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

Zhiting Xu

Displaying similar documents to “On the oscillation of forced second order mixed-nonlinear elliptic equations”

Oscillation properties for a scalar linear difference equation of mixed type

Leonid Berezansky, Sandra Pinelas (2016)

Mathematica Bohemica

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The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $Δ x (n) + \sum_{k = - p}^{q} a_{k} (n) x (n + k) = 0, n > n_{0},$ where $Δ x (n) = x (n + 1) - x (n)$ is the difference operator and ${a_{k} (n)}$ are sequences of real numbers for $k = - p, ..., q$ , and $p > 0$ , $q \geq 0$ . We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

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 the existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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An example of the equation $u_{t} = u_{x x} + f (x, t, u)$ with distinct maximum and minimum solutions of a mixed problem

W. Mlak (1963)

Annales Polonici Mathematici

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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.

T-p(x)-solutions for nonlinear elliptic equations with an L¹-dual datum

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf (2012)

Applicationes Mathematicae

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We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $- d i v (a (x, u, \nabla u)) + g (x, u) = f - d i v F$ in Ω, where Ω is a bounded open domain of $ℝ^{N}$ , N ≥ 2 and $a : Ω \times ℝ \times ℝ^{N} \to ℝ^{N}$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

Forced oscillation of third order nonlinear dynamic equations on time scales

Baoguo Jia (2010)

Annales Polonici Mathematici

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Consider the third order nonlinear dynamic equation $x^{Δ Δ Δ} (t) + p (t) f (x) = g (t)$ , (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $Δ ³ x (n) + n^{α} {| x |}^{γ} s g n (n) = (- 1) ⁿ n^{c}$ , where α ≥ -1, γ > 0, c > 3, is oscillatory.

Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Arun K. Tripathy, Shyam S. Santra (2021)

Mathematica Bohemica

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In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type ${(r (t) {(z^{'} (t))}^{γ})}^{'} + \sum_{i = 1}^{m} q_{i} (t) x^{α_{i}} (σ_{i} (t)) = 0, t \geq t_{0},$ where $z (t) = x (t) + p (t) x (τ (t))$ . Under the assumption $\int^{\infty} {(r (η))}^{- 1 / γ} d η = \infty$ , we consider two cases when $γ > α_{i}$ and $γ < α_{i}$ . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

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 criteria for nonlinear differential equations with $p (t)$ -Laplacian

Yutaka Shoukaku (2016)

Mathematica Bohemica

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Recently there has been an increasing interest in studying $p (t)$ -Laplacian equations, an example of which is given in the following form $(| u^{'} {(t) |}^{p (t) - 2} u^{'} {(t))}^{'} + c (t) {| u (t) |}^{q (t) - 2} u (t) = 0, t > 0 .$ In particular, the first study of sufficient conditions for oscillatory solution of $p (t)$ -Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations...

Integral averaging technique for oscillation of damped half-linear oscillators

Yukihide Enaka, Masakazu Onitsuka (2018)

Czechoslovak Mathematical Journal

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This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${(a (t) φ_{p} (x^{'}))}^{'} + b (t) φ_{p} (x^{'}) + c (t) φ_{p} (x) = 0$ , where $φ_{p} (x) = {| x |}^{p - 1} sgn x$ for $x \in ℝ$ and $p > 1$ . A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang (2008)

Studia Mathematica

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We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$ , there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$ . Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field $a$ on the lattice $ℤ^{d}$ . The distribution $〈 \cdot 〉$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G (t, x, y)$ satisfy optimal annealed estimates which are $L^{2}$ and $L^{1}$ , respectively, in probability, i.e., they obtained bounds on $〈 | \nabla_{x} G (t, x, y) {|^{2} 〉}^{1 / 2}$ and $〈 | \nabla_{x} \nabla_{y} G (t, x, y) | 〉$ . In particular, the elliptic Green’s function $G (x, y)$ satisfies optimal annealed bounds. In their recent work,...

Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Youssef Akdim, Mohammed Belayachi, Hassane Hjiaj (2023)

Mathematica Bohemica

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This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $\begin{matrix} t 2 & - {div (b (| u |) | \nabla u |}^{p - 2} {\nabla u) + d (| u |) | \nabla u |}^{p} = f - {div (c (x) | u |}^{α}) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} t$ where $Ω$ is a bounded open set of $ℝ^{N}$ ( $N \geq 2$ ) with $1 < p < N$ and $f \in L^{1} (Ω),$ under some growth conditions on the function $b (\cdot)$ and $d (\cdot),$ where $c (\cdot)$ is assumed to be in $L^{\frac{N}{(p - 1)}} (Ω) .$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study nonlinear elliptic equations of the form $F (D^{2} u) = f (u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $ξ \in ℝ^{n}$ . We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.