Abstract The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.
{"title":"Front propagation in a double degenerate equation with delay","authors":"Wei-Jian Bo, Shiliang Wu, Li-Jun Du","doi":"10.1515/anona-2022-0313","DOIUrl":"https://doi.org/10.1515/anona-2022-0313","url":null,"abstract":"Abstract The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.","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":"44528231","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 the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right. where N ≥ 3 Nge 3 , 0 < β < 2 0lt beta lt 2 , and v 0 {v}_{0} is a continuous function in R N {{mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -left(2+beta )log | x| as ∣ x ∣ → ∞ | x| to infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .
摘要本文研究以下问题:v t = Δ v +∣x∣β e v, x∈rn, t > 0, v (x, 0) = v 0 (x)， x∈rn, left ｛phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right。其中N≥3n ge 3, 0 < β < 20 ltbetalt 2, {v0 }v＿0{是R N中的连续函数}{{mathbb{R}}} ^{N}。在v 0 {v＿0}衰减速率为- (2+ β) log∣x∣- {}left (2+ beta) log | x|为∣x∣→∞| x| toinfty的情况下，证明了前向自相似解的存在性和渐近性。特别地，我们得到了初始值{v0 }v＿0{的最优衰减界。}
{"title":"Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity","authors":"Dong-sheng Gao, Jun Wang, Xuan Wang","doi":"10.1515/anona-2022-0290","DOIUrl":"https://doi.org/10.1515/anona-2022-0290","url":null,"abstract":"Abstract In the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right. where N ≥ 3 Nge 3 , 0 < β < 2 0lt beta lt 2 , and v 0 {v}_{0} is a continuous function in R N {{mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -left(2+beta )log | x| as ∣ x ∣ → ∞ | x| to infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .","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":"44574952","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 formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H(x) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.
{"title":"Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment","authors":"Jinliang Wang, W. Wu, Chunyang Li","doi":"10.1515/anona-2022-0295","DOIUrl":"https://doi.org/10.1515/anona-2022-0295","url":null,"abstract":"Abstract In this article, we formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H(x) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.","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":"46452230","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 a nonlinear singular Dirichlet problem driven by the ( p , q ) left(p,q) -Laplacian and a reaction where the singular term u − η {u}^{-eta } is multiplied by a strictly positive Carathéodory function f ( z , u ) fleft(z,u) . By using a topological approach, based on the Leray-Schauder alternative principle, we show the existence of a smooth positive solution.
{"title":"Positive solutions for a class of singular (p, q)-equations","authors":"S. Leonardi, Nikolaos S. Papageorgiou","doi":"10.1515/anona-2022-0300","DOIUrl":"https://doi.org/10.1515/anona-2022-0300","url":null,"abstract":"Abstract We consider a nonlinear singular Dirichlet problem driven by the ( p , q ) left(p,q) -Laplacian and a reaction where the singular term u − η {u}^{-eta } is multiplied by a strictly positive Carathéodory function f ( z , u ) fleft(z,u) . By using a topological approach, based on the Leray-Schauder alternative principle, we show the existence of a smooth positive solution.","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":"46083169","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 following general Kirchhoff type equation: − M ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + u = a ( x ) f ( u ) in R 3 , -Mleft(mathop{int }limits_{{{mathbb{R}}}^{3}}| nabla u{| }^{2}{rm{d}}xright)Delta u+u=aleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{3}, where inf R + M > 0 {inf }_{{{mathbb{R}}}^{+}}Mgt 0 and f f is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known Ambrosetti-Rabinowitz type condition.
摘要在这篇文章中，我们研究了以下一般的Kirchhoff型方程：在R3中，−MŞR 3ŞõuŞ2 d xΔu+u=a（x）f（u），-Mleft（mathop｛int｝limits_{｛mathbb｛R｝｝}^{3}｝|nabla u｛|｝^｛2｝｛rm｛d｝}xright）Delta u+u=aleft R}}}^{3}，其中inf R+M>0｛inf｝_｛｛mathbb｛R｝｝｝^｛+｝Mgt 0并且f是超线性次临界项。利用Pohozлev流形，在不存在Ambrosetti-Rabinowitz型条件的情况下，我们得到了上述方程的高能解的存在性。
{"title":"High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition","authors":"Jian Zhang, Hui Liu, J. Zuo","doi":"10.1515/anona-2022-0311","DOIUrl":"https://doi.org/10.1515/anona-2022-0311","url":null,"abstract":"Abstract In this article, we study the following general Kirchhoff type equation: − M ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + u = a ( x ) f ( u ) in R 3 , -Mleft(mathop{int }limits_{{{mathbb{R}}}^{3}}| nabla u{| }^{2}{rm{d}}xright)Delta u+u=aleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{3}, where inf R + M > 0 {inf }_{{{mathbb{R}}}^{+}}Mgt 0 and f f is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known 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":"48895824","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 work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.
{"title":"Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term","authors":"Yuepeng Li, Z. Fang","doi":"10.1515/anona-2022-0303","DOIUrl":"https://doi.org/10.1515/anona-2022-0303","url":null,"abstract":"Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.","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":"47243870","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 stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.
{"title":"Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects","authors":"Kun‐Peng Jin, Li Wang","doi":"10.1515/anona-2022-0285","DOIUrl":"https://doi.org/10.1515/anona-2022-0285","url":null,"abstract":"Abstract We are concerned with the stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.","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":"41880206","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 focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming that the wave speed of the hyperbolic equation is a positive function, we show that its smooth solution will break down in finite time even for an arbitrarily small initial energy. Based on an estimate of the solution for the heat equation, we use the method of characteristics to control the wave speed and its derivative so that the wave speed does not degenerate and its derivative does not change sign in a period of time.
{"title":"Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals","authors":"Yan-bo Hu","doi":"10.1515/anona-2022-0268","DOIUrl":"https://doi.org/10.1515/anona-2022-0268","url":null,"abstract":"Abstract This article focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming that the wave speed of the hyperbolic equation is a positive function, we show that its smooth solution will break down in finite time even for an arbitrarily small initial energy. Based on an estimate of the solution for the heat equation, we use the method of characteristics to control the wave speed and its derivative so that the wave speed does not degenerate and its derivative does not change sign in a period of time.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43080526","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 paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.
{"title":"On a strongly damped semilinear wave equation with time-varying source and singular dissipation","authors":"Yi Yang, Z. Fang","doi":"10.1515/anona-2022-0267","DOIUrl":"https://doi.org/10.1515/anona-2022-0267","url":null,"abstract":"Abstract This paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44715711","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 Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , left{begin{array}{c}-Delta u+{V}_{1}left(x)u=frac{{eta }_{1}}{{eta }_{1}+{eta }_{2}}frac{{| u| }^{{eta }_{1}-2}u{| v| }^{{eta }_{2}}}{| x^{prime} | }+frac{alpha }{alpha +beta }Qleft(x)| u{| }^{alpha -2}u| v{| }^{beta }, -Delta v+{V}_{2}left(x)v=frac{{eta }_{2}}{{eta }_{1}+{eta }_{2}}frac{{| v| }^{{eta }_{2}-2}v{| u| }^{{eta }_{1}}}{| x^{prime} | }+frac{beta }{alpha +beta }Qleft(x){| v| }^{beta -2}v{| u| }^{alpha },end{array}right. where n ≥ 3 nge 3 , 2 ≤ m < n 2le mlt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= left(x^{prime} ,{x}^{^{primeprime} })in {{mathbb{R}}}^{m}times {{mathbb{R}}}^{n-m} , η 1 , η 2 > 1 {eta }_{1},{eta }_{2}gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {eta }_{1}+{eta }_{2}=frac{2left(n-1)}{n-2} , α , β > 1 alpha ,beta gt 1 and α + β < 2 n n − 2 alpha +beta lt frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}left(x),{V}_{2}left(x),Qleft(x)in Cleft({{mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.
{"title":"A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms","authors":"Lu Shun Wang, T. Yang, Xiao Long Yang","doi":"10.1515/anona-2022-0276","DOIUrl":"https://doi.org/10.1515/anona-2022-0276","url":null,"abstract":"Abstract In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , left{begin{array}{c}-Delta u+{V}_{1}left(x)u=frac{{eta }_{1}}{{eta }_{1}+{eta }_{2}}frac{{| u| }^{{eta }_{1}-2}u{| v| }^{{eta }_{2}}}{| x^{prime} | }+frac{alpha }{alpha +beta }Qleft(x)| u{| }^{alpha -2}u| v{| }^{beta }, -Delta v+{V}_{2}left(x)v=frac{{eta }_{2}}{{eta }_{1}+{eta }_{2}}frac{{| v| }^{{eta }_{2}-2}v{| u| }^{{eta }_{1}}}{| x^{prime} | }+frac{beta }{alpha +beta }Qleft(x){| v| }^{beta -2}v{| u| }^{alpha },end{array}right. where n ≥ 3 nge 3 , 2 ≤ m < n 2le mlt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= left(x^{prime} ,{x}^{^{primeprime} })in {{mathbb{R}}}^{m}times {{mathbb{R}}}^{n-m} , η 1 , η 2 > 1 {eta }_{1},{eta }_{2}gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {eta }_{1}+{eta }_{2}=frac{2left(n-1)}{n-2} , α , β > 1 alpha ,beta gt 1 and α + β < 2 n n − 2 alpha +beta lt frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}left(x),{V}_{2}left(x),Qleft(x)in Cleft({{mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49547812","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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