Calculus of Variations and Partial Differential Equations最新文献

In this paper, we study the fractional critical Schrödinger–Poisson system

having prescribed mass

where ( s, t in (0, 1)) satisfy (2,s+2t> 3, qin (2,2^*_s), a>0) and (lambda ,mu >0) parameters and (alpha in {mathbb {R}}) is an undetermined parameter. For this problem, under the (L^2)-subcritical perturbation (mu |u|^{q-2}u, qin (2,2+frac{4,s}{3})), we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the (L^2)-supercritical perturbation (mu |u|^{q-2}u,qin (2+frac{4,s}{3}, 2^*_s)), we prove two different results of normalized solutions when parameters (lambda ,mu ) satisfy different assumptions, by applying the constrained variational methods and the mountain pass theorem. Our results extend and improve some previous ones of Zhang et al. (Adv Nonlinear Stud 16:15–30, 2016); and of Teng (J Differ Equ 261:3061–3106, 2016), since we are concerned with normalized solutions.

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng [5] recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition ((Ric-epsilon Rgge 0)) must be compact. This answers higher dimensional Hamilton’s pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng’s theorem.

For each (k = 0,dots ,n) we construct a continuous phase (f_k), with (f_k(0) = (n-2k)frac{pi }{2}), and viscosity sub- and supersolutions (v_k), (u_k), of the elliptic PDE (sum _{i=1}^n arctan (lambda _i(mathcal {H}w)) = f_k(x)) such that (v_k-u_k) has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases (f:mathbb {R}^nsupseteq Omega rightarrow (-npi /2,npi /2)). Our examples show it does not.

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

In this article, we consider the Einstein-scalar field Lichnerowicz equation

on any connected finite graph (G=(V,E)), where A, B, h are given functions on V with (Age 0), (Anot equiv 0) on V, and (p>2) is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely (h>0), (h<0) and (h=0) respectively.

We look for solutions to the Schrödinger equation

coupled with the mass constraint (int _{mathbb {R}^N}|u|^2,dx = rho ^2), with (Nge 2). The behaviour of g at the origin is allowed to be strongly sublinear, i.e., (lim _{srightarrow 0}g(s)/s = -infty ), which includes the case

with (alpha > 0) and (mu in mathbb {R}), (2 < p le 2^*) properly chosen. We consider a family of approximating problems that can be set in (H^1(mathbb {R}^N)) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of (H^1(mathbb {R}^N)), we prove the existence of infinitely, many solutions.

We study the behaviour of global minimizers of a continuum Landau–de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of (mathbb {S}^1)-equivariant configurations. It is known from our previous paper (Dipasquale et al. in J Funct Anal 286:110314, 2024) that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of torus or of split type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler–Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimization among all competitors.

Let (Gamma ) be a compact codimension-two submanifold of ({mathbb {R}}^n), and let L be a nontrivial real line bundle over (X = {mathbb {R}}^n {setminus } Gamma ). We study the Allen–Cahn functional,

on the space of sections u of L. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to (Gamma ). We first show that, for a family of critical sections with uniformly bounded energy, in the limit as (varepsilon rightarrow 0), the associated family of energy measures converges to an integer rectifiable ((n-1))-varifold V. Moreover, V is stationary with respect to any variation which leaves (Gamma ) fixed. Away from (Gamma ), this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary (Gamma ). Under additional hypotheses, we can say more about V. When V arises as a limit of critical sections with uniformly bounded Morse index, (Sigma := {{,textrm{supp},}}Vert VVert ) is a minimal hypersurface, smooth away from (Gamma ) and a singular set of Hausdorff dimension at most (n-8). If the sections are globally energy minimizing and (n = 3), then (Sigma ) is a smooth surface with boundary, (partial Sigma = Gamma ) (at least if L is chosen correctly), and (Sigma ) has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau’s problem admits a solution for every boundary curve in ({mathbb {R}}^3). This also works if (4 le nle 7) and (Gamma ) is assumed to lie in a strictly convex hypersurface.

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are (C^{1,gamma }). No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when (p(x)equiv p), i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.