Pub Date : 2021-08-11DOI: 10.57262/die035-0708-411
V. Giri, M. Sorella
In this work we prove the existence of an autonomous Hamiltonian vector field in W^{1,r}(T^d;R^d) with r=4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio superposition principle, we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover we show that the Hamiltonian is not constant along these integral curves.
{"title":"Non-uniqueness of integral curves for autonomous Hamiltonian vector fields","authors":"V. Giri, M. Sorella","doi":"10.57262/die035-0708-411","DOIUrl":"https://doi.org/10.57262/die035-0708-411","url":null,"abstract":"In this work we prove the existence of an autonomous Hamiltonian vector field in W^{1,r}(T^d;R^d) with r=4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio superposition principle, we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover we show that the Hamiltonian is not constant along these integral curves.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46257691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-28DOI: 10.57262/die036-0708-573
A. Salort, Belem Schvager, Analía Silva
In this article we study some optimal design problems related to nonstandard growth eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $Omegasubset R^n$ and $alpha,c>0$ we consider the optimization problem $inf { lambda_Omega(alpha,E)colon Esubset Omega, |E|=c }$, where $lambda_Omega(alpha,E)$ is related to the first eigenvalue to $$ -text{div}(g( |nabla u |)tfrac{nabla u}{|nabla u|}) + g(u)tfrac{u}{|u|}+ alpha chi_E g(u)tfrac{u}{|u|} quad text{ in }Omega $$ subject to Dirichlet, Neumann or Steklov boundary conditions. We analyze existence of an optimal configuration, symmetry properties of them, and the asymptotic behavior as $alpha$ approaches $+infty$.
{"title":"Nonstandard growth optimization problems with volume constraint","authors":"A. Salort, Belem Schvager, Analía Silva","doi":"10.57262/die036-0708-573","DOIUrl":"https://doi.org/10.57262/die036-0708-573","url":null,"abstract":"In this article we study some optimal design problems related to nonstandard growth eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $Omegasubset R^n$ and $alpha,c>0$ we consider the optimization problem $inf { lambda_Omega(alpha,E)colon Esubset Omega, |E|=c }$, where $lambda_Omega(alpha,E)$ is related to the first eigenvalue to $$ -text{div}(g( |nabla u |)tfrac{nabla u}{|nabla u|}) + g(u)tfrac{u}{|u|}+ alpha chi_E g(u)tfrac{u}{|u|} quad text{ in }Omega $$ subject to Dirichlet, Neumann or Steklov boundary conditions. We analyze existence of an optimal configuration, symmetry properties of them, and the asymptotic behavior as $alpha$ approaches $+infty$.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45414308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-28DOI: 10.57262/die036-1112-947
C. O. Alves, E. D. S. Boer, O. Miyagaki
In the present work we are concerned with the existence of normalized solutions to the following Schr"odinger-Poisson System $$ left{ begin{array}{ll} -Delta u + lambda u + mu (ln|cdot|ast |u|^{2})u = f(u) textrm{ in } mathbb{R}^2 , intR |u(x)|^2 dx = c, c>0 , end{array} right. $$ for $mu in R $ and a nonlinearity $f$ with exponential critical growth. Here $ lambdain R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in cite{Ji} and cite{[cjj]}.
{"title":"Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity","authors":"C. O. Alves, E. D. S. Boer, O. Miyagaki","doi":"10.57262/die036-1112-947","DOIUrl":"https://doi.org/10.57262/die036-1112-947","url":null,"abstract":"In the present work we are concerned with the existence of normalized solutions to the following Schr\"odinger-Poisson System $$ left{ begin{array}{ll} -Delta u + lambda u + mu (ln|cdot|ast |u|^{2})u = f(u) textrm{ in } mathbb{R}^2 , intR |u(x)|^2 dx = c, c>0 , end{array} right. $$ for $mu in R $ and a nonlinearity $f$ with exponential critical growth. Here $ lambdain R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in cite{Ji} and cite{[cjj]}.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43015765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.57262/die035-1112-795
A. Salort, E. Vecchi
In this paper we consider a H'{e}non-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s to 1$.
{"title":"On the mixed local-nonlocal Hénon equation","authors":"A. Salort, E. Vecchi","doi":"10.57262/die035-1112-795","DOIUrl":"https://doi.org/10.57262/die035-1112-795","url":null,"abstract":"In this paper we consider a H'{e}non-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s to 1$.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43752632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: (−∆)u = −λ|u|u+ au+ b
{"title":"Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity","authors":"S. Rawat, K. Sreenadh","doi":"10.57262/die035-0102-89","DOIUrl":"https://doi.org/10.57262/die035-0102-89","url":null,"abstract":"In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: (−∆)u = −λ|u|u+ au+ b ","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43135110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-08DOI: 10.57262/die035-0506-299
Xiaobing H. Feng, Mitchell Sutton
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.
{"title":"On a new class of fractional calculus of variations and related fractional differential equations","authors":"Xiaobing H. Feng, Mitchell Sutton","doi":"10.57262/die035-0506-299","DOIUrl":"https://doi.org/10.57262/die035-0506-299","url":null,"abstract":"This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43015411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a given Finsler-Minkowski norm $mathcal{F}$ in $mathbb{R}^N$ and a bounded smooth domain $Omegasubsetmathbb{R}^N$ $big(Ngeq 2big)$, we establish the following weighted anisotropic Sobolev inequality $$ Sleft(int_{Omega}|u|^q f,dxright)^frac{1}{q}leqleft(int_{Omega}mathcal{F}(nabla u)^p w,dxright)^frac{1}{p},quadforall,uin W_0^{1,p}(Omega,w)leqno{mathcal{(P)}} $$ where $W_0^{1,p}(Omega,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Omega$. We discuss the case $0
{"title":"Weighted anisotropic Sobolev inequality with extremal and associated singular problems","authors":"K. Bal, Prashanta Garain","doi":"10.57262/die036-0102-59","DOIUrl":"https://doi.org/10.57262/die036-0102-59","url":null,"abstract":"For a given Finsler-Minkowski norm $mathcal{F}$ in $mathbb{R}^N$ and a bounded smooth domain $Omegasubsetmathbb{R}^N$ $big(Ngeq 2big)$, we establish the following weighted anisotropic Sobolev inequality $$ Sleft(int_{Omega}|u|^q f,dxright)^frac{1}{q}leqleft(int_{Omega}mathcal{F}(nabla u)^p w,dxright)^frac{1}{p},quadforall,uin W_0^{1,p}(Omega,w)leqno{mathcal{(P)}} $$ where $W_0^{1,p}(Omega,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Omega$. We discuss the case $0<q<1$ and observe that $$ mu(Omega):=inf_{uin W_{0}^{1,p}(Omega,w)}Bigg{int_{Omega}mathcal{F}(nabla u)^p w,dx:int_{Omega}|u|^{q}f,dx=1Bigg}leqno{mathcal{(Q)}} $$ is associated with singular weighted anisotropic $p$-Laplace equations. To this end, we also study existence and regularity properties of solutions for weighted anisotropic $p$-Laplace equations under the mixed and exponential singularities.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42815386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-30DOI: 10.57262/die035-0506-241
H. Walther
Let r > 0, n ∈ N,k ∈ N. Consider the delay differential equation x(t) = g(x(t− d1(Lxt)), . . . , x(t− dk(Lxt))) for g : (R) ⊃ V → R continuously differentiable, L a continuous linear map from C([−r, 0],R) into a finite-dimensional vectorspace F , each dk : F ⊃ W → [0, r], k = 1, . . . ,k, continuously differentiable, and xt(s) = x(t + s). The solutions define a semiflow of continuously differentiable solution operators on the submanifold Xf ⊂ C([−r, 0],R) which is given by the compatibility condition φ′(0) = f(φ) with f(φ) = g(φ(−d1(Lφ)), . . . , φ(−dk(Lφ))). We prove that Xf has a finite atlas of at most 2 k manifold charts, whose domains are almost graphs over X0. The size of the atlas depends solely on the zerosets of the delay functions dk.
{"title":"A finite atlas for solution manifolds of differential systems with discrete state-dependent delays","authors":"H. Walther","doi":"10.57262/die035-0506-241","DOIUrl":"https://doi.org/10.57262/die035-0506-241","url":null,"abstract":"Let r > 0, n ∈ N,k ∈ N. Consider the delay differential equation x(t) = g(x(t− d1(Lxt)), . . . , x(t− dk(Lxt))) for g : (R) ⊃ V → R continuously differentiable, L a continuous linear map from C([−r, 0],R) into a finite-dimensional vectorspace F , each dk : F ⊃ W → [0, r], k = 1, . . . ,k, continuously differentiable, and xt(s) = x(t + s). The solutions define a semiflow of continuously differentiable solution operators on the submanifold Xf ⊂ C([−r, 0],R) which is given by the compatibility condition φ′(0) = f(φ) with f(φ) = g(φ(−d1(Lφ)), . . . , φ(−dk(Lφ))). We prove that Xf has a finite atlas of at most 2 k manifold charts, whose domains are almost graphs over X0. The size of the atlas depends solely on the zerosets of the delay functions dk.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48833202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.57262/die034-0708-425
Yonggeun Cho, Kiyoen Lee
We revisit the Cauchy problem of nonlinear massive Dirac equation with Yukawa type potentials F [ (b + |ξ|) ] in 2 dimensions. The authors of [10, 4] obtained small data scattering and large data global well-posedness in H for s > 0, respectively. In this paper we show that the small data scattering occurs in L x (R). This can be done by combining bilinear estimates and modulation estimates of [12, 10].
{"title":"Small data scattering of Dirac equations with Yukawa type\u0000 potentials in $L_x^2(mathbb R^2)$","authors":"Yonggeun Cho, Kiyoen Lee","doi":"10.57262/die034-0708-425","DOIUrl":"https://doi.org/10.57262/die034-0708-425","url":null,"abstract":"We revisit the Cauchy problem of nonlinear massive Dirac equation with Yukawa type potentials F [ (b + |ξ|) ] in 2 dimensions. The authors of [10, 4] obtained small data scattering and large data global well-posedness in H for s > 0, respectively. In this paper we show that the small data scattering occurs in L x (R). This can be done by combining bilinear estimates and modulation estimates of [12, 10].","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48670688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.57262/die034-0708-337
Hyungjin Huh
{"title":"Remarks on the Ginzburg-Landau-Chern-Simons\u0000 equations","authors":"Hyungjin Huh","doi":"10.57262/die034-0708-337","DOIUrl":"https://doi.org/10.57262/die034-0708-337","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43113101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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