Calculus of Variations and Partial Differential Equations最新文献

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.

In this paper, we classify the hypersurfaces of (mathbb {S}^2times mathbb {S}^2) with constant sectional curvature. We prove that the constant sectional curvature can only be (frac{1}{2}). We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in (mathbb {S}^2times mathbb {S}^2), and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous “sinh-Gordon equation” ( left( frac{partial ^2}{partial u^2}+frac{partial ^2}{partial v^2}right) h =-tfrac{1}{sqrt{2}}sinh left( sqrt{2}hright) ).

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 (uequiv 0). 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.

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential

where (Nge 3), (2<p<2^*={2N}/({N-2})) and (varepsilon >0). We mainly focus on the normalized ground state solitary waves of the form (psi (t,x)=e^{imu t}u_{varepsilon ,rho }(x)), where (u_{varepsilon ,rho }(x)) is radially symmetric-decreasing and (int _{mathbb {R}^N}|u_{varepsilon ,rho }|^2,dx=rho ). Firstly, we prove the existence and nonexistence of normalized ground states under the (L^2)-subcritical, (L^2)-critical ((p=4/N +2)) and (L^2)-supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states (u_{varepsilon ,rho }) as (varepsilon rightarrow 0^+) and find that the (varepsilon )-blow-up phenomenon happens for (rho ge rho _c=Vert QVert ^2_{L^2}), where Q is a positive radially symmetric ground state of (-Delta u+u-(|x|^{-2}*|u|^2)u=0) in (mathbb {R}^N). We prove that (int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }(x)|^2,dxsim varepsilon ^{-frac{4}{N(p-2)+4}}) for (rho =rho _c) and (2<p<2^*), while (int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }|^2,dxsim varepsilon ^{-frac{4}{N(p-2)-4}}) for (rho >rho _c) and (4/N+2<p<2^*), and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as (varepsilon rightarrow +infty ), which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer (u^{TF}=left[ mu ^{TF}-|x|^2 right] ^{frac{1}{p-2}}_{+}), where (mu ^{TF}) is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for (u_{varepsilon , rho }) that (Vert u_{varepsilon , rho }Vert _{L^{infty }}sim varepsilon ^{-frac{N}{N(p-2)+4}}) as (varepsilon rightarrow +infty ).

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.

We consider a quasilinear equation involving the (n-)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.

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.

In this paper we consider the following critical Schrödinger–Bopp–Podolsky system

in the unknowns (u,phi :mathbb {R}^{3}rightarrow mathbb {R}) and where (varepsilon , a>0) are parameters. The functions V, K, Q satisfy suitable assumptions as well as the nonlinearity h which is subcritical. For any fixed (a>0), we show existence of “small” solutions in the semiclassical limit, namely whenever (varepsilon rightarrow 0). We give also estimates of the norm of this solutions in terms of (varepsilon ). Moreover, we show also that fixed (varepsilon ) suitably small, when (arightarrow 0) the solutions found strongly converge to solutions of the Schrödinger-Poisson system.

We establish a dynamical nonlinear instability of liquid Lane–Emden stars in ({mathbb {R}}^{3}) whose adiabatic exponents take values in ([1,frac{4}{3})). 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.

For fixed positive integer n, (pin [0,1)), (ain (0,1)), we prove that if a function (g:{mathbb {S}}^{n-1}rightarrow {mathbb {R}}) is sufficiently close to 1, in the (C^a) sense, then there exists a unique convex body K whose (L_p) curvature function equals g. This was previously established for (n=3), (p=0) 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 (p=0) and (n=4) or (nle 3) and (pin [0,1)), and the (L_p) curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies (lambda ^{-1}le gle lambda ), for some (lambda >1), then (max _{xin {mathbb {S}}^{n-1}}h_K(x)le C(p,lambda )), for some constant (C(p,lambda )>0) that depends only on p and (lambda ). 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 (L_p) surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the (L_p)-Minkowksi problem, for (-n<p<0).