Calculus of Variations and Partial Differential Equations最新文献

{"title":"Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs","authors":"Wilfrid Gangbo, Chenchen Mou, Andrzej Święch","doi":"10.1007/s00526-024-02758-w","DOIUrl":"https://doi.org/10.1007/s00526-024-02758-w","url":null,"abstract":"<p>In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron’s method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a $$mathbb {C}$$ -convex ring","authors":"Jingchen Hu","doi":"10.1007/s00526-024-02764-y","DOIUrl":"https://doi.org/10.1007/s00526-024-02764-y","url":null,"abstract":"<p>Suppose <span>(Omega _0,Omega _1)</span> are two bounded strongly <span>(mathbb {C})</span>-convex domains in <span>(mathbb {C}^n)</span>, with <span>(nge 2)</span> and <span>(Omega _1supset overline{Omega _0})</span>. Let <span>(mathcal {R}=Omega _1backslash overline{Omega _0})</span>. We call <span>(mathcal {R})</span> a <span>(mathbb {C})</span>-convex ring. We will show that for a solution <span>(Phi )</span> to the homogenous complex Monge–Ampère equation in <span>(mathcal {R})</span>, with <span>(Phi =1)</span> on <span>(partial Omega _1)</span> and <span>(Phi =0)</span> on <span>(partial Omega _0)</span>, <span>(sqrt{-1}partial {overline{partial }}Phi )</span> has rank <span>(n-1)</span> and the level sets of <span>(Phi )</span> are strongly <span>(mathbb {C})</span>-convex.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Morse index of concentrated solutions for the nonlinear Schrödinger equation with a very degenerate potential","authors":"Peng Luo, Kefan Pan, Shuangjie Peng","doi":"10.1007/s00526-024-02766-w","DOIUrl":"https://doi.org/10.1007/s00526-024-02766-w","url":null,"abstract":"<p>We revisit the following nonlinear Schrödinger equation </p><span>$$begin{aligned} -varepsilon ^2Delta u+ V(x)u=u^{p},quad u&gt;0,;; uin H^1({mathbb {R}}^N), end{aligned}$$</span><p>where <span>(varepsilon &gt;0)</span> is a small parameter, <span>(Nge 2)</span> and <span>(1&lt;p&lt;2^*-1)</span>. It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive <i>k</i>-peak solutions to above problem when the critical points of <i>V</i>(<i>x</i>) are non-isolated and degenerate. We also give a specific formula for the Morse index of <i>k</i>-peak solutions when the critical point set of <i>V</i>(<i>x</i>) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential <i>V</i>(<i>x</i>). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the optimality and decay of p-Hardy weights on graphs","authors":"Florian Fischer","doi":"10.1007/s00526-024-02754-0","DOIUrl":"https://doi.org/10.1007/s00526-024-02754-0","url":null,"abstract":"<p>We construct optimal Hardy weights to subcritical energy functionals <i>h</i> associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight <i>w</i> is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional <span>(h-w)</span> is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hypersurfaces of $$mathbb {S}^2times mathbb {S}^2$$ with constant sectional curvature","authors":"Haizhong Li, Luc Vrancken, Xianfeng Wang, Zeke Yao","doi":"10.1007/s00526-024-02765-x","DOIUrl":"https://doi.org/10.1007/s00526-024-02765-x","url":null,"abstract":"<p>In this paper, we classify the hypersurfaces of <span>(mathbb {S}^2times mathbb {S}^2)</span> with constant sectional curvature. We prove that the constant sectional curvature can only be <span>(frac{1}{2})</span>. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in <span>(mathbb {S}^2times mathbb {S}^2)</span>, and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous “sinh-Gordon equation” <span>( left( frac{partial ^2}{partial u^2}+frac{partial ^2}{partial v^2}right) h =-tfrac{1}{sqrt{2}}sinh left( sqrt{2}hright) )</span>. </p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Palais–Smale sequences for the prescribed Ricci curvature functional","authors":"Artem Pulemotov, Wolfgang Ziller","doi":"10.1007/s00526-024-02776-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02776-8","url":null,"abstract":"<p>We obtain a complete description of divergent Palais–Smale sequences for the prescribed Ricci curvature functional on compact homogeneous spaces. As an application, we prove the existence of saddle points on generalized Wallach spaces and several types of generalized flag manifolds. We also describe the image of the Ricci map in some of our examples.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Liouville theorem for elliptic equations with a potential on infinite graphs","authors":"Stefano Biagi, Giulia Meglioli, Fabio Punzo","doi":"10.1007/s00526-024-02768-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02768-8","url":null,"abstract":"<p>We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is <span>(uequiv 0)</span>. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perturbation limiting behaviors of normalized ground states to focusing mass-critical Hartree equations with Local repulsion","authors":"Deke Li, Qingxuan Wang","doi":"10.1007/s00526-024-02772-y","DOIUrl":"https://doi.org/10.1007/s00526-024-02772-y","url":null,"abstract":"<p>In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential </p><span>$$begin{aligned} ipartial _tpsi =-Delta psi +|x|^2psi -(|x|^{-2}*|psi |^2) psi +varepsilon |psi |^{p-2}psi , text {in} mathbb {R}^+ times mathbb {R}^N, end{aligned}$$</span><p>where <span>(Nge 3)</span>, <span>(2&lt;p&lt;2^*={2N}/({N-2}))</span> and <span>(varepsilon &gt;0)</span>. We mainly focus on the normalized ground state solitary waves of the form <span>(psi (t,x)=e^{imu t}u_{varepsilon ,rho }(x))</span>, where <span>(u_{varepsilon ,rho }(x))</span> is radially symmetric-decreasing and <span>(int _{mathbb {R}^N}|u_{varepsilon ,rho }|^2,dx=rho )</span>. Firstly, we prove the existence and nonexistence of normalized ground states under the <span>(L^2)</span>-subcritical, <span>(L^2)</span>-critical (<span>(p=4/N +2)</span>) and <span>(L^2)</span>-supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states <span>(u_{varepsilon ,rho })</span> as <span>(varepsilon rightarrow 0^+)</span> and find that the <span>(varepsilon )</span>-blow-up phenomenon happens for <span>(rho ge rho _c=Vert QVert ^2_{L^2})</span>, where <i>Q</i> is a positive radially symmetric ground state of <span>(-Delta u+u-(|x|^{-2}*|u|^2)u=0)</span> in <span>(mathbb {R}^N)</span>. We prove that <span>(int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }(x)|^2,dxsim varepsilon ^{-frac{4}{N(p-2)+4}})</span> for <span>(rho =rho _c)</span> and <span>(2&lt;p&lt;2^*)</span>, while <span>(int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }|^2,dxsim varepsilon ^{-frac{4}{N(p-2)-4}})</span> for <span>(rho &gt;rho _c)</span> and <span>(4/N+2&lt;p&lt;2^*)</span>, and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as <span>(varepsilon rightarrow +infty )</span>, which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer <span>(u^{TF}=left[ mu ^{TF}-|x|^2 right] ^{frac{1}{p-2}}_{+})</span>, where <span>(mu ^{TF})</span> is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for <span>(u_{varepsilon , rho })</span> that <span>(Vert u_{varepsilon , rho }Vert _{L^{infty }}sim varepsilon ^{-frac{N}{N(p-2)+4}})</span> as <span>(varepsilon rightarrow +infty )</span>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A class of fully nonlinear equations on Riemannian manifolds with negative curvature","authors":"Li Chen, Yan He","doi":"10.1007/s00526-024-02756-y","DOIUrl":"https://doi.org/10.1007/s00526-024-02756-y","url":null,"abstract":"<p>In this paper, we consider a class of fully nonlinear equations on Riemannian manifolds with negative curvature which naturally arise in conformal geometry. Moreover, we prove the a priori estimates for solutions to these equations and establish the existence results. Our results can be viewed as an extension of previous results given by Gursky–Viaclovsky and Li–Sheng.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Harnack inequalities and quantization properties for the $$n-$$ Liouville equation","authors":"Pierpaolo Esposito, Marcello Lucia","doi":"10.1007/s00526-024-02777-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02777-7","url":null,"abstract":"<p>We consider a quasilinear equation involving the <span>(n-)</span>Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtained by the first author in Esposito (Ann. Inst. H. Poincaré Anal. Non Linéaire 35(3), 781–801, 2018). A fundamental tool is provided here by some Harnack inequality of “sup+inf\" type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}