首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 31 毫秒
1.
We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu?f(u)l(|∇u|) and Δφu?f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality Δu?f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.  相似文献   

2.
We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|m−2u)=p(|x|)g(v), div(|∇v|n−2v)=q(|x|)f(u) on , where f and g are positive and non-decreasing functions on (0,∞) satisfying the Keller-Osserman condition.  相似文献   

3.
In this note we investigate the existence of positive solutions vanishing at +∞ to the elliptic equation Δu+f(x,u)+g(|x|)x⋅∇u=0, |x|>A>0, in Rn (n?3) under mild restrictions on the functions f, g.  相似文献   

4.
We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δu+|∇u|=p(|x|)f(u,v), Δv+|∇v|=q(|x|)g(u,v) on RN, N?3, provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.  相似文献   

5.
We prove the one-dimensional symmetry of solutions to elliptic equations of the form ?div(e G(x) a(|?u|)?u) = f(u) e G(x), under suitable energy conditions. Our results holds without any restriction on the dimension of the ambient space.  相似文献   

6.
In this article, we consider uniqueness of positive radial solutions to the elliptic system Δu+a(|x|)f(u,v)=0, Δv+b(|x|)g(u,v)=0, subject to the Dirichlet boundary condition on the open unit ball in RN (N?2). Our uniqueness results applies to, for instance, f(u,v)=uqvp, g(u,v)=upvq, p,q>0, p+q<1 or more general cases.  相似文献   

7.
We study the 3×3 elliptic systems ∇(a(x)∇×u)−∇(b(x)∇⋅u)=f, where the coefficients a(x) and b(x) are positive scalar functions that are measurable and bounded away from zero and infinity. We prove that weak solutions of the above system are Hölder continuous under some minimal conditions on the inhomogeneous term f. We also present some applications and discuss several related topics including estimates of the Green?s functions and the heat kernels of the above systems.  相似文献   

8.
This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form Δφuf(u)l(|0u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δφ is the φ-Laplace operator, a natural generalization of the p-Laplace operator which has recently been studied in the context of Carnot groups. We extend to general Carnot groups the results proved in Magliaro et al. (2011) [7] for the Heisenberg group, showing the validity of Liouville-type theorems under a suitable Keller-Osserman condition. In doing so, we also prove a maximum principle for inequality Δφuf(u)l(|0u|). Finally, we show sharpness of our results for a general φ-Laplacian.  相似文献   

9.
We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|p−2u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b0(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any mR. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.  相似文献   

10.
By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation ut = ∇(a(u)b(x)c(t)∇u) + f(xuqt) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.  相似文献   

11.
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2u), with initial data u0Lr(Ω) is proved under r>N(qp)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f.  相似文献   

12.
Existence and some regularity results of global attractor in Lq, q?1, for m-Laplacian type quasilinear parabolic equation with a perturbation like a(x)(α|u|uβ|u|u)+f(x) with α>β?0, a(x)?0 are proved. For the proofs Moser's technique is used extensively.  相似文献   

13.
Under the simple conditions on f and g, we show that entire positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)f(v), Δv=q(|x|)g(u), xRN, N?3, where the functions are continuous.  相似文献   

14.
The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u|Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.  相似文献   

15.
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,u)=g−div(f), where a(x,u,u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where gL1(Ω) and f∈(Lp(Ω))N.  相似文献   

16.
We are concerned with singular elliptic equations of the form −Δu=p(x)(g(u)+f(u)+a|∇u|) in RN (N?3), where p is a positive weight and 0<a<1. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.  相似文献   

17.
Let ΩRN(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|u|q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)= as dist(x,Ω)→0,f is Γ-varying at . Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution.  相似文献   

18.
We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation
{ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ¥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.  相似文献   

19.
We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:
tu−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)  相似文献   

20.
We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)?1?Lq((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of Lp(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x0)) u(x0), (Fu)(x) = ∫xLf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L2(0, L) function.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号