Calculus of Variations and Partial Differential Equations最新文献

{"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":"49 1","pages":""},"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":"49 1","pages":""},"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":"15 1","pages":""},"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":"9 1","pages":""},"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}

{"title":"Minimal measures beyond Mather","authors":"Min Zhou","doi":"10.1007/s00526-024-02759-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02759-9","url":null,"abstract":"<p>For a positive definite Lagrangian, the minimal measure was defined in terms of first homology or cohomology class. For a configuration manifold that has a larger fundamental group than its first homology group, it makes a difference to define minimal measure in terms of path in fundamental group. Unlike Mather measures that are supported only on the level set not below the Mañé critical value in autonomous case, it is found in this paper that newly defined minimal measures are supported on the level sets not only above but also below the Mañé critical value. In particular, the support of the measure for a commutator looks like a figure of four petals that persists when the energy crosses the critical value.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515267","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 nonlinear instability of liquid Lane–Emden stars","authors":"Zeming Hao, Shuang Miao","doi":"10.1007/s00526-024-02761-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02761-1","url":null,"abstract":"<p>We establish a dynamical nonlinear instability of liquid Lane–Emden stars in <span>({mathbb {R}}^{3})</span> whose adiabatic exponents take values in <span>([1,frac{4}{3}))</span>. Our proof relies on a priori estimates for the free boundary problem of a compressible self-gravitating liquid, as well as a quantitative analysis of the competition between the fastest linear growing mode and the source.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"28 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515268","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":"Critical Schrödinger–Bopp–Podolsky systems: solutions in the semiclassical limit","authors":"Heydy M. Santos Damian, Gaetano Siciliano","doi":"10.1007/s00526-024-02775-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02775-9","url":null,"abstract":"<p>In this paper we consider the following critical Schrödinger–Bopp–Podolsky system </p><span>$$begin{aligned} {left{ begin{array}{ll} -epsilon ^2 Delta u+ V(x)u+Q(x)phi u=h(x,u)+K(x)vert u vert ^{4}u&amp;{} text{ in } mathbb {R}^3 - Delta phi + a^{2}Delta ^{2} phi = 4pi Q(x) u^{2}&amp;{} text{ in } mathbb {R}^3 end{array}right. } end{aligned}$$</span><p>in the unknowns <span>(u,phi :mathbb {R}^{3}rightarrow mathbb {R})</span> and where <span>(varepsilon , a&gt;0)</span> are parameters. The functions <i>V</i>, <i>K</i>, <i>Q</i> satisfy suitable assumptions as well as the nonlinearity <i>h</i> which is subcritical. For any fixed <span>(a&gt;0)</span>, we show existence of “small” solutions in the semiclassical limit, namely whenever <span>(varepsilon rightarrow 0)</span>. We give also estimates of the norm of this solutions in terms of <span>(varepsilon )</span>. Moreover, we show also that fixed <span>(varepsilon )</span> suitably small, when <span>(arightarrow 0)</span> the solutions found strongly converge to solutions of the Schrödinger-Poisson system.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552098","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":"Uniqueness when the $$L_p$$ curvature is close to be a constant for $$pin [0,1)$$","authors":"Károly J. Böröczky, Christos Saroglou","doi":"10.1007/s00526-024-02763-z","DOIUrl":"https://doi.org/10.1007/s00526-024-02763-z","url":null,"abstract":"<p>For fixed positive integer <i>n</i>, <span>(pin [0,1))</span>, <span>(ain (0,1))</span>, we prove that if a function <span>(g:{mathbb {S}}^{n-1}rightarrow {mathbb {R}})</span> is sufficiently close to 1, in the <span>(C^a)</span> sense, then there exists a unique convex body <i>K</i> whose <span>(L_p)</span> curvature function equals <i>g</i>. This was previously established for <span>(n=3)</span>, <span>(p=0)</span> by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if <span>(p=0)</span> and <span>(n=4)</span> or <span>(nle 3)</span> and <span>(pin [0,1))</span>, and the <span>(L_p)</span> curvature function <i>g</i> of a (sufficiently regular, containing the origin) convex body <i>K</i> satisfies <span>(lambda ^{-1}le gle lambda )</span>, for some <span>(lambda &gt;1)</span>, then <span>(max _{xin {mathbb {S}}^{n-1}}h_K(x)le C(p,lambda ))</span>, for some constant <span>(C(p,lambda )&gt;0)</span> that depends only on <i>p</i> and <span>(lambda )</span>. This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the <span>(L_p)</span> surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the <span>(L_p)</span>-Minkowksi problem, for <span>(-n&lt;p&lt;0)</span>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"174 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515269","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":"Existence results for the higher-order Q-curvature equation","authors":"Saikat Mazumdar, Jérôme Vétois","doi":"10.1007/s00526-024-02757-x","DOIUrl":"https://doi.org/10.1007/s00526-024-02757-x","url":null,"abstract":"<p>We obtain existence results for the <i>Q</i>-curvature equation of order 2<i>k</i> on a closed Riemannian manifold of dimension <span>(nge 2k+1)</span>, where <span>(kge 1)</span> is an integer. We obtain these results under the assumptions that the Yamabe invariant of order 2<i>k</i> is positive and the Green’s function of the corresponding operator is positive, which are satisfied in particular when the manifold is Einstein with positive scalar curvature. In the case where <span>(2k+1le nle 2k+3)</span> or the manifold is locally conformally flat, we assume moreover that the operator has positive mass. In the case where <span>(nge 2k+4)</span> and the manifold is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on <i>n</i> and <i>k</i>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552099","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":"Uniqueness of weak solutions of the Plateau flow","authors":"Christopher Wright","doi":"10.1007/s00526-024-02760-2","DOIUrl":"https://doi.org/10.1007/s00526-024-02760-2","url":null,"abstract":"<p>In this paper, we study the uniqueness of weak solutions of the heat flow of half-harmonic maps, which was first introduced by Wettstein as a half-Laplacian heat flow and recently studied by Struwe using more classical techniques. On top of its similarity with the two dimensional harmonic map flow, this geometric gradient flow is of interest due to its links with free boundary minimal surfaces and the Plateau problem, leading Struwe to propose the name Plateau flow, which we adopt throughout. We obtain uniqueness of weak solutions of this flow under a natural condition on the energy, which answers positively a question raised by Struwe.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530775","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}