Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -Delta u+Vleft(x)u=fleft(x,u)+lambda aleft(x)| u{| }^{q-2}u,hspace{1em}xin {{mathbb{R}}}^{2}, where λ > 0 lambda gt 0 , q ∈ ( 1 , 2 ) qin left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) ain {L}^{2text{/}left(2-q)}left({{mathbb{R}}}^{2}) , V ( x ) Vleft(x) , and f ( x , t ) fleft(x,t) are 1-periodic with respect to x x , and f ( x , t ) fleft(x,t) has critical exponential growth at t = ∞ t=infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 lambda gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.
摘要本文考虑了形式为(0.1)−Δ u + V (x) u = f (x, u) + λ a (x)∣u∣q−2 u, x∈r2， -的非线性椭圆方程解的存在性Delta u+Vleftu=fleft(x,u)+lambda aleft(x)| u{| }＾{q-2}你，hspace{1em}xin {{mathbb{R}}}＾{2}，其中λ > 0 lambda gt 0, q∈(1,2)qin left(1,2)， a∈l2 /(2−q) (r2) ain {l}＾{2text{/}left(2-q)}left（{{mathbb{R}}}＾{2})， V (x) Vleft(x) f (x, t) fleft(x,t)是关于x x的1周期函数，f (x,t) fleft(x,t)在t=∞处具有临界指数增长infty 。结合变分方法、Trudinger-Moser不等式和一些新的技术，详细估计了能量泛函的极大极小水平，证明了上述方程在一些弱假设下的非平凡解的存在性。我们的结果表明，凹项(即λ > 0)的存在 lambda gt 0)在某种意义上对方程(0.1)非平凡解的存在性有很大帮助。
{"title":"On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth","authors":"Xiaoyan Lin, Xianhua Tang","doi":"10.1515/anona-2022-0257","DOIUrl":"https://doi.org/10.1515/anona-2022-0257","url":null,"abstract":"Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -Delta u+Vleft(x)u=fleft(x,u)+lambda aleft(x)| u{| }^{q-2}u,hspace{1em}xin {{mathbb{R}}}^{2}, where λ > 0 lambda gt 0 , q ∈ ( 1 , 2 ) qin left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) ain {L}^{2text{/}left(2-q)}left({{mathbb{R}}}^{2}) , V ( x ) Vleft(x) , and f ( x , t ) fleft(x,t) are 1-periodic with respect to x x , and f ( x , t ) fleft(x,t) has critical exponential growth at t = ∞ t=infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 lambda gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"169 - 181"},"PeriodicalIF":4.2,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41399651","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 evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V = 1 α ψ ( x ) κ α V=frac{1}{alpha }psi left(x){kappa }^{alpha } for α < 0 alpha lt 0 or α > 1 alpha gt 1 , where x ∈ [ 0 , 2 m π ] xin left[0,2mpi ] is the tangential angle at the point on evolving curves. For − 1 ≤ α < 0 -1le alpha lt 0 , we show the flow exists globally and the rescaled flow has a full-time convergence. For α < − 1 alpha lt -1 or α > 1 alpha gt 1 , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.
{"title":"The evolution of immersed locally convex plane curves driven by anisotropic curvature flow","authors":"Yaping Wang, Xiaoliu Wang","doi":"10.1515/anona-2022-0245","DOIUrl":"https://doi.org/10.1515/anona-2022-0245","url":null,"abstract":"Abstract In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V = 1 α ψ ( x ) κ α V=frac{1}{alpha }psi left(x){kappa }^{alpha } for α < 0 alpha lt 0 or α > 1 alpha gt 1 , where x ∈ [ 0 , 2 m π ] xin left[0,2mpi ] is the tangential angle at the point on evolving curves. For − 1 ≤ α < 0 -1le alpha lt 0 , we show the flow exists globally and the rescaled flow has a full-time convergence. For α < − 1 alpha lt -1 or α > 1 alpha gt 1 , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"117 - 131"},"PeriodicalIF":4.2,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42611462","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 concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space R d {{mathbb{R}}}^{d} , a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.
在本文中，我们关注的是Keller-Segel模型的一个修正版本。Keller-Segel是一个偏微分方程系统，用于模拟化学趋向性，其中化学物质影响流动物种的运动。为了考虑模型的记忆效应，我们用卡普托意义上的时间分数阶导数代替了经典的关于时间变量的导数。在此基础上，重点讨论了与该模型相关的柯西问题的适定性。精确地说，当空间变量在空间R d {{mathbb{R}}}^{d}中考虑时，假设初始基准足够小，在临界齐次Besov空间中得到全局解。对于有界域情况，利用Neumann Laplace算子的离散谱，给出了Hilbert尺度空间中温和解的存在唯一性。此外，还研究了爆破行为。为了克服问题的挑战并获得上述所有结果，我们使用了Banach不动点定理，以及Mainardi函数和Mittag-Leffler函数等特殊函数及其性质。
{"title":"On Cauchy problem for fractional parabolic-elliptic Keller-Segel model","authors":"A. Nguyen, N. Tuan, Chao Yang","doi":"10.1515/anona-2022-0256","DOIUrl":"https://doi.org/10.1515/anona-2022-0256","url":null,"abstract":"Abstract In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space R d {{mathbb{R}}}^{d} , a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"97 - 116"},"PeriodicalIF":4.2,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41684866","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 Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ rho in a power law ρ δ {rho }^{delta } (with 0 < δ < 1 0lt delta lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.
{"title":"On regular solutions to compressible radiation hydrodynamic equations with far field vacuum","authors":"Hao Li, Shengguo Zhu","doi":"10.1515/anona-2022-0264","DOIUrl":"https://doi.org/10.1515/anona-2022-0264","url":null,"abstract":"Abstract The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ rho in a power law ρ δ {rho }^{delta } (with 0 < δ < 1 0lt delta lt 1 ), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"54 - 96"},"PeriodicalIF":4.2,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44673197","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 investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.
{"title":"Identification of discontinuous parameters in double phase obstacle problems","authors":"Shengda Zeng, Yunru Bai, Patrick Winkert, J. Yao","doi":"10.1515/anona-2022-0223","DOIUrl":"https://doi.org/10.1515/anona-2022-0223","url":null,"abstract":"Abstract In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"1 - 22"},"PeriodicalIF":4.2,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42326369","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 will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m Delta {n}^{m} for m ≥ 65 63 mge frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m > 7 6 mgt frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m > 9 8 mgt frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.
{"title":"Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity","authors":"Yu Tian, Zhaoyin Xiang","doi":"10.1515/anona-2022-0228","DOIUrl":"https://doi.org/10.1515/anona-2022-0228","url":null,"abstract":"Abstract In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m Delta {n}^{m} for m ≥ 65 63 mge frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m > 7 6 mgt frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m > 9 8 mgt frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"23 - 53"},"PeriodicalIF":4.2,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45929222","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 asymptotic behavior of solutions for n n -component Ginzburg-Landau equations as ε → 0 varepsilon to 0 . We prove that the minimizers converge locally in any C k {C}^{k} -norm to a solution of a system of generalized harmonic map equations.
研究了n个n分量Ginzburg-Landau方程在ε→0 varepsilon to 0时解的渐近性质。证明了极小值在任意ck {C}^{k}范数上局部收敛于一个广义调和映射方程系统的解。
{"title":"On a system of multi-component Ginzburg-Landau vortices","authors":"R. Hadiji, Jongmin Han, Juhee Sohn","doi":"10.1515/anona-2022-0315","DOIUrl":"https://doi.org/10.1515/anona-2022-0315","url":null,"abstract":"Abstract We study the asymptotic behavior of solutions for n n -component Ginzburg-Landau equations as ε → 0 varepsilon to 0 . We prove that the minimizers converge locally in any C k {C}^{k} -norm to a solution of a system of generalized harmonic map equations.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45340084","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 validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case p s > 1 psgt 1 for fractional W 0 s , p ( Ω ) {W}_{0}^{s,p}left(Omega ) Sobolev fields in open and bounded C 1 {C}^{1} -regular domains Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} . Also, in the case p s < 1 pslt 1 , for any open bounded C 1 {C}^{1} domain Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} , we construct counterexamples to the inequality, i.e., Korn’s first inequality fails to hold in bounded domains. The proof of the inequality in the case p s > 1 psgt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case p s < 1 pslt 1 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.
{"title":"On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1","authors":"D. Harutyunyan, H. Mikayelyan","doi":"10.1515/anona-2022-0283","DOIUrl":"https://doi.org/10.1515/anona-2022-0283","url":null,"abstract":"Abstract The validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case p s > 1 psgt 1 for fractional W 0 s , p ( Ω ) {W}_{0}^{s,p}left(Omega ) Sobolev fields in open and bounded C 1 {C}^{1} -regular domains Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} . Also, in the case p s < 1 pslt 1 , for any open bounded C 1 {C}^{1} domain Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} , we construct counterexamples to the inequality, i.e., Korn’s first inequality fails to hold in bounded domains. The proof of the inequality in the case p s > 1 psgt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case p s < 1 pslt 1 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43672152","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 multiple solutions to a generalized p ( ⋅ ) pleft(cdot ) -Laplace equation with two parameters involving critical growth. More precisely, we give sufficient “local” conditions, which mean that growths between the main operator and nonlinear term are locally assumed for p ( ⋅ ) pleft(cdot ) -sublinear, p ( ⋅ ) pleft(cdot ) -superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p p -Laplacian and ( p , q ) left(p,q) -Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p ( ⋅ ) pleft(cdot ) -sublinear and p ( ⋅ ) pleft(cdot ) -superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature.
摘要本文研究了一类具有两个参数的p(⋅)pleft(cdot) -拉普拉斯方程的多重解的存在性。更准确地说，我们给出了充分的“局部”条件，这意味着对于p(⋅)pleft(cdot) -次线性，p(⋅)pleft(cdot) -超线性和三明治型情况，主算子和非线性项之间的增长是局部假设的。与常指数问题(例如，p p -Laplacian和(p,q) left(p,q) -Laplacian)相比，这是研究变指数问题的特点。我们通过对p(⋅)pleft(cdot) -次线性和p(⋅)pleft(cdot) -超线性情况应用山口定理的变体来证明这一点，并构造了三明治型情况下属理论中由极大极小参数定义的临界值。此外，我们还得到了改变参数作用的三明治型情况的非平凡非负解。我们的工作是对文献中已有的几部作品的概括。
{"title":"On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth","authors":"Ky Ho, Inbo Sim","doi":"10.1515/anona-2022-0269","DOIUrl":"https://doi.org/10.1515/anona-2022-0269","url":null,"abstract":"Abstract In this article, we study the existence of multiple solutions to a generalized p ( ⋅ ) pleft(cdot ) -Laplace equation with two parameters involving critical growth. More precisely, we give sufficient “local” conditions, which mean that growths between the main operator and nonlinear term are locally assumed for p ( ⋅ ) pleft(cdot ) -sublinear, p ( ⋅ ) pleft(cdot ) -superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p p -Laplacian and ( p , q ) left(p,q) -Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p ( ⋅ ) pleft(cdot ) -sublinear and p ( ⋅ ) pleft(cdot ) -superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":"182 - 209"},"PeriodicalIF":4.2,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43872171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz s s -fractional gradient and peridynamics, in the sense that its integration domain depends on a ball of radius δ > 0 delta gt 0 (horizon of interaction among particles, in the terminology of peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for non-local models in calculus of variations and partial differential equations. Our motivation is to develop the proper functional analysis framework to tackle non-local models in continuum mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s s -fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the L p {L}^{p} norms of the function and its non-local gradient. Among the results showed in this investigation, we highlight a non-local version of the fundamental theorem of calculus (namely, a representation formula where a function can be recovered from its non-local gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger, and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this non-local situation are also established, and those can be viewed as a new class of non-local partial differential equations in bounded domains.
{"title":"Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings","authors":"J. C. Bellido, J. Cueto, C. Mora-Corral","doi":"10.1515/anona-2022-0316","DOIUrl":"https://doi.org/10.1515/anona-2022-0316","url":null,"abstract":"Abstract In this article, we develop a new set of results based on a non-local gradient jointly inspired by the Riesz s s -fractional gradient and peridynamics, in the sense that its integration domain depends on a ball of radius δ > 0 delta gt 0 (horizon of interaction among particles, in the terminology of peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for non-local models in calculus of variations and partial differential equations. Our motivation is to develop the proper functional analysis framework to tackle non-local models in continuum mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s s -fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the L p {L}^{p} norms of the function and its non-local gradient. Among the results showed in this investigation, we highlight a non-local version of the fundamental theorem of calculus (namely, a representation formula where a function can be recovered from its non-local gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger, and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this non-local situation are also established, and those can be viewed as a new class of non-local partial differential equations in bounded domains.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43151875","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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