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1.
In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian
((1.λ))
and
((2.λ))
are studied and some new multiplicity results of solutions for systems (1.λ) and (2.λ) are obtained. Moreover, by using the KKM principle we give also two new existence results of solutions for systems (1.1) and (2.1). This Work is supported in part by the National Natural Science Foundation of China (10561011).  相似文献   

2.
In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation
in the unit ball of , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2). F. Dickstein was partially supported by CNPq (Brazil).  相似文献   

3.
We consider the problem
where Ω is a bounded smooth domain in , 1  <  p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem has always a nodal solution u ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω. Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily many negative peaks on the boundary. The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.  相似文献   

4.
In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E f,p among the maps in which coincide with on .   相似文献   

5.
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N  =  2), we prove that the problem has a solution if ∫Ω f dx  = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.  相似文献   

6.
Solutions of elliptic problems with nonlinearities of linear growth   总被引:1,自引:0,他引:1  
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I m is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity f and . Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013) and the Doctoral Programme Foundation of NEM of China (20070028004).  相似文献   

7.
Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.   相似文献   

8.
In this paper we establish an existence and regularity result for solutions to the problem
for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that the solution is Lipschitz continuous and that, in addition, is bounded.  相似文献   

9.
We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) m u  =  u q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .   相似文献   

10.
We study the initial-boundary value problem for nonlinear nonlocal equations on a finite interval where λ > 0 and pseudodifferential operator is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find the main term of the asymptotic representation in the case of the large initial data.  相似文献   

11.
Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.  相似文献   

12.
In this paper, we prove that the weak solutions u∈Wloc^1, p (Ω) (1 〈p〈∞) of the following equation with vanishing mean oscillation coefficients A(x): -div[(A(x)△↓u·△↓u)p-2/2 A(x)△↓u+│F(x)│^p-2 F(x)]=B(x, u, △↓u), belong to Wloc^1, q (Ω)(A↓q∈(p, ∞), provided F ∈ Lloc^q(Ω) and B(x, u, h) satisfies proper growth conditions where Ω ∪→R^N(N≥2) is a bounded open set, A(x)=(A^ij(x)) N×N is a symmetric matrix function.  相似文献   

13.
We consider nonlinear elliptic problems whose prototype is
, with Ω bounded open subset of and p > 1. When several notions of solutions have been introduced; we refer to distributional solutions which can be obtained by an approximation procedure and point out that the question can be faced by a new method which uses symmetrization techniques. In this way we prove both a priori estimates and a continuity with respect to data result which allow us to deduce existence and uniqueness of the solution.   相似文献   

14.
We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality
where c >  0, α ≥ 0 and p >  0 are constants, and is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation
where V and f are Borel measurable functions conditioned by the generalized Kato class.  相似文献   

15.
A circular distribution is a Galois equivariant map ψ from the roots of unity μ to an algebraic closure of such that ψ satisfies product conditions, for ϵμ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ l and μ s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P s and units C s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).  相似文献   

16.
Our first basic model is the fully nonlinear dual porous medium equation with source
for which we consider the Cauchy problem with given nonnegative bounded initial data u0. For the semilinear case m=1, the critical exponent was obtained by H. Fujita in 1966. For p ∈(1, p0] any nontrivial solution blows up in finite time, while for p > p0 there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations with differential operators in divergent form, where classical techniques associated with weak solutions and integration by parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches to calculating critical Fujita exponents in different functional settings. The second models with a “semi-divergent” diffusion operator is the thin film equation with source
for which the critical exponent is shown to be   相似文献   

17.
We study the vector p-Laplacian
We prove that there exists a sequence (u n ) of solutions of (*) such that u n is a critical point of ϕ and another sequence (u n * ) of solutions of (*) such that u n * is a local minimum point of ϕ, where ϕ is a functional defined below. The research is supported by NNSF of China (10301033).  相似文献   

18.
Let X1 and X2 be subspaces of quotients of R OH and C OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product into the predual of a sufficiently large QWEP von Neumann algebra. As an immediate application, given any 1 < q ≤ 2, our result produces a completely isomorphic embedding of (equipped with its natural operator space structure) into with a QWEP von Neumann algebra. Received: June 2006, Revision: June 2007, Accepted: September 2007  相似文献   

19.
In this note, we consider the problem
on a smooth bounded domain Ω in for p > 1. Let u p be a positive solution of the above problem with Morse index less than or equal to . We prove that if u p further satisfies the assumption as p → ∞, then the number of maximum points of u p is less than or equal to m for p sufficiently large. If Ω is convex, we also show that a solution of Morse index one satisfying the above assumption has a unique critical point and the level sets are star-shaped for p sufficiently large.   相似文献   

20.
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” (a)=m satisfies
with constants q>1, ρ 1<ρ 2<⋅⋅⋅<ρ r =ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ ρ≥1, and τ α ρ≥1. Then for a multiplicative function f(a) on the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ ρ mean-value
exists with m f ≠0 and the limit
exists with M v (α)>0.   相似文献   

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