The Oscillatory Behavior of Second Order Nonlinear Elliptic Equations

Some oscillation criteria are established for the nonlinear damped elliptic dieren tial equation of second order N X i; j=1 Di( aij(x)Djy ) + N X i=1 bi(x)Diy + p(x)f(y) = 0; (E) which are dieren t from most known ones in the sense that they are based on a new weighted function H(r; s; l) dened in the sequel. Both the cases when Dibi(x) exists for all i and when it does not exist for some i are considered.


Introduction and Preliminaries
In this paper, we are concerned with the oscillatory behavior of the general nonlinear damped elliptic differential equation of second order where , Ω(a) = {x ∈ R N : |x| ≥ a} for some a > 0.
In what follows, the solution of Eq.(1.1) is every function of the class C 2+µ loc (Ω(a), R), 0 < µ < 1, which satisfies Eq.(1.1) almost everywhere on Ω(a).We consider only nontrivial solution of Eq.(1.1) which is defined for all large |x| ( cf [1] ).The oscillation is considered in the usual sense, i.e., a solution y(x) of Eq.(1.1) is said to be oscillatory if it has zero on Ω(b) for every b ≥ a. Equation (1.1) is said to be oscillatory if every solution ( if any exists ) is oscillatory.Conversely, Equation(1.1) is nonoscillatory if there exists a solution which is not oscillatory.E-mail address: xztxhyyj@pub.guangzhou.gd.cnEJQTDE, 2005 No. 8, p. 1 Equation (1.1) is an very important type of partial differential equations, which has wide applications in various problems dealing with physics, biology and glaciology, etc., see [1].In the qualitative theory of nonlinear partial differential equations, one of the important problems is to determine whether or not solutions of the equation under consideration are oscillatory.For the similinear elliptic equation the oscillation theory is fully developed by [ 6,9,[11][12][13]15 ] where further references can be found.In particular, Noussair and Swanson [6] first gave Fite-Leighton type oscillation criteria [2,5] for Eq.(1.2).For a related study, we refer to [12] in which a classical Kamenev theorem [3] ( as extended and improved by Phiols [7] and Yan [14] ) is to be extended to Eq.(1.2).However, as far as we know that the equation (1.1) in general form has never been the subject of systematic investigations.
In the case when N = 1, a ij (x) = 1 for all i, j, f (y) = y, Eq.(1.2) reduces second order ordinary differential equation Recently, by using an weighted function, Sun [8] gave an interesting result.More precisely, Sun proved the following theorem.
Early similar results were proved by Kamenev [3], Kong [4], Philos [7], Wintner [10] and Yan [14].But, Theorem 1.1 is simpler and more sharper than that of previous results.It is noting that Theorem 1.1 is given in [8] for a differential equation which is more general than Eq.(1.3).But, the above particular form of Sun's theorem is the basic one.
In present paper, one main objective is to extend Theorem 1.1 to Eq.(1.1).In section 2, by using an weighted function H(r, s, l), we shall establish some oscillation criteria for Eq.(1.1) for the case when D i b i (x) exists for all i .Then in section 3, we deal with the oscillation of Eq.(1.1) for the case when D i b i (x) does not exist for some i .Finally in section 4, we will show the application of our oscillation criteria by several examples.
To formulate our results we shall use the following notations.
Following Sun [8], we shall define a class of functions H.For this purpose, we first define the sets.
An weighted function H ∈ C(D, R) is said to belong to the class H defined by H ∈ H if (H 1 ) H(r, r, l) = 0, H(r, l, l) = 0 for r > l ≥ a, and H(r, s, l) = 0 for (r, s, l) ∈ D 0 .(H 2 ) H(r, s, l) has a continuous partial derivative on D with respect to the second variable, and there is a function h We now define an integral operator T r l in terms of H(r, s, l) and φ(s) as It is easily seen that T r l satisfies the following: By choosing specific functions H(r, s, l), it is possible to derive several oscillation criteria for Eq.(1.1).For instance, for an arbitrary positive function ξ ∈ C([a, ∞), R + ), define the kernel function where ξ(τ ) = 1, H(r, s, l) = (r−s) α (s−l) β , and when ξ(τ ) = τ, H(r, s, l) = ( ln r/s) α ( ln s/l) β .It is easily verified that the kernel function (1.6) satisfies (H 1 ) and (H 2 ).

Oscillation results for the case when D i b i (x) exists for all i
In this section, we establish oscillation theorems which extend Theorem 1.1 to Eq.(1.1) for the case when D i b i (x) exists for all i .For this purpose, we shall impose the following conditions: loc (Ω(a), R) for all i, j, µ ∈ (0, 1).Denote by λ max (x) the largest eigenvalue of the matrix A. We suppose that there exists Theorem 2.1.Let (C 1 )-(C 4 ) hold.Suppose that for each l ≥ a, there exist functions where and , σ denotes the measure on S r , ω N denotes the surface area of the unit sphere in R N , i.e., ω N = 2π N/2 /Γ(N/2).Then Eq.(1.1) is oscillatory.
Proof.Let y = y(x) be a nonoscillatory solution of Eq.(1.1), and suppose that there exists a b ≥ a such that y = y(x) = 0 for all x ∈ Ω(b).Define where 2) with respect to x i gives for all i.Summation over i, using of Eq.(1.1) and (2.2), leads to where ν(x) = x/r, r = |x| = 0, denotes the outward unit normal to S r .By means of the Green formula and (2.3), we have In view of (C 4 ), we have that (2.6) The Schwartz inequality yields (2.7) Thus, by (2.5)-(2.7),we obtain Applying the operator T r l to (2.8), we have the following inequality In view of (H 2 ), (A 1 ) and (A 2 ), we get that (2.9) Clearly, inequality (2.9) contradicts (2.1).
For the case H(r, s, l) = H 1 (r, s)H 2 (s, l), by Theorem 2.1, we have the following theorem.
for some α > 1, then Based on the above results, we obtain the following Kamenev type oscillation criteria.
3. Oscillation results for the case when D i b i (x) does not exist for some i In this section, we establish oscillation criteria for Eq.(1.1) in case when D i b i (x) does not exist for some i.We begin with the following lemma, the proof of this lemma is easy and thus omitted.
Lemma 3.1.For two n-dimensional vectors u, v ∈ R N , and a positive constant c, then (Ω(a), R) for all i, µ ∈ (0, 1) hold.Suppose that for each l ≥ a, there exist functions η where and S r , dσ, ω N are defined as in Theorem 2.1.Then Eq.(1.1) is oscillatory.Proof.Let y = y(x) be a nonoscillatory solution of Eq.(1.1), and suppose that there exists a b ≥ a such that y = y(x) = 0 for all x ∈ Ω(b).Define Differentiation of the i−th component of (3.3) with respect to x i gives