Pub Date : 2021-04-01DOI: 10.57262/ade026-0506-259
Xiaofei Cao, B. Ge, Beilei Zhang
{"title":"On a class of $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions","authors":"Xiaofei Cao, B. Ge, Beilei Zhang","doi":"10.57262/ade026-0506-259","DOIUrl":"https://doi.org/10.57262/ade026-0506-259","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49285739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.57262/ade026-0506-229
G. Bonanno, G. D'Aguí, A. Sciammetta
. This paper deals with the existence of nontrivial solutions for a class of nonlinear elliptic equations driven by an anisotropic Laplacian operator. In particular, the existence of two nontrivial solutions is obtained, adapting a two critical point result to a suitable functional framework that involves the anisotropic Sobolev spaces.
{"title":"Existence of two positive solutions for anisotropic nonlinear elliptic equations","authors":"G. Bonanno, G. D'Aguí, A. Sciammetta","doi":"10.57262/ade026-0506-229","DOIUrl":"https://doi.org/10.57262/ade026-0506-229","url":null,"abstract":". This paper deals with the existence of nontrivial solutions for a class of nonlinear elliptic equations driven by an anisotropic Laplacian operator. In particular, the existence of two nontrivial solutions is obtained, adapting a two critical point result to a suitable functional framework that involves the anisotropic Sobolev spaces.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46605541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-25DOI: 10.57262/ade027-1112-683
M. Bravin, F. Sueur
A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich’s paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on a priori bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a Lp integrability in space, with p in [1,+∞], and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where p > 4 3 , in the renormalized sense in the case where p > 1, and in a symmetrized sense in the case where p = 1.
{"title":"Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks","authors":"M. Bravin, F. Sueur","doi":"10.57262/ade027-1112-683","DOIUrl":"https://doi.org/10.57262/ade027-1112-683","url":null,"abstract":"A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich’s paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on a priori bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a Lp integrability in space, with p in [1,+∞], and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where p > 4 3 , in the renormalized sense in the case where p > 1, and in a symmetrized sense in the case where p = 1.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42295545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.57262/ade026-0304-107
Huite Jiang, Caidi Zhao
{"title":"Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth","authors":"Huite Jiang, Caidi Zhao","doi":"10.57262/ade026-0304-107","DOIUrl":"https://doi.org/10.57262/ade026-0304-107","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42054359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-08DOI: 10.57262/ade027-0708-467
Jacopo Schino
We look for ground state solutions to the Schödinger-type system −∆uj + λjuj = ∂jF (u) ∫ R u2j dx = a 2 j (λj , uj) ∈ R×H1(RN ) j ∈ {1, . . . ,M} with N ≥ 1 and 1 ≤ M < 2 + 4/N , where a = (a1, . . . , aM ) ∈]0,∞[M is prescribed and (λ, u) = (λ1, . . . , λM , u1, . . . uM ) is the unknown. We provide generic assumptions on the nonlinearity F which correspond to the L-subcritical and L-critical cases, i.e., when the energy is bounded from below for all or some values of a. Making use of a recent idea, we minimize the energy over the constraint { |uj |L2 ≤ aj for all j } and then provide further assumptions that ensure |uj |L2 = aj .
{"title":"Normalized ground states to a cooperative system of Schrödinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity","authors":"Jacopo Schino","doi":"10.57262/ade027-0708-467","DOIUrl":"https://doi.org/10.57262/ade027-0708-467","url":null,"abstract":"We look for ground state solutions to the Schödinger-type system −∆uj + λjuj = ∂jF (u) ∫ R u2j dx = a 2 j (λj , uj) ∈ R×H1(RN ) j ∈ {1, . . . ,M} with N ≥ 1 and 1 ≤ M < 2 + 4/N , where a = (a1, . . . , aM ) ∈]0,∞[M is prescribed and (λ, u) = (λ1, . . . , λM , u1, . . . uM ) is the unknown. We provide generic assumptions on the nonlinearity F which correspond to the L-subcritical and L-critical cases, i.e., when the energy is bounded from below for all or some values of a. Making use of a recent idea, we minimize the energy over the constraint { |uj |L2 ≤ aj for all j } and then provide further assumptions that ensure |uj |L2 = aj .","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44222008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-02DOI: 10.57262/ade026-0910-397
Marco Gallo
Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrodinger equation $$ varepsilon^{2s}(- Delta)^s u+ V(x) u= f(u), quad x in mathbb{R}^N,$$ where $s in (0,1)$, $N geq 2$, $V in C(mathbb{R}^N,mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for $varepsilon>0$ small, where the number of solutions is related to the cup-length of a set of local minima of $V$. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. We highlight that these results are new also in the limiting local setting $s=1$ and $Ngeq 3$, with an exponential decay of the solutions.
本文的目的是研究非线性薛定谔方程$$ varepsilon^{2s}(- Delta)^s u+ V(x) u= f(u), quad x in mathbb{R}^N,$$的正半经典解，其中$s in (0,1)$, $N geq 2$, $V in C(mathbb{R}^N,mathbb{R})$为正势，$f$为临界且满足一般Berestycki-Lions型条件。我们得到了$varepsilon>0$小问题的存在性和多重性，其中解的个数与$V$的一组局部极小值的杯长有关。此外，这些解被证明集中在势井中，表现出多项式衰减。我们强调，这些结果在极限局部设置$s=1$和$Ngeq 3$中也是新的，解呈指数衰减。
{"title":"Multiplicity and concentration results for local and fractional NLS equations with critical growth","authors":"Marco Gallo","doi":"10.57262/ade026-0910-397","DOIUrl":"https://doi.org/10.57262/ade026-0910-397","url":null,"abstract":"Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrodinger equation $$ varepsilon^{2s}(- Delta)^s u+ V(x) u= f(u), quad x in mathbb{R}^N,$$ where $s in (0,1)$, $N geq 2$, $V in C(mathbb{R}^N,mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for $varepsilon>0$ small, where the number of solutions is related to the cup-length of a set of local minima of $V$. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. We highlight that these results are new also in the limiting local setting $s=1$ and $Ngeq 3$, with an exponential decay of the solutions.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48146472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Strong attractors and their continuity for the semilinear wave equations with fractional damping","authors":"Yanan Li, Zhijian Yang","doi":"10.57262/ade/1610420434","DOIUrl":"https://doi.org/10.57262/ade/1610420434","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential 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":"42691774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence and $W^{1,p}$ estimates of certain Maxwell type equations in Reifenberg domains","authors":"Zhihong Chen, Dongsheng Li","doi":"10.57262/ade/1610420435","DOIUrl":"https://doi.org/10.57262/ade/1610420435","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential 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":"46159920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Navier-Stokes-Darcy-Boussinesq system that models the thermal convection of a fluid overlying a saturated porous medium in a general decomposed domain. In both two and three spatial dimensions, we prove existence of global weak solutions to the initial boundary value problem subject to the Lions and Beavers-Joseph-Saffman-Jones interface conditions. The proof is based on a proper time-implicit discretization scheme combined the compactness argument. Next, we establish a weak-strong uniqueness result such that a weak solution coincides with a strong solution emanating from the same initial data as long as the latter exists.
{"title":"Global weak solutions to the Navier-Stokes-Darcy-Boussinesq system for thermal convection in coupled free and porous media flows","authors":"Xiaoming Wang, Hao Wu","doi":"10.57262/ade/1610420433","DOIUrl":"https://doi.org/10.57262/ade/1610420433","url":null,"abstract":"We study the Navier-Stokes-Darcy-Boussinesq system that models the thermal convection of a fluid overlying a saturated porous medium in a general decomposed domain. In both two and three spatial dimensions, we prove existence of global weak solutions to the initial boundary value problem subject to the Lions and Beavers-Joseph-Saffman-Jones interface conditions. The proof is based on a proper time-implicit discretization scheme combined the compactness argument. Next, we establish a weak-strong uniqueness result such that a weak solution coincides with a strong solution emanating from the same initial data as long as the latter exists.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47483482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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