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1.
We study herein the Camassa–Holm-type equation, which can be considered as a model in the shallow water for the long-crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. This quasi-linear equation is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solutions in the energy space H1 to this model equation.  相似文献   

2.
This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp O(ε) convergence rate in Wm?1,p0 with p0=2dd?1 in a bounded Lipschitz domain in Rd as well as the uniform large-scale interior Cm?1,1 estimate. With additional smoothness assumptions, the uniform interior Cm?1,1, Wm,p and Cm?1,α estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.  相似文献   

3.
In this paper, we study semilinear elliptic systems with critical nonlinearity of the form
(0.1)Δu=Q(x,u,?u),
for u:RnRK, Q has quadratic growth in ?u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When n=2, such a system does not have smooth regularity in general for W1,2 weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for n=2) and F. Béthuel (for n3), assert that a W1,n weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1), that a W1,n weak solution of the system is smooth for n3. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a W2,n/2 weak solution of such system is always smooth, for n5. We also construct various examples, and these examples show that our regularity results are optimal in various sense.  相似文献   

4.
We consider a smooth solution u>0 of the singular minimal surface equation 1+|Du|2 div(Du/1+|Du|2)=α/u defined in a bounded strictly convex domain of R2 with constant boundary condition. If α<0, we prove the existence a unique critical point of u. We also derive some C0 and C1 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 α<0.  相似文献   

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7.
We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth
?Δu?λc(x)u?κα(Δ(|u|2α))|u|2α?2u=|u|q?2u+|u|2??2u,uD1,2(RN),
via variational methods, where λ0, c:RNR+, κ>0, 0<α<1/2, 2<q<2?. It is interesting that we do not need to add a weight function to control |u|q?2u.  相似文献   

8.
Our main task in this note is to prove the existence and to classify the exact growth at infinity of radial positive C6-solutions of (?Δ)3u=up in Rn, where n?15 and p is bounded from below by the sixth-order Joseph–Lundgren exponent. Following the main work of Winkler, we introduce the sub- and super-solution method and comparison principle to conclude the asymptotic behavior of solutions.  相似文献   

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10.
We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R2, where the intraspecies interaction (?a1,?a2) and the interspecies interaction ?β are both attractive, i.e, a1, a2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L2-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as ββ?=a?+(a??a1)(a??a2), where 0<ai<a?:=6w622 (i=1,2) is fixed and w is the unique positive solution of Δw?w+w3=0 in R2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].  相似文献   

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