Annual oscillation criteria for second-order nonlinear elliptic differential equations

Abstract

In this paper, we present some new oscillation criteria for the second-order nonlinear elliptic differential equationwhere is an exterior domain in R N . These criteria are different from most known ones in the sense that they are based on the information only on a sequence of annuals of in R N , rather than on the whole exterior domain . Also information about the distribution of the zero of solutions is obtained.

Oscillation of second-order elliptic equations with damping terms

“…The aim of this paper is to study oscillation properties of (1.1) via modified Riccati technique and obtain extensions of Leighton [7] and Kamenev [8] for this equation, thereby improving results of Noussair and Swanson [9], Xu [12] and Zhang et al [17]. It is to emphasized that the obtained oscillation criteria here are new even for (E 2 ).…”

“…Without this term, our method cannot be applied to damped elliptic equation (1.1) (cf. [9,[11][12][13][14][15][16][17]). …”

“…In 1980, employing an N -dimensional vector Riccati transformation, Noussair and Swanson [9] first extended the Leighton theorem to the semilinear elliptic equation The survey paper by Swanson [11] contains a complete bibliography up to 1979. Recently, this technique was explored further by Xu [12] and Zhang et al [17], who established Kamenev-type oscillation criteria [7] for (E 2 ), respectively. For (E 2 ), as more late work in this direction we refer the reader to the papers [13][14][15][16] and references cited therein.…”

Fite and Kamenev type oscillation criteria for second order elliptic equations

“…In the qualitative theory of nonlinear PDE, one of the important themes is to determine whether or not solutions of the equation under consideration are oscillatory. In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions for different classes of second order elliptic differential equations (see for example, [1,2,[7][8][9][10][11][12][13][14][16][17][18][19][20][21][22] and the references therein). In particular, for the semilinear elliptic differential equation…”

Oscillation Theorems for Nonlinear Second Order Elliptic Equations

“…In 1980, Noussair and Swanson [11] first employed an N -dimensional vector Riccati transformation and established Leighton [7] -type criteria for (1.2). Recently, Xu [15] and Zhang et al [24] gave some Kamenev [8] -type theorem for (1.2), independently. On the other hand, in 1998, Usami [14] obtained further extensions of Leighton [7] criteria for (1.3).…”