This paper is concerned with the following quasilinear Schr"{o}dinger system in the entire space $mathbb R^{N}$($Ngeq3$): $$left{begin{align}&-Delta u+A(x)u-frac{1}{2}triangle(u^{2})u = frac{2alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&-Delta v+Bv-frac{1}{2}triangle(v^{2})v=frac{2beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v.end{align}right. $$ By establishing a suitable constraint set and studying related minimization problem, we prove the existence of ground state solution for $alpha,beta>1$, $2
{"title":"Existence of ground state solution of Nehari-Pohožaev type for a quasilinear Schrödinger system","authors":"Jianqing Chen, Qian Zhang","doi":"10.57262/die/1610420451","DOIUrl":"https://doi.org/10.57262/die/1610420451","url":null,"abstract":"This paper is concerned with the following quasilinear Schr\"{o}dinger system in the entire space $mathbb R^{N}$($Ngeq3$): $$left{begin{align}&-Delta u+A(x)u-frac{1}{2}triangle(u^{2})u = frac{2alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&-Delta v+Bv-frac{1}{2}triangle(v^{2})v=frac{2beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v.end{align}right. $$ By establishing a suitable constraint set and studying related minimization problem, we prove the existence of ground state solution for $alpha,beta>1$, $2<alpha+beta<frac{4N}{N-2}$. Our results can be looked on as a generalization to results by Guo and Tang (Ground state solutions for quasilinear Schr\"{o}dinger systems, J. Math. Anal. Appl. 389 (2012) 322).","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42275765","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}
{"title":"On the nonlinear Schrödinger equation in spaces of infinite mass and low regularity","authors":"Vanessa Barros, Simão Correia, Filipe Oliveira","doi":"10.57262/die035-0708-371","DOIUrl":"https://doi.org/10.57262/die035-0708-371","url":null,"abstract":"We study the nonlinear Schr\"odinger equation with initial data in $mathcal{Z}^s_p(mathbb{R}^d)=dot{H}^s(mathbb{R}^d)cap L^p(mathbb{R}^d)$, where $0<s<min{d/2,1}$ and $2<p<2d/(d-2s)$. After showing that the linear Schr\"odinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44182612","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}
{"title":"Well-posedness of the initial-boundary value problem for the Schrödinger-Boussinesq system","authors":"B. Guo, Rudong Zheng","doi":"10.57262/die/1605150096","DOIUrl":"https://doi.org/10.57262/die/1605150096","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44443245","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}
{"title":"Nontrivial solutions for a quasilinear elliptic system with weight functions","authors":"Xiyou Cheng, Zhaosheng Feng, Lei Wei","doi":"10.57262/die/1605150095","DOIUrl":"https://doi.org/10.57262/die/1605150095","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42635326","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 : 2020-10-09DOI: 10.57262/die034-0506-295
J. Holmes, F. Tiglay, R. Thompson
For Besov spaces $B^s_{p,r}(rr)$ with $s>max{ 2 + frac1p , frac52} $, $p in (1,infty]$ and $r in [1 , infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(rr)$ to $C([0,T]; B^s_{p,r}(rr))$. The proof of non-uniform dependence is based on approximate solutions and the Littlewood-Paley decomposition.
对于含有$s>max{ 2 + frac1p , frac52} $、$p in (1,infty]$和$r in [1 , infty)$的Besov空间$B^s_{p,r}(rr)$，证明了FORQ方程的数据-解映射从$B^s_{p,r}(rr)$到$C([0,T]; B^s_{p,r}(rr))$不是一致连续的。非一致相关性的证明是基于近似解和Littlewood-Paley分解。
{"title":"Continuity of the data-to-solution map for the FORQ equation in Besov spaces","authors":"J. Holmes, F. Tiglay, R. Thompson","doi":"10.57262/die034-0506-295","DOIUrl":"https://doi.org/10.57262/die034-0506-295","url":null,"abstract":"For Besov spaces $B^s_{p,r}(rr)$ with $s>max{ 2 + frac1p , frac52} $, $p in (1,infty]$ and $r in [1 , infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(rr)$ to $C([0,T]; B^s_{p,r}(rr))$. The proof of non-uniform dependence is based on approximate solutions and the Littlewood-Paley decomposition.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46006285","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 : 2020-10-04DOI: 10.57262/die034-0708-383
S. Takeuchi, Kohtaro Watanabe
This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian begin{align*} begin{cases} left(vert u'vert^{p-2}u'right)'+left(lambda+r(x)right)vert uvert ^{p-2}u=0,,, xin (0,pi_{p}), u(0)=u(pi_{p})=0, end{cases} end{align*} where $1
{"title":"Lyapunov-type inequalities for a Sturm-Liouville problem of the\u0000 one-dimensional p-Laplacian","authors":"S. Takeuchi, Kohtaro Watanabe","doi":"10.57262/die034-0708-383","DOIUrl":"https://doi.org/10.57262/die034-0708-383","url":null,"abstract":"This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian begin{align*} begin{cases} left(vert u'vert^{p-2}u'right)'+left(lambda+r(x)right)vert uvert ^{p-2}u=0,,, xin (0,pi_{p}), u(0)=u(pi_{p})=0, end{cases} end{align*} where $1<p<infty$, $pi_{p}$ is the generalized $pi$ given by $pi_{p}=2pi/left(psin(pi/p)right)$, $rin C[0,pi_{p}]$ and $lambda<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41580286","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 : 2020-09-04DOI: 10.57262/die034-0506-245
Takayuki Kobayashi, M. Murata
In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As a result, we also prove the decay estimates of the solutions to the nonliner problem. In order to obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for a low frequencies.
{"title":"The global well-posedness of the compressible fluid model of Korteweg type for the critical case","authors":"Takayuki Kobayashi, M. Murata","doi":"10.57262/die034-0506-245","DOIUrl":"https://doi.org/10.57262/die034-0506-245","url":null,"abstract":"In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As a result, we also prove the decay estimates of the solutions to the nonliner problem. In order to obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for a low frequencies.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44687760","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}
{"title":"Fast diffusion equations on Riemannian manifolds","authors":"S. Bakim, G. Goldstein, J. Goldstein, I. Kombe","doi":"10.57262/die/1600135324","DOIUrl":"https://doi.org/10.57262/die/1600135324","url":null,"abstract":"","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45040660","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}
The objective of the paper is to characterize the Gaussian free field as a stationary solution of the heat equation with additive space-time white noise. In the case of whole space, the investigation leads to other types of Gaussian fields, as well as interesting phenomena in dimensions one and two.
{"title":"Gaussian fields and stochastic heat equations","authors":"S. Lototsky, Apoorva Shah","doi":"10.57262/die/1600135325","DOIUrl":"https://doi.org/10.57262/die/1600135325","url":null,"abstract":"The objective of the paper is to characterize the Gaussian free field as a stationary solution of the heat equation with additive space-time white noise. In the case of whole space, the investigation leads to other types of Gaussian fields, as well as interesting phenomena in dimensions one and two.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44890078","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 : 2020-07-06DOI: 10.57262/die036-0910-813
Jin Deng, W. Long, Jianfu Yang
In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrodinger system begin{equation}label{eqabstr} left{begin{array}{ll} -ihD_0Psi-h^2(D_1D_1+D_2D_2)Psi+VPsi=|Psi|^{p-2}Psi, partial_0A_1-partial_1A_0=-frac 12ih[overline{Psi}D_2Psi-Psioverline{D_2Psi}], partial_0A_2-partial_2A_0=frac 12ih[overline{Psi}D_1Psi-Psioverline{D_1Psi}], partial_1A_2-partial_2A_1=-frac12|Psi|^2, end{array} right. end{equation} where $p>2$ and non-radial potential $V(x)$ satisfies some certain conditions. We show that for every positive integer $k$, there exists $h_0>0$ such that for $0
{"title":"Multi-Peak solutions to Chern-Simons-Schrödinger systems with non-radial potential","authors":"Jin Deng, W. Long, Jianfu Yang","doi":"10.57262/die036-0910-813","DOIUrl":"https://doi.org/10.57262/die036-0910-813","url":null,"abstract":"In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrodinger system begin{equation}label{eqabstr} left{begin{array}{ll} -ihD_0Psi-h^2(D_1D_1+D_2D_2)Psi+VPsi=|Psi|^{p-2}Psi, partial_0A_1-partial_1A_0=-frac 12ih[overline{Psi}D_2Psi-Psioverline{D_2Psi}], partial_0A_2-partial_2A_0=frac 12ih[overline{Psi}D_1Psi-Psioverline{D_1Psi}], partial_1A_2-partial_2A_1=-frac12|Psi|^2, end{array} right. end{equation} where $p>2$ and non-radial potential $V(x)$ satisfies some certain conditions. We show that for every positive integer $k$, there exists $h_0>0$ such that for $0<h<h_0$, problem eqref{eqabstr} has a nontrivial static solution $(Psi_h, A_0^h, A_1^h,A_2^h)$. Moreover, $Psi_h$ is a positive non-radial function with $k$ positive peaks, which approach to the local maximum point of $V(x)$ as $hto 0^+$.","PeriodicalId":50581,"journal":{"name":"Differential and Integral Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48592208","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：