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Oscillation theorems of quasilinear elliptic equations with arbitrary nonlinearities
Abstract
We establish oscillation criteria for solutions of quasilinear second order elliptic
equations. We do not impose any additional conditions on the nonlinear terms except for
the continuity. In particular, we can characterize the oscillation property of every
solution for autonomous equations.
In this paper we consider the second-order nonlinear differential equation (∗)(tα−1Φ(x′))′+tα−1−pf(x)=0,Φ(x)=|x|p−2x,p>1,α∈R,
with f satisfying xf(x)>0, x≠0. We analyze the difference between the cases α<p, α>p, and α=p. In each case we give a condition on the function f which guarantees that solutions of Eq. (∗) are (non)oscillatory. The principal methods used in this paper are the Riccati technique and its modifications. The results of our paper complement and extend several previously obtained results on the subject.
We consider quasilinear ordinary differential equations with sub-homogeneity near infinity. A necessary and sufficient condition is obtained for the equations to have slowly decaying positive solutions. Asymptotic forms of such positive solutions are established. As an application of these results, we obtain Liouville-type theorems for quasilinear elliptic problems.
Our concern is to solve the nonlinear perturbation problem for the semilinear elliptic equation in an exterior domain of RN with N ≥ 3. The lower limit of the nonlinear perturbed term is given for all nontrivial solutions to be oscillatory. The tools for obtaining our theorems are the so-called "supersolution-subsolution" method and some results concerning the oscillation and nonoscillation of solutions of the ordinary differential equation associated with the elliptic equation. A simple example is given to illustrate the main results.
These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form
(1)
in unbounded domains Ω in n -dimensional Euclidean space R ⁿ . Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.
Synopsis
Sufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain of E ⁿ . Such results apply in particular to the n -dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.
Vazquez in 1984 established a strong maximum principle for the classical m-Laplace differential inequalityΔmu−f(u)≤0,where Δmu=div(|Du|m−2Du) and f(u) is a non-decreasing continuous function with f(0)=0. We extend this principle to a wide class of singular inequalities involving quasilinear divergence structure elliptic operators, and also consider the converse problem of compact support solutions in exterior domains.
On considere des solutions u∈H 1,p (Ω)∧L ∞ (Ω) (1<p<∞) de l'equation differentielle: ∫ Ω Σ j=1 n {a j (x,u,⊇u)•φ xj }−a(x,u,⊇u)•φdx=0, ∀φΩC c ∞ (Ω). Ω est un sous-ensemble ouvert de R n . On demontre que les derivees de ces solutions sont Holder-continues dans l'interieur du domaine Ω
The second order elliptic equation with damping∑i,j=1dDi[Aij(x)Djy]+∑i=1dbi(x)Diy+p(x)f(y)=0is considered in an exterior domain Ω⊂Rd, where the coefficients bi(x), i=1,…,d and p(x) are not be nonnegative in Ω. By the generalized partial Riccati transformation and the integral averaging technique, some new oscillation theorems are established.