Averaging techniques and oscillation of quasilinear elliptic equations

Zhi-Ting Xu; Bao-Guo Jia; Shao-Yuan Xu

Displaying similar documents to “Averaging techniques and oscillation of quasilinear elliptic equations”

Some new oscillation criteria for second order elliptic equations with damping

Rong-Kun Zhuang, Zheng-an Yao (2005)

Annales Polonici Mathematici

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Some new oscillation criteria are obtained for second order elliptic differential equations with damping $\sum_{i, j = 1}^{n} D_{i} [A_{i j} (x) D_{j} y] + \sum_{i = 1}^{n} b_{i} (x) D_{i} y + q (x) f (y) = 0$ , x ∈ Ω, where Ω is an exterior domain in ℝⁿ. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of subdomains of Ω ⊂ ℝⁿ, rather than on the whole exterior domain Ω. Our results are more natural in view of the Sturm Separation Theorem.

Fite and Kamenev type oscillation criteria for second order elliptic equations

Zhiting Xu (2007)

Annales Polonici Mathematici

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Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation $\sum_{i, j = 1}^{N} D_{i} [a_{i j} (x) D_{j} y] + \sum_{i = 1}^{N} b_{i} (x) D_{i} y + p (x) f (y) = 0$ are obtained. Our results are extensions of those for ordinary differential equations and improve some known oscillation criteria in the literature. Several examples are given to show the significance of the results.

On the oscillation of forced second order mixed-nonlinear elliptic equations

Zhiting Xu (2010)

Annales Polonici Mathematici

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Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${d i v (A (x) | | \nabla y | |}^{p - 1} {\nabla y) + ⟨ b (x), | | \nabla y | |}^{p - 1} \nabla y ⟩ + C (x, y) = e (x)$ , ⎨ ⎩ $C (x, y) = {c (x) | y |}^{p - 1} y + \sum_{i = 1}^{m} c_{i} (x) {| y |}^{p_{i} - 1} y$ under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with mixed nonlinearities, and include some known oscillation results in the literature

Positive coefficients case and oscillation

Ján Ohriska (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and $\int^{\infty} [1 / r (t)] d t$ converges.

A Note on the Uniqueness and Structure of Solutions to the Dirichlet Problem for Some Elliptic Systems

Chern, Jang-Long, Yotsutani, Shoji, Kawano, Nichiro

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In this note, we consider some elliptic systems on a smooth domain of $R^{n}$ . By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.

A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Akihito Unai (2018)

Applications of Mathematics

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We shall prove a weak comparison principle for quasilinear elliptic operators $- div (a (x, \nabla u))$ that includes the negative $p$ -Laplace operator, where $a : Ω \times ℝ^{N} \to ℝ^{N}$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

Optimal $L^{p}$ -properties of Green’s functions for non-divergence elliptic equations in two dimensions

Gioconda Moscariello, Carlo Sbordone (2005)

Studia Mathematica

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A sharp integrability result for non-negative adjoint solutions to planar non-divergence elliptic equations is proved. A uniform estimate is also given for the Green's function.

Infinitely many solutions for a semilinear elliptic equation in $ℝ^{N}$ via a perturbation method

Marino Badiale (2002)

Annales Polonici Mathematici

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We introduce a method to treat a semilinear elliptic equation in $ℝ^{N}$ (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of $ℝ^{N}$ but requires an oscillatory behavior of the potential b.

Oscillation of second-order quasilinear retarded difference equations via canonical transform

George E. Chatzarakis, Deepalakshmi Rajasekar, Saravanan Sivagandhi, Ethiraju Thandapani (2024)

Mathematica Bohemica

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We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $Δ (p (n) {(Δ y (n))}^{α}) + η (n) y^{β} (n - k) = 0$ under the condition $\sum_{n = n_{0}}^{\infty} p^{- \frac{1}{α}} (n) < \infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

Bounded oscillation of nonlinear neutral differential equations of arbitrary order

Yeter Ş. Yilmaz, Ağacik Zafer (2001)

Czechoslovak Mathematical Journal

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The paper is concerned with oscillation properties of $n$ -th order neutral differential equations of the form ${[x (t) + c x (τ (t))]}^{(n)} + q (t) f (x (σ (t))) = 0, t \geq t_{0} > 0,$ where $c$ is a real number with $| c | \neq 1$ , $q \in C ([t_{0}, \infty), ℝ)$ , $f \in C (ℝ, ℝ)$ , $τ, σ \in C ([t_{0}, \infty), ℝ_{+})$ with $τ (t) < t$ and ${lim}_{t \to \infty} τ (t) = {lim}_{t \to \infty} σ (t) = \infty$ . Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations...

$L^{p, Φ}$ spaces and their application to elliptic partial differential operators

Magdalena Jaroszewska (1983)

Annales Polonici Mathematici

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A note on the oscillation problems for differential equations with $p (t)$ -Laplacian

Kōdai Fujimoto (2023)

Archivum Mathematicum

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This paper deals with the oscillation problems on the nonlinear differential equation $(a (t) | x^{'} |^{p (t) - 2} x^{'})^{'} + b (t) {| x |}^{λ - 2} x = 0$ involving $p (t)$ -Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong (2014)

Acta Arithmetica

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Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h ̂_{E_{ω}}$ of the specialized elliptic curve $E_{ω}$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $(h ̂_{E_{ω}} (P_{ω})) / h (ω)$ over all nontorsion P ∈ E(K).

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

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We study a general class of nonlinear elliptic problems associated with the differential inclusion $β (u) - d i v (a (x, D u) + F (u)) ∋ f$ in Ω where $f \in L^{\infty} (Ω)$ . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty}$ -data.

On oscillation of solutions of forced nonlinear neutral differential equations of higher order II

N. Parhi, R. N. Rath (2003)

Annales Polonici Mathematici

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Sufficient conditions are obtained so that every solution of ${[y (t) - p (t) y (t - τ)]}^{(n)} + Q (t) G (y (t - σ)) = f (t)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t \to \infty$ . Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that $\int_{0}^{\infty} Q (t) d t = \infty$ . Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Convex integration and the $L^{p}$ theory of elliptic equations

Kari Astala, Daniel Faraco, László Székelyhidi Jr. (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper deals with the $L^{p}$ theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general $L^{p}$ theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...