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Saddle solutions for the planar Schrödinger–Poisson system with exponential growth 具有指数增长的平面薛定谔-泊松系统的鞍解法
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-15 DOI: 10.1007/s00030-024-00980-1
Liying Shan, Wei Shuai
{"title":"Saddle solutions for the planar Schrödinger–Poisson system with exponential growth","authors":"Liying Shan, Wei Shuai","doi":"10.1007/s00030-024-00980-1","DOIUrl":"https://doi.org/10.1007/s00030-024-00980-1","url":null,"abstract":"<p>In this paper, we are interested in the following planar Schrödinger–Poisson system </p><span>$$begin{aligned} left{ begin{array}{ll} -Delta u+a(x)u+2pi phi u=|u|^{p-2}ue^{alpha _0|u|^gamma }, {} &amp;{} xin {mathbb {R}}^2, Delta phi =u^2, {} &amp;{} xin {mathbb {R}}^2, end{array} right. end{aligned}$$</span>(0.1)<p>where <span>(p&gt;2)</span>, <span>(alpha _0&gt;0)</span> and <span>(0&lt;gamma le 2)</span>, the potential <span>(a:{mathbb {R}}^2rightarrow {mathbb {R}})</span> is invariant under the action of a closed subgroup of the orthogonal transformation group <i>O</i>(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if <span>(0&lt;gamma &lt;2)</span> and <span>(p&gt;2)</span>. Furthermore, in the critical case <span>(gamma =2)</span> and <span>(p&gt;4)</span>, we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games 海森堡周期情况下的连续性方程：表示公式及平均场博弈的应用
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-11 DOI: 10.1007/s00030-024-00967-y
Alessandra Cutrì, Paola Mannucci, Claudio Marchi, Nicoletta Tchou
{"title":"The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games","authors":"Alessandra Cutrì, Paola Mannucci, Claudio Marchi, Nicoletta Tchou","doi":"10.1007/s00030-024-00967-y","DOIUrl":"https://doi.org/10.1007/s00030-024-00967-y","url":null,"abstract":"<p>We provide a representation of the weak solution of the continuity equation on the Heisenberg group <span>({mathbb {H}}^1)</span> with periodic data (the periodicity is suitably adapted to the group law). This solution is the push forward of a measure concentrated on the flux associated with the drift of the continuity equation. Furthermore, we shall use this interpretation for proving that weak solutions to first order Mean Field Games on <span>({mathbb {H}}^1)</span> are also mild solutions.\u0000</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$$C^{1,alpha }$$ regularity for stationary mean-field games with logarithmic coupling 具有对数耦合的静态均场博弈的 $$C^{1,α }$$ 正则性
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-10 DOI: 10.1007/s00030-024-00976-x
Tigran Bakaryan, Giuseppe Di Fazio, Diogo A. Gomes
{"title":"$$C^{1,alpha }$$ regularity for stationary mean-field games with logarithmic coupling","authors":"Tigran Bakaryan, Giuseppe Di Fazio, Diogo A. Gomes","doi":"10.1007/s00030-024-00976-x","DOIUrl":"https://doi.org/10.1007/s00030-024-00976-x","url":null,"abstract":"<p>This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of <span>(C^{1,alpha })</span> solutions to address the research gap between low-regularity results for bounded and measurable diffusions and the smooth results modeled by the Laplacian. We use the Hopf-Cole transformation to convert the MFG system into a scalar elliptic equation. Then, we apply Morrey space methods to establish the existence and regularity of solutions. The introduction of Morrey space methods offers a novel approach to address regularity issues in the context of MFGs.\u0000</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global stability of three trophic levels predator–prey model with alarm-taxis 带有警报-税率的三个营养级捕食者-猎物模型的全球稳定性
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-10 DOI: 10.1007/s00030-024-00978-9
Qingshan Zhang, Chao Chen
{"title":"Global stability of three trophic levels predator–prey model with alarm-taxis","authors":"Qingshan Zhang, Chao Chen","doi":"10.1007/s00030-024-00978-9","DOIUrl":"https://doi.org/10.1007/s00030-024-00978-9","url":null,"abstract":"<p>This paper is concerned with the three trophic levels predator–prey system with alarm-taxis </p><span>$$begin{aligned} left{ begin{array}{lll} u_{t}=d_{1} Delta u+uleft( 1-u-frac{a v}{v+rho }right) , &amp;{} x in Omega , &amp;{} t&gt;0, v_{t}=d_{2} Delta v+vleft( frac{b u}{v+rho }-alpha -frac{c w}{w+sigma }right) , &amp;{} x in Omega , &amp;{} t&gt;0, w_{t}=d_{3} Delta w-chi nabla cdot left( wnabla (uv)right) +wleft( frac{m v}{w+sigma }-beta right) , &amp;{} x in Omega , &amp;{} t&gt;0 end{array}right. end{aligned}$$</span><p>under homogeneous Neumann boundary condition in smooth bounded domains <span>(Omega subset {mathbb {R}}^n (nge 1))</span>. We prove that the system possesses a unique global bounded classical solution for all sufficiently smooth initial data. Moreover, we show the large time behavior of the solution with convergence rates and perform some numerical simulations to verify the analytic results.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{textbf {p}}}(cdot )$$ -Laplacian 各向异性 $${{textbf {p}}(cdot )$$ - 拉普拉卡方的弱解和粘性解的等价性和正则性
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-09 DOI: 10.1007/s00030-024-00981-0
Pablo Ochoa, Federico Ramos Valverde
{"title":"Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{textbf {p}}}(cdot )$$ -Laplacian","authors":"Pablo Ochoa, Federico Ramos Valverde","doi":"10.1007/s00030-024-00981-0","DOIUrl":"https://doi.org/10.1007/s00030-024-00981-0","url":null,"abstract":"<p>In this paper, we state the equivalence between weak and viscosity solutions for non-homogeneous problems involving the anisotropic <span>({{textbf {p}}}(cdot ))</span>-Laplacian. The proof that viscosity solutions are weak solutions is performed by the inf-convolution technique. However, due to the anisotropic nature of the <span>({{textbf {p}}}(cdot ))</span>-Laplacian we adapt the definition of inf-convolution to the non-homogeneity of this operator. For the converse, we develop comparison principles for weak solutions. Since the locally Lipschitz assumption is crucial to get the viscosity-weak implication, we prove that a class of bounded viscosity solutions are indeed locally Lipschitz.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Smooth solutions in a three-dimensional chemotaxis-Stokes system involving Dirichlet boundary conditions for the signal 涉及信号迪里希特边界条件的三维趋化-斯托克斯系统中的平滑解
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-09 DOI: 10.1007/s00030-024-00982-z
Yulan Wang, Michael Winkler, Zhaoyin Xiang
{"title":"Smooth solutions in a three-dimensional chemotaxis-Stokes system involving Dirichlet boundary conditions for the signal","authors":"Yulan Wang, Michael Winkler, Zhaoyin Xiang","doi":"10.1007/s00030-024-00982-z","DOIUrl":"https://doi.org/10.1007/s00030-024-00982-z","url":null,"abstract":"<p>In a smoothly bounded domain <span>(Omega subset mathbb {R}^3)</span>, the chemotaxis-Stokes system </p><span>$$begin{aligned} left{ begin{array}{l} n_t + ucdot nabla n = Delta n - nabla cdot (nnabla c), c_t + ucdot nabla c =Delta c - nc, u_t = Delta u + nabla P + nnabla phi , qquad nabla cdot u =0 end{array} right. end{aligned}$$</span><p>is considered along with the boundary conditions </p><span>$$begin{aligned} big (nabla n - nnabla cbig )cdot nu = 0, quad c=c_star , quad u=0, quad xin partial Omega , ,, t&gt;0, end{aligned}$$</span><p>where <span>(c_star ge 0)</span> is a given constant. It is shown that under a smallness condition on <span>(c(cdot ,0))</span> and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.\u0000</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiplicity results for elliptic problems with critical exponential growth 具有临界指数增长的椭圆问题的多重性结果
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-04 DOI: 10.1007/s00030-024-00973-0
Kanishka Perera
{"title":"Multiplicity results for elliptic problems with critical exponential growth","authors":"Kanishka Perera","doi":"10.1007/s00030-024-00973-0","DOIUrl":"https://doi.org/10.1007/s00030-024-00973-0","url":null,"abstract":"<p>We prove new multiplicity results for some elliptic problems with critical exponential growth. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter <span>(mu &gt; 0)</span>. In particular, the number of solutions goes to infinity as <span>(mu rightarrow infty )</span>. The proof is based on an abstract critical point theorem.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Normalized ground states for a coupled Schrödinger system: mass super-critical case 耦合薛定谔系统的归一化基态：质量超临界情况
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-07-03 DOI: 10.1007/s00030-024-00972-1
Louis Jeanjean, Jianjun Zhang, Xuexiu Zhong
{"title":"Normalized ground states for a coupled Schrödinger system: mass super-critical case","authors":"Louis Jeanjean, Jianjun Zhang, Xuexiu Zhong","doi":"10.1007/s00030-024-00972-1","DOIUrl":"https://doi.org/10.1007/s00030-024-00972-1","url":null,"abstract":"<p>We consider the existence of solutions <span>((lambda _1,lambda _2, u, v)in mathbb {R}^2times (H^1(mathbb {R}^N))^2)</span> to systems of coupled Schrödinger equations </p><span>$$begin{aligned} {left{ begin{array}{ll} -Delta u+lambda _1 u=mu _1 u^{p-1}+beta r_1 u^{r_1-1}v^{r_2}&amp;{}hbox {in}quad mathbb {R}^N, -Delta v+lambda _2 v=mu _2 v^{q-1}+beta r_2 u^{r_1}v^{r_2-1}&amp;{}hbox {in}quad mathbb {R}^N, 0&lt;u,vin H^1(mathbb {R}^N),quad 1le Nle 4,&amp;{} end{array}right. } end{aligned}$$</span><p>satisfying the normalization </p><span>$$begin{aligned} int _{mathbb {R}^N}u^2 textrm{d}x=a quad text{ and } quad int _{mathbb {R}^N}v^2 textrm{d}x=b. end{aligned}$$</span><p>Here <span>(mu _1,mu _2,beta &gt;0)</span> and the prescribed masses <span>(a,b&gt;0)</span>. We focus on the coupled purely mass super-critical case, i.e., </p><span>$$begin{aligned} 2+frac{4}{N}&lt;p,q,r_1+r_2&lt;2^* end{aligned}$$</span><p>with <span>(2^*=frac{2N}{(N-2)_+}, 1le Nle 4)</span> and give a partial affirmative answer to one open question in Bartsch et al. (J Math Pures Appl (9), 106(4):583–614, 2016). In particular, for <span>(N=3,4)</span> with <span>(r_1,r_2in (1,2))</span>, our result indicates the existence for all <span>(a,b&gt;0)</span> and <span>(beta &gt;0)</span>.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stackelberg exact controllability for the Boussinesq system 布辛斯克系统的堆栈伯格精确可控性
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-06-22 DOI: 10.1007/s00030-024-00971-2
Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang
{"title":"Stackelberg exact controllability for the Boussinesq system","authors":"Takéo Takahashi, Luz de Teresa, Yingying Wu-Zhang","doi":"10.1007/s00030-024-00971-2","DOIUrl":"https://doi.org/10.1007/s00030-024-00971-2","url":null,"abstract":"<p>We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the “leader” control is the null-controllability property whereas the objective of the “follower” control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Well-posedness of a bulk-surface convective Cahn–Hilliard system with dynamic boundary conditions 具有动态边界条件的体表对流卡恩-希利亚德系统的良好拟合性
Nonlinear Differential Equations and Applications (NoDEA) Pub Date : 2024-06-22 DOI: 10.1007/s00030-024-00970-3
Patrik Knopf, Jonas Stange
{"title":"Well-posedness of a bulk-surface convective Cahn–Hilliard system with dynamic boundary conditions","authors":"Patrik Knopf, Jonas Stange","doi":"10.1007/s00030-024-00970-3","DOIUrl":"https://doi.org/10.1007/s00030-024-00970-3","url":null,"abstract":"<p>We consider a general class of bulk-surface convective Cahn–Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters <span>(K,Lin [0,infty ])</span>, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case <span>(K,Lin (0,infty ))</span> by means of a Faedo–Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where <i>K</i> and <i>L</i> are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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