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1.
In this paper, we study the existence and concentration behavior of minimizers for iV(c)=infuSc?IV(u), here Sc={uH1(RN)|RNV(x)|u|2<+,|u|2=c>0} and
IV(u)=12RN(a|?u|2+V(x)|u|2)+b4(RN|?u|2)2?1pRN|u|p,
where N=1,2,3 and a,b>0 are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of iV(c) for 2<p<2? when V(x)0, V(x)Lloc(RN) and lim|x|+?V(x)=+. For the case p(2,2N+8N)\{4}, we prove that the global constraint minimizers uc of iV(c) behave like
uc(x)c|Qp|2(mcc)N2Qp(mccx?zc),
for some zcRN when c is large, where Qp is, up to translations, the unique positive solution of ?N(p?2)4ΔQp+2N?p(N?2)4Qp=|Qp|p?2Qp in RN and mc=(a2D12?4bD2i0(c)+aD12bD2)12, D1=Np?2N?42N(p?2) and D2=2N+8?Np4N(p?2).  相似文献   

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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   

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Let M be a random m×n rank-r matrix over the binary field F2, and let wt(M) be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as m,n+ with r fixed and m/n tending to a constant, we have thatwt(M)12r2mn2r(12r)4(m+n)mn converges in distribution to a standard normal random variable.  相似文献   

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We consider the equation Δgu+hu=|u|2??2u in a closed Riemannian manifold (M,g), where hC0,θ(M), θ(0,1) and 2?=2nn?2, n:=dim?(M)3. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that n7 and h<n?24(n?1)Scalg in M, where Scalg is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h.  相似文献   

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In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:
(?Δ)su=K(x)uN+2sN?2s,u>0inRN,
where s(0,1) and N>2+2s, K>0 is periodic in (x1,,xk) with 1k<N?2s2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in Rk, the restriction on 1k<N?2s2 is in some sense optimal, since we can show that for kN?2s2, no such solutions exist.  相似文献   

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We explicitly determine generators of cyclic codes over a non-Galois finite chain ring Zp[u]/u3 of length pk, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of Zp[u]/u3 and four types of non-principal ideals of Zp[u]/u3, which are associated with cyclic codes over Zp[u]/u3 of length pk. We then obtain a mass formula for cyclic codes over Zp[u]/u3 of length pk.  相似文献   

13.
We study the non-linear minimization problem on H01(Ω)?Lq with q=2nn?2, α>0 and n4:
infuH01(Ω)6u6Lq=1?Ωa(x,u)|?u|2?λΩ|u|2
where a(x,s) presents a global minimum α at (x0,0) with x0Ω. In order to describe the concentration of u(x) around x0, one needs to calibrate the behavior of a(x,s) with respect to s. The model case is
infuH01(Ω)6u6Lq=1?Ω(α+|x|β|u|k)|?u|2?λΩ|u|2.
In a previous paper dedicated to the same problem with λ=0, we showed that minimizers exist only in the range β<kn/q, which corresponds to a dominant non-linear term. On the contrary, the linear influence for βkn/q prevented their existence. The goal of this present paper is to show that for 0<λαλ1(Ω), 0kq?2 and β>kn/q+2, minimizers do exist.  相似文献   

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The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by ?(?Δ)αu and ?(?Δ)βb, respectively, the resulting equations with α54 and α+β52 then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion ?(?Δh)54b, where Δh=?12+?22.  相似文献   

19.
Consider the integral equation
fq?1(x)=Ωf(y)|x?y|n?αdy,f(x)>0,xΩ,
where Ω?Rn is a smooth bounded domain. For 1<α<n, the existence of energy maximizing positive solution in the subcritical case 2<q<2nn+α, and nonexistence of energy maximizing positive solution in the critical case q=2nn+α are proved in [6]. For α>n, the existence of energy minimizing positive solution in the subcritical case 0<q<2nn+α, and nonexistence of energy minimizing positive solution in the critical case q=2nn+α are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as q(2nn+α)+ (in the case of 1<α<n), and the blowup behaviour of energy minimizing positive solution as q(2nn+α)? (in the case of α>n) are analyzed. We see that for 1<α<n the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for α>n, different phenomena appear.  相似文献   

20.
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 R3. We establish that, in the inviscid resistive case, the energy 6b(t)622 vanishes and 6u(t)622 converges to a constant as time tends to infinity provided the velocity is bounded in W1?α,3α(R3); in the viscous non-resistive case, the energy 6u(t)622 vanishes and 6b(t)622 converges to a constant provided the magnetic field is bounded in W1?β,(R3). In summary, one single diffusion, being as weak as (?Δ)αb or (?Δ)βu with small enough α,β, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.  相似文献   

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