title":"Positive solutions for a nonhomogeneous Schrödinger-Poisson system","authors":"Jing Zhang, Rui Niu, Xiumei Han","doi":"10.1515/anona-2022-0238","DOIUrl":"https://doi.org/10.1515/anona-2022-0238","url":null,"abstract":"Abstract In this article, we consider the following Schrödinger-Poisson system: − Δ u + u + k ( x ) ϕ ( x ) u = f ( x ) ∣ u ∣ p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = k ( x ) u 2 , x ∈ R 3 , left{begin{array}{ll}-Delta u+u+kleft(x)phi left(x)u=fleft(x)| u{| }^{p-1}u+gleft(x),& xin {{mathbb{R}}}^{3}, -Delta phi =kleft(x){u}^{2},& xin {{mathbb{R}}}^{3},end{array}right. with p ∈ ( 3 , 5 ) pin left(3,5) . Under suitable assumptions on potentials f ( x ) fleft(x) , g ( x ) gleft(x) and k ( x ) kleft(x) , then at least four positive solutions for the above system can be obtained for sufficiently small ‖ g ‖ H − 1 ( R 3 ) Vert g{Vert }_{{H}^{-1}left({{mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1201 - 1222"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45635289","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 the fractional Rayleigh-Stokes problem with the nonlinearity term satisfies certain critical conditions. The local existence, uniqueness and continuous dependence upon the initial data of ε varepsilon -regular mild solutions are obtained. Furthermore, a unique continuation result and a blow-up alternative result of ε varepsilon -regular mild solutions are given in the end.
{"title":"Well-posedness and blow-up results for a class of nonlinear fractional Rayleigh-Stokes problem","authors":"J. Wang, A. Alsaedi, B. Ahmad, Yong Zhou","doi":"10.1515/anona-2022-0249","DOIUrl":"https://doi.org/10.1515/anona-2022-0249","url":null,"abstract":"Abstract In this article, we consider the fractional Rayleigh-Stokes problem with the nonlinearity term satisfies certain critical conditions. The local existence, uniqueness and continuous dependence upon the initial data of ε varepsilon -regular mild solutions are obtained. Furthermore, a unique continuation result and a blow-up alternative result of ε varepsilon -regular mild solutions are given in the end.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1579 - 1597"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41788452","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 consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFk F mapsto Delta F' + sumnolimits_{k geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that ∑k⩾1bk⩽1 sumnolimits_{k geqslant 1} b_k leqslant 1 . It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.
{"title":"Continuous flows driving branching processes and their nonlinear evolution equations","authors":"L. Beznea, Cătălin Vrabie","doi":"10.1515/anona-2021-0229","DOIUrl":"https://doi.org/10.1515/anona-2021-0229","url":null,"abstract":"Abstract We consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFk F mapsto Delta F' + sumnolimits_{k geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that ∑k⩾1bk⩽1 sumnolimits_{k geqslant 1} b_k leqslant 1 . It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"921 - 936"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46128056","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 consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert’s conjecture about the multiplicity of brake orbits to Hamiltonian functions of the classical type.
{"title":"Brake orbits for Hamiltonian systems of the classical type via geodesics in singular Finsler metrics","authors":"Dario Corona, F. Giannoni","doi":"10.1515/anona-2022-0222","DOIUrl":"https://doi.org/10.1515/anona-2022-0222","url":null,"abstract":"Abstract We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert’s conjecture about the multiplicity of brake orbits to Hamiltonian functions of the classical type.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1223 - 1248"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49532853","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 existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.
{"title":"Asymptotic proximity to higher order nonlinear differential equations","authors":"I. Astashova, M. Bartusek, Z. Došlá, M. Marini","doi":"10.1515/anona-2022-0254","DOIUrl":"https://doi.org/10.1515/anona-2022-0254","url":null,"abstract":"Abstract The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1598 - 1613"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43510347","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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