共查询到20条相似文献,搜索用时 468 毫秒
1.
This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space . We establish that, in the inviscid resistive case, the energy vanishes and converges to a constant as time tends to infinity provided the velocity is bounded in ; in the viscous non-resistive case, the energy vanishes and converges to a constant provided the magnetic field is bounded in . In summary, one single diffusion, being as weak as or with small enough , is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system. 相似文献
2.
In this paper, we study semilinear elliptic systems with critical nonlinearity of the form
(0.1)
for , Q has quadratic growth in ?u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When , such a system does not have smooth regularity in general for weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for ) and F. Béthuel (for ), assert that a weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1), that a weak solution of the system is smooth for . For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a weak solution of such system is always smooth, for . We also construct various examples, and these examples show that our regularity results are optimal in various sense. 相似文献
3.
Zhouxin Li 《Journal of Differential Equations》2019,266(11):7264-7290
We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth via variational methods, where , , , , . It is interesting that we do not need to add a weight function to control . 相似文献
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We derive the sharp convergence rate in in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in settings derived in [7], [28] for second order parabolic systems with time-dependent coefficients. 相似文献
6.
Rafael López 《Journal of Differential Equations》2019,266(7):3927-3941
We consider a smooth solution of the singular minimal surface equation defined in a bounded strictly convex domain of with constant boundary condition. If , we prove the existence a unique critical point of u. We also derive some and estimates of u by using the theory of maximum principles of Payne and Philippin for a certain family of Φ-functions. Finally we deduce an existence theorem of the Dirichlet problem when . 相似文献
7.
In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth where and . For the case of . We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of , we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of , we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for with some . The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case. 相似文献
8.
Qingbo Huang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(7):1869-1902
We develop interior and regularity theories for -viscosity solutions to fully nonlinear elliptic equations , where T is approximately convex at infinity. Particularly, regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of are at least as . regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of regularity theory is dense in the space of fully nonlinear uniformly elliptic operators. 相似文献
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We study the non-linear minimization problem on with , and : where presents a global minimum α at with . In order to describe the concentration of around , one needs to calibrate the behavior of with respect to s. The model case is In a previous paper dedicated to the same problem with , we showed that minimizers exist only in the range , which corresponds to a dominant non-linear term. On the contrary, the linear influence for prevented their existence. The goal of this present paper is to show that for , and , minimizers do exist. 相似文献
11.
We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases: where ε is a small positive parameter, , ? denotes the convolution, and is the angle between the dipole axis determined by and the vector x. Under certain assumptions on , we construct a family of positive solutions which concentrates around the local minima of V as . Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation. 相似文献
12.
We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with boundary, i.e., positive solutions u to in so that and along ?Ω, the unit outward normal along ?Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez–Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem.When is the standard sphere and f is a function so that and for any , we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki–Caffarelli–Nirenberg conjecture in for this choice of f. More precisely, this shows that if u is a positive solution to on a topological disk with boundary so that and along ?Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33], [35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in . 相似文献
13.
We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in , where the intraspecies interaction and the interspecies interaction ?β are both attractive, , , and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as , where () is fixed and w is the unique positive solution of in . The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. 相似文献
14.
In this paper, we study the existence and concentration behavior of minimizers for , here and where and are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of for when , and . For the case , we prove that the global constraint minimizers of behave like for some when c is large, where is, up to translations, the unique positive solution of in and , and . 相似文献
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Zihua Guo Xingxing Liu Luc Molinet Zhaoyang Yin 《Journal of Differential Equations》2019,266(2-3):1698-1707
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space and even in the Besov space for . Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]). 相似文献
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Considering the radial nonlinear Schrödinger equation
()
we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem () is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem () to an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator restricted to , the space of radially symmetric functions in , for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness. 相似文献