R. Clemente, J. Marcos do Ó, Esteban da Silva, E. Shamarova
Abstract We study general problems modeling electrostatic microelectromechanical systems devices (Pλ ) φ ( r , − u ′ ( r ) ) = λ ∫ 0 r f ( s ) g ( u ( s ) ) d s , r ∈ ( 0 , 1 ) , 0 < u ( r ) < 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , left{begin{array}{ll}varphi (r,-u^{prime} left(r))=lambda underset{0}{overset{r}{displaystyle int }}frac{fleft(s)}{gleft(uleft(s))}{rm{d}}s,hspace{1.0em}& rin left(0,1), 0lt uleft(r)lt 1,hspace{1.0em}& rin left(0,1), uleft(1)=0,hspace{1.0em}end{array}right. where φ varphi , g g , and f f are some functions on [ 0 , 1 ] left[0,1] and λ > 0 lambda gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to ( P λ {P}_{lambda } ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ ( r , v ) = r α ∣ v ∣ β v varphi left(r,v)={r}^{alpha }{| v| }^{beta }v , i.e., when the associated differential equation involves the operator r − γ ( r α ∣ u ′ ∣ β u ′ ) ′ {r}^{-gamma }left({r}^{alpha }{| u^{prime} | }^{beta }u^{prime} )^{prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.
摘要研究静电微机电系统器件(Pλ) φ (r， - u ' (r)) = λ∫0 r f (s) g (u (s)) d s, r∈(0,1)，0 < u (r) < 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , left ｛ begin{array}{ll}varphi (r,-u^{prime} left(r))=lambda underset{0}{overset{r}{displaystyle int }}frac{fleft(s)}{gleft(uleft(s))}{rm{d}}s,hspace{1.0em}& rin left(0,1), 0lt uleft(r)lt 1,hspace{1.0em}& rin left(0,1), uleft(1)=0,hspace{1.0em}end{array}right . where φ varphi , g g , and f f are some functions on [ 0 , 1 ] left[0,1] and λ >< 1, r∈(0,1)，u (1) = 0, ｛。其中φ , g g, f f是[> 0 lambdagt 0是参数。我们得到了(P λ P＿ {}{lambda)触地解的存在性和规律性}，并和下界。在特殊情况下，当φ (r,v)=r α∣v∣β v varphileft (r{,}v)=r^ {alpha | v| ^ }{}{beta v，即当相关}微分{方程涉及算子r−γ (r α∣}u '{∣β u ') ' r^- gamma}left (r^ {}{alpha | u^ }{{prime} | ^ }{beta u^ }{prime})^ {prime}时，我们得到了在原点附近的触地解的精确渐近行为。
{"title":"Touchdown solutions in general MEMS models","authors":"R. Clemente, J. Marcos do Ó, Esteban da Silva, E. Shamarova","doi":"10.1515/anona-2023-0102","DOIUrl":"https://doi.org/10.1515/anona-2023-0102","url":null,"abstract":"Abstract We study general problems modeling electrostatic microelectromechanical systems devices (Pλ ) φ ( r , − u ′ ( r ) ) = λ ∫ 0 r f ( s ) g ( u ( s ) ) d s , r ∈ ( 0 , 1 ) , 0 < u ( r ) < 1 , r ∈ ( 0 , 1 ) , u ( 1 ) = 0 , left{begin{array}{ll}varphi (r,-u^{prime} left(r))=lambda underset{0}{overset{r}{displaystyle int }}frac{fleft(s)}{gleft(uleft(s))}{rm{d}}s,hspace{1.0em}& rin left(0,1), 0lt uleft(r)lt 1,hspace{1.0em}& rin left(0,1), uleft(1)=0,hspace{1.0em}end{array}right. where φ varphi , g g , and f f are some functions on [ 0 , 1 ] left[0,1] and λ > 0 lambda gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to ( P λ {P}_{lambda } ) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ ( r , v ) = r α ∣ v ∣ β v varphi left(r,v)={r}^{alpha }{| v| }^{beta }v , i.e., when the associated differential equation involves the operator r − γ ( r α ∣ u ′ ∣ β u ′ ) ′ {r}^{-gamma }left({r}^{alpha }{| u^{prime} | }^{beta }u^{prime} )^{prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48254089","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 A nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate dependent viscosity is considered. This problem is nonlinear and nonlocal in time and inverse to the nonlinear heat equation. The provided mathematical analysis includes the proof of the existence, uniqueness, regularity, and stability of the velocity and the pressure slope for a given flux carrier and of the exponential decay of the solution as the time variable goes to infinity for the exponentially decaying flux.
{"title":"Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity","authors":"G. Panasenko, K. Pileckas","doi":"10.1515/anona-2022-0259","DOIUrl":"https://doi.org/10.1515/anona-2022-0259","url":null,"abstract":"Abstract A nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate dependent viscosity is considered. This problem is nonlinear and nonlocal in time and inverse to the nonlinear heat equation. The provided mathematical analysis includes the proof of the existence, uniqueness, regularity, and stability of the velocity and the pressure slope for a given flux carrier and of the exponential decay of the solution as the time variable goes to infinity for the exponentially decaying flux.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43701764","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 By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are constructed by stable small solitons.
{"title":"Small solitons and multisolitons in the generalized Davey-Stewartson system","authors":"M. Bai, Jian Zhang, Shihui Zhu","doi":"10.1515/anona-2022-0266","DOIUrl":"https://doi.org/10.1515/anona-2022-0266","url":null,"abstract":"Abstract By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are constructed by stable small solitons.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44153443","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 paper, we consider stochastic dynamics of a two-dimensional stochastic differential equation with additive noise. When the strength of the noise is zero, this equation undergoes a Bautin bifurcation. We obtain the main conclusions including the existence and uniqueness of the solution, synchronization of system and property of the random equilibrium, where going through some processes like deducing the stationary probability density of the equation and calculating the Lyapunov exponent. For better understanding of the effect under noise, we make a clear comparison between the stochastic system and the deterministic one and make precise numerical simulations to show the slight changes at Bautin bifurcation point. Furthermore, we take a real model as an example to present the application of our theoretical results.
{"title":"Bautin bifurcation with additive noise","authors":"Diandian Tang, Jingli Ren","doi":"10.1515/anona-2022-0277","DOIUrl":"https://doi.org/10.1515/anona-2022-0277","url":null,"abstract":"Abstract In this paper, we consider stochastic dynamics of a two-dimensional stochastic differential equation with additive noise. When the strength of the noise is zero, this equation undergoes a Bautin bifurcation. We obtain the main conclusions including the existence and uniqueness of the solution, synchronization of system and property of the random equilibrium, where going through some processes like deducing the stationary probability density of the equation and calculating the Lyapunov exponent. For better understanding of the effect under noise, we make a clear comparison between the stochastic system and the deterministic one and make precise numerical simulations to show the slight changes at Bautin bifurcation point. Furthermore, we take a real model as an example to present the application of our theoretical results.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48943050","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 to study nonlinear problem driven by the p ( t ) pleft(t) -Laplacian with nonsmooth potential. We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue-Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the Palais-Smale sequence. To our best knowledge, our theorems appear to be the first such result about homoclinic solution for differential inclusion system involving the p ( t ) pleft(t) -Laplacian.
{"title":"Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian","authors":"Jun Cheng, Peng Chen, Limin Zhang","doi":"10.1515/anona-2022-0272","DOIUrl":"https://doi.org/10.1515/anona-2022-0272","url":null,"abstract":"Abstract The aim of this article is to study nonlinear problem driven by the p ( t ) pleft(t) -Laplacian with nonsmooth potential. We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue-Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the Palais-Smale sequence. To our best knowledge, our theorems appear to be the first such result about homoclinic solution for differential inclusion system involving the p ( t ) pleft(t) -Laplacian.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49354391","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 the existence and nonexistence of the standing wave solution for the generalized Jackiw-Pi model by using variational method. Depending on interaction strength λ lambda , we have three different situations. The existence and nonexistence of the standing wave solution correspond to 1 < λ 1lt lambda and 0 < λ < 1 0lt lambda lt 1 , respectively. We have the explicit solution of self-dual equation for the borderline λ = 1 lambda =1 .
{"title":"Standing wave solution for the generalized Jackiw-Pi model","authors":"Hyungjin Huh, Yuanfeng Jin, You Ma, Guanghui Jin","doi":"10.1515/anona-2022-0261","DOIUrl":"https://doi.org/10.1515/anona-2022-0261","url":null,"abstract":"Abstract We study the existence and nonexistence of the standing wave solution for the generalized Jackiw-Pi model by using variational method. Depending on interaction strength λ lambda , we have three different situations. The existence and nonexistence of the standing wave solution correspond to 1 < λ 1lt lambda and 0 < λ < 1 0lt lambda lt 1 , respectively. We have the explicit solution of self-dual equation for the borderline λ = 1 lambda =1 .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"369 - 382"},"PeriodicalIF":4.2,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44867375","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 the homogeneous Dirichlet problem for the parabolic equations u t − div ( A ( z , ∣ ∇ u ∣ ) ∇ u ) = F ( z , u , ∇ u ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , {u}_{t}-{rm{div}}left({mathcal{A}}left(z,| nabla u| )nabla u)=Fleft(z,u,nabla u),hspace{1.0em}z=left(x,t)in Omega times left(0,T), with the double phase flux A ( z , ∣ ∇ u ∣ ) ∇ u = ( ∣ ∇ u ∣ p ( z ) − 2 + a ( z ) ∣ ∇ u ∣ q ( z ) − 2 ) ∇ u {mathcal{A}}left(z,| nabla u| )nabla u=(| nabla u{| }^{pleft(z)-2}+aleft(z)| nabla u{| }^{qleft(z)-2})nabla u and the nonlinear source F F . The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a a , p p , and q q are Lipschitz-continuous, a ( z ) aleft(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p p , and q q satisfy the balance conditions 2 N N + 2 < p − ≤ p ( z ) ≤ q ( z ) < p ( z ) + r ∗ 2 frac{2N}{N+2}lt {p}^{-}le pleft(z)le qleft(z)lt pleft(z)+frac{{r}^{ast }}{2} with r ∗ = r ∗ ( p − , N ) {r}^{ast }={r}^{ast }left({p}^{-},N) , p − = min Q ¯ T p ( z ) {p}^{-}={min }_{{overline{Q}}_{T}}hspace{0.33em}pleft(z) . It is shown that under suitable conditions on the growth of F ( z , u , ∇ u ) Fleft(z,u,nabla u) with respect to the second and third arguments, the problem has a solution u u with the following properties: u t ∈ L 2 ( Q T ) , ∣ ∇ u ∣ p ( z ) + δ ∈ L 1 ( Q T ) for every 0 ≤ δ < r ∗ , ∣ ∇ u ∣ s ( z ) , a ( z ) ∣ ∇ u ∣ q ( z ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) with s ( z ) = max { 2 , p ( z ) } . begin{array}{l}{u}_{t}in {L}^{2}left({Q}_{T}),hspace{1.0em}| nabla u{| }^{pleft(z)+delta }in {L}^{1}left({Q}_{T})hspace{1.0em}hspace{0.1em}text{for every}hspace{0.1em}hspace{0.33em}0le delta lt {r}^{ast }, | nabla u{| }^{sleft(z)},hspace{0.33em}aleft(z)| nabla u{| }^{qleft(z)}in {L}^{infty }left(0,T;hspace{0.33em}{L}^{1}left(Omega ))hspace{1em}{rm{with}}hspace{0.33em}sleft(z)=max left{2,pleft(z)right}.end{array} Uniqueness is proven under stronger assumptions on the source F F . The same results are established for the equations with the regularized flux A ( z , ( ε 2 + ∣ ∇ u ∣ 2 ) 1 / 2 ) ∇ u {mathcal{A}}(z,{({varepsilon }^{2}+| nabla u{| }^{2})}^{1text{/}2})nabla u , ε > 0 varepsilon gt 0 .
{"title":"Double-phase parabolic equations with variable growth and nonlinear sources","authors":"R. Arora, S. Shmarev","doi":"10.1515/anona-2022-0271","DOIUrl":"https://doi.org/10.1515/anona-2022-0271","url":null,"abstract":"Abstract We study the homogeneous Dirichlet problem for the parabolic equations u t − div ( A ( z , ∣ ∇ u ∣ ) ∇ u ) = F ( z , u , ∇ u ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , {u}_{t}-{rm{div}}left({mathcal{A}}left(z,| nabla u| )nabla u)=Fleft(z,u,nabla u),hspace{1.0em}z=left(x,t)in Omega times left(0,T), with the double phase flux A ( z , ∣ ∇ u ∣ ) ∇ u = ( ∣ ∇ u ∣ p ( z ) − 2 + a ( z ) ∣ ∇ u ∣ q ( z ) − 2 ) ∇ u {mathcal{A}}left(z,| nabla u| )nabla u=(| nabla u{| }^{pleft(z)-2}+aleft(z)| nabla u{| }^{qleft(z)-2})nabla u and the nonlinear source F F . The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a a , p p , and q q are Lipschitz-continuous, a ( z ) aleft(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p p , and q q satisfy the balance conditions 2 N N + 2 < p − ≤ p ( z ) ≤ q ( z ) < p ( z ) + r ∗ 2 frac{2N}{N+2}lt {p}^{-}le pleft(z)le qleft(z)lt pleft(z)+frac{{r}^{ast }}{2} with r ∗ = r ∗ ( p − , N ) {r}^{ast }={r}^{ast }left({p}^{-},N) , p − = min Q ¯ T p ( z ) {p}^{-}={min }_{{overline{Q}}_{T}}hspace{0.33em}pleft(z) . It is shown that under suitable conditions on the growth of F ( z , u , ∇ u ) Fleft(z,u,nabla u) with respect to the second and third arguments, the problem has a solution u u with the following properties: u t ∈ L 2 ( Q T ) , ∣ ∇ u ∣ p ( z ) + δ ∈ L 1 ( Q T ) for every 0 ≤ δ < r ∗ , ∣ ∇ u ∣ s ( z ) , a ( z ) ∣ ∇ u ∣ q ( z ) ∈ L ∞ ( 0 , T ; L 1 ( Ω ) ) with s ( z ) = max { 2 , p ( z ) } . begin{array}{l}{u}_{t}in {L}^{2}left({Q}_{T}),hspace{1.0em}| nabla u{| }^{pleft(z)+delta }in {L}^{1}left({Q}_{T})hspace{1.0em}hspace{0.1em}text{for every}hspace{0.1em}hspace{0.33em}0le delta lt {r}^{ast }, | nabla u{| }^{sleft(z)},hspace{0.33em}aleft(z)| nabla u{| }^{qleft(z)}in {L}^{infty }left(0,T;hspace{0.33em}{L}^{1}left(Omega ))hspace{1em}{rm{with}}hspace{0.33em}sleft(z)=max left{2,pleft(z)right}.end{array} Uniqueness is proven under stronger assumptions on the source F F . The same results are established for the equations with the regularized flux A ( z , ( ε 2 + ∣ ∇ u ∣ 2 ) 1 / 2 ) ∇ u {mathcal{A}}(z,{({varepsilon }^{2}+| nabla u{| }^{2})}^{1text{/}2})nabla u , ε > 0 varepsilon gt 0 .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"304 - 335"},"PeriodicalIF":4.2,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46586260","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 establish a viscosity method for random homogenization of an obstacle problem with nondivergence structure. We study the asymptotic behavior of the viscosity solution u ε {u}_{varepsilon } of fully nonlinear equations in a perforated domain with the stationary ergodic condition. By capturing the behavior of the homogeneous solution, analyzing the characters of the corresponding obstacle problem, and finding the capacity-like quantity through the construction of appropriate barriers, we prove that the limit profile u u of u ε {u}_{varepsilon } satisfies a homogenized equation without obstacles.
{"title":"Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles","authors":"Ki-ahm Lee, Se-Chan Lee","doi":"10.1515/anona-2022-0273","DOIUrl":"https://doi.org/10.1515/anona-2022-0273","url":null,"abstract":"Abstract In this article, we establish a viscosity method for random homogenization of an obstacle problem with nondivergence structure. We study the asymptotic behavior of the viscosity solution u ε {u}_{varepsilon } of fully nonlinear equations in a perforated domain with the stationary ergodic condition. By capturing the behavior of the homogeneous solution, analyzing the characters of the corresponding obstacle problem, and finding the capacity-like quantity through the construction of appropriate barriers, we prove that the limit profile u u of u ε {u}_{varepsilon } satisfies a homogenized equation without obstacles.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"266 - 303"},"PeriodicalIF":4.2,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45655902","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 paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M ( ‖ u ‖ 2 ) ( − Δ H u + V ( ξ ) u ) = ∫ H N ∣ u ( η ) ∣ Q λ ∗ ∣ η − 1 ξ ∣ λ d η ∣ u ∣ Q λ ∗ − 2 u + μ f ( ξ , u ) , Mleft(Vert u{Vert }^{2})left(-{Delta }_{{mathbb{H}}}uleft+Vleft(xi )u)=left(mathop{int }limits_{{{mathbb{H}}}^{N}}frac{| uleft(eta ){| }^{{Q}_{lambda }^{ast }}}{| {eta }^{-1}xi {| }^{lambda }}{rm{d}}eta right)| u{| }^{{Q}_{lambda }^{ast }-2}u+mu fleft(xi ,u), where M M is the Kirchhoff function, Δ H {Delta }_{{mathbb{H}}} is the Kohn Laplacian on the Heisenberg group H N {{mathbb{H}}}^{N} , f f is a Carathéodory function, μ > 0 mu gt 0 is a parameter and Q λ ∗ = 2 Q − λ Q − 2 {Q}_{lambda }^{ast }=frac{2Q-lambda }{Q-2} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We first establish a new version of the concentration-compactness principle for the Choquard equation on the Heisenberg group. Then, combining with the mountain pass theorem, we obtain the existence of nontrivial solutions to the aforementioned problem in the case of nondegenerate and degenerate cases.
{"title":"On the critical Choquard-Kirchhoff problem on the Heisenberg group","authors":"Xueqi Sun, Yueqiang Song, Sihua Liang","doi":"10.1515/anona-2022-0270","DOIUrl":"https://doi.org/10.1515/anona-2022-0270","url":null,"abstract":"Abstract In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M ( ‖ u ‖ 2 ) ( − Δ H u + V ( ξ ) u ) = ∫ H N ∣ u ( η ) ∣ Q λ ∗ ∣ η − 1 ξ ∣ λ d η ∣ u ∣ Q λ ∗ − 2 u + μ f ( ξ , u ) , Mleft(Vert u{Vert }^{2})left(-{Delta }_{{mathbb{H}}}uleft+Vleft(xi )u)=left(mathop{int }limits_{{{mathbb{H}}}^{N}}frac{| uleft(eta ){| }^{{Q}_{lambda }^{ast }}}{| {eta }^{-1}xi {| }^{lambda }}{rm{d}}eta right)| u{| }^{{Q}_{lambda }^{ast }-2}u+mu fleft(xi ,u), where M M is the Kirchhoff function, Δ H {Delta }_{{mathbb{H}}} is the Kohn Laplacian on the Heisenberg group H N {{mathbb{H}}}^{N} , f f is a Carathéodory function, μ > 0 mu gt 0 is a parameter and Q λ ∗ = 2 Q − λ Q − 2 {Q}_{lambda }^{ast }=frac{2Q-lambda }{Q-2} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We first establish a new version of the concentration-compactness principle for the Choquard equation on the Heisenberg group. Then, combining with the mountain pass theorem, we obtain the existence of nontrivial solutions to the aforementioned problem in the case of nondegenerate and degenerate cases.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"210 - 236"},"PeriodicalIF":4.2,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49095822","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 large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].
{"title":"Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data","authors":"W. Dong, Zhenhua Guo","doi":"10.1515/anona-2022-0246","DOIUrl":"https://doi.org/10.1515/anona-2022-0246","url":null,"abstract":"Abstract In this article, we study the large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"132 - 168"},"PeriodicalIF":4.2,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67260661","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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