Symmetry Results for Semilinear Elliptic Systems in the Whole Space
نویسندگان
چکیده
where n ≥ 1, N ≥ 2 are arbitrary integers. In the case of a bounded domain, related results for autonomous systems were established by Troy [17] (see also de Figueiredo [4], Shaker [16]). Under additional hypotheses on the asymptotic behaviour of the solutions at infinity, in the spirit of Gidas, Ni and Nirenberg [11], a symmetry result in IR was obtained by Shaker. We remark that the case of a single equation has been extensively studied since the work of Gidas, Ni and Nirenberg (see for instance C. Li [12], Y. Li and W.-M. Ni [13]). In a recent paper, D.G. de Figueiredo and J. Yang [8] studied the symmetry of the positive solutions of systems of two equations, under some restrictive hypotheses on the nonlinearities (see Section 2.1). Using variational methods, de Figueiredo and Yang also proved existence and decay at infinity of positive solutions of such systems. More general results about existence and decay can be found in [15], as well as an application of our symmetry result to the existence of a ground state of the system. We note u = (u1, . . . , un) ∈ IR+ = (0,∞) and
منابع مشابه
Existence and multiplicity of positive solutions for a class of semilinear elliptic system with nonlinear boundary conditions
This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.
متن کاملA two-phase free boundary problem for a semilinear elliptic equation
In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary. We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...
متن کاملEntire Solutions of Semilinear Elliptic Equations in R and a Conjecture of De Giorgi
This paper is concerned with the study of bounded solutions of semilinear elliptic equations ∆u − F ′(u) = 0 in the whole space R, under the assumption that u is monotone in one direction, say, ∂nu > 0 in R. The goal is to establish the one-dimensional character or symmetry of u, namely, that u only depends on one variable or, equivalently, that the level sets of u are hyperplanes. This type of...
متن کاملAsymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: two approaches
In this paper, we study the asymptotic behavior as x1 → +∞ of solutions of semilinear elliptic equations in quarteror half-spaces, for which the value at x1 = 0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum prin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998