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1.
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   

2.
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   

3.
In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth
?u+(u2?1|4πx|)u=μ|u|p?1u+|u|4u,inR3,
where μ>0 and p(11/7,5). For the case of p(2,5). 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 p=2, 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 p(11/7,2), 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 μ(0,μ?) with some μ?>0. 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   

4.
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   

5.
We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:
{?ε2Δu+V(x)u+λ1|u|2u+λ2(K?|u|2)u=0 in R3,u>0,uH1(R3),
where ε is a small positive parameter, λ1,λ2R, ? denotes the convolution, K(x)=1?3cos2?θ|x|3 and θ=θ(x) is the angle between the dipole axis determined by (0,0,1) and the vector x. Under certain assumptions on (λ1,λ2)R2, we construct a family of positive solutions uεH1(R3) which concentrates around the local minima of V as ε0. 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   

6.
7.
We develop interior W2,p,μ and W2,BMO regularity theories for Ln-viscosity solutions to fully nonlinear elliptic equations T(D2u,x)=f(x), where T is approximately convex at infinity. Particularly, W2,BMO regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of D2T(M) are at least ?C6M6?(1+σ0) as M. W2,BMO regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of W2,BMO regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.  相似文献   

8.
We are concerned with the existence of blowing-up solutions to the following boundary value problem
?Δu=λa(x)eu?4πNδ0 in Ω,u=0 on ?Ω,
where Ω is a smooth and bounded domain in R2 such that 0Ω, a(x) is a positive smooth function, N is a positive integer and λ>0 is a small parameter. Here δ0 defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution uλ blowing up at 0 and satisfying λΩa(x)euλ8π(N+1) as λ0+.  相似文献   

9.
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   

10.
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   

11.
12.
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   

13.
This paper is concerned with the Cauchy problem for the Hartree equation on Rn,nN with the nonlinearity of type (|?|?γ?|u|2)u,0<γ<n. It is shown that a global solution with some twisted persistence property exists for data in the space LpL2,1p2 under some suitable conditions on γ and spatial dimension nN. It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map t?u(t) is well defined and continuous from R?{0} to Lp, which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat Lp-spaces are also presented. The local and global results are proved by combining arguments by Carles–Mouzaoui with a new functional framework introduced by Zhou. Furthermore, it is also shown that the global results can be improved via generalized dispersive estimates in the case of one space dimension.  相似文献   

14.
We prove the existence of solutions to the nonlinear Schrödinger equation ε2(i?+A)2u+V(y)u?|u|p?1u=0 in R2 with a magnetic potential A=(A1,A2). Here V represents the electric potential, the index p is greater than 1. Along some sequence {εn} tending to zero we exhibit complex-value solutions that concentrate along some closed curves.  相似文献   

15.
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   

16.
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   

17.
This paper deals with positive solutions of the fully parabolic system
{ut=Δu?χ??(u?v)inΩ×(0,),τ1vt=Δv?v+winΩ×(0,),τ2wt=Δw?w+uinΩ×(0,)
under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω?R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).Global existence and boundedness of solutions were shown if 6u06L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying 6u06L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.  相似文献   

18.
We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm (ΩFN(?u)dx)1N on W01,N(Ω) for N2. Here F is convex and homogeneous of degree 1, and its polar Fo represents a Finsler metric on RN. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.  相似文献   

19.
20.
With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem
(?Δ+m2)σu+Vu=(W?F(u))f(u)in RN,
where 0<σ<1, V is a bounded continuous potential and F the primitive of f. We also show results about the regularity of any solution of this problem.  相似文献   

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