共查询到20条相似文献,搜索用时 35 毫秒
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In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1, u>0, u∈H1(RN), p∈(2,2N/(N-2)) was proved under assumption b(x)?b∞?lim|x|→∞b(x). In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x)<b∞. For any periodic lattice L in RN and for any b∈C(RN) satisfying b(x)<b∞, b∞>0, there is a finite set Y⊂L and a convex combination bY of b(·-y), y∈Y, such that the problem -Δu+u=bY(x)up-1 has a positive solution u∈H1(RN). 相似文献
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Daniele Cassani Bernhard Ruf Cristina Tarsi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
We study optimal embeddings for the space of functions whose Laplacian Δu belongs to L1(Ω), where Ω⊂RN is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω). On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L1(Ω), u=0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N?3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions. 相似文献
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We describe the orbit space of the action of the group Sp(2)Sp(1) on the real Grassmann manifolds Grk(H2) in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H2 which are invariant under the action of the group Sp(2)Sp(1). 相似文献
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This paper treats some variational principles for solutions of inhomogeneous p -Laplacian boundary value problems on exterior regions U?RN with dimension N?3. Existence-uniqueness results when p∈(1,N) are provided in a space E1,p(U) of functions that contains W1,p(U). Functions in E1,p(U) are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an Lp-imbedding theorem. Also an existence result for a general variational problem of this type is obtained. 相似文献
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We study a multi-dimensional nonlocal active scalar equation of the form ut+v⋅∇u=0 in R+×Rd, where v=Λ−2+α∇u with Λ=(−Δ)1/2. We show that when α∈(0,2] certain radial solutions develop gradient blowup in finite time. In the case when α=0, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces. 相似文献
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We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2) dissipation α(−Δ): If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R2) with δ>1−2α on the time interval [t0,t], then it is actually a classical solution on (t0,t]. 相似文献
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Given a rank-r binary matroid we construct a system of O(r3) linear equations in O(r2) variables that has a solution over GF(2) if and only if the matroid is graphic. 相似文献
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In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献
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João Marcos do Ó Manassés de SouzaEveraldo de Medeiros Uberlandio Severo 《Journal of Differential Equations》2014
In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W1,n(Rn), n?2, into the Orlicz space LΦα determined by the Young function Φα(s) behaving like eα|s|n/(n−1)−1 as |s|→+∞. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rn. 相似文献
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Mohamed Ali Toumi 《Expositiones Mathematicae》2010,28(3):269-275
Let A be an Archimedean f -algebra and let N(A) be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A is a derivation if and only if d(A)⊂N(A) and d(A2)={0}, where A2 is the set of all products ab in A. 相似文献
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For a non-degenerate convex subset Y of the n -dimensional Euclidean space Rn, let F(Y) be the family of all fuzzy sets of Rn which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y . We show that the space F(Y) with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?2 if Y is compact; and the space F(Rn) is also homeomorphic to ?2. 相似文献
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In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of Rd on Lp(X)-spaces are convergent for d?3 and p>d/(d-1). 相似文献
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It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1 solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution u=U(x/t) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. 相似文献
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We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−Δ)su=g in Ω , u≡0 in Rn\Ω, for some s∈(0,1) and g∈L∞(Ω), then u is Cs(Rn) and u/δs|Ω is Cα up to the boundary ∂Ω for some α∈(0,1), where δ(x)=dist(x,∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. 相似文献
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We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X,d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X,d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X,d) which admits a one-to-one continuous map f:X→Rn for some n such that d0(f(x),f(y))=?(d(x,y)) (where d0 is the Euclidean metric) for some homeomorphism ? of R+. 相似文献