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A population biological model with a singular nonlinearity

Sayyed Hashem Rasouli (2014)

Applications of Mathematics

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form - div ( | x | - α p | u | p - 2 u ) = | x | - ( α + 1 ) p + β a u p - 1 - f ( u ) - c u γ , x Ω , u = 0 , x Ω , where Ω is a bounded smooth domain of N with 0 Ω , 1 < p < N , 0 α < ( N - p ) / p , γ ( 0 , 1 ) , and a , β , c and λ are positive parameters. Here f : [ 0 , ) is a continuous function. This model arises in the studies of population biology of one species with u representing the concentration of the species. We discuss the existence of a positive solution when f satisfies certain additional conditions. We use the method of sub-supersolutions...

Averaging techniques and oscillation of quasilinear elliptic equations

Zhi-Ting Xu, Bao-Guo Jia, Shao-Yuan Xu (2004)

Annales Polonici Mathematici

By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order i , j = 1 N D i [ A i j ( x ) | D y | p - 2 D j y ] + p ( x ) f ( y ) = 0 are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a ( | u ( x ) | ) u ( x ) + f ( u ( x ) ) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem

Baoqing Liu, Qikui Du (2014)

Applications of Mathematics

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence....

Dynamic Programming Principle for tug-of-war games with noise

Juan J. Manfredi, Mikko Parviainen, Julio D. Rossi (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

Entropy solutions for nonhomogeneous anisotropic Δ p ( · ) problems

Elhoussine Azroul, Abdelkrim Barbara, Mohamed Badr Benboubker, Hassane Hjiaj (2014)

Applicationes Mathematicae

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

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 L ( 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 -data.

Existence of entropy solutions for degenerate quasilinear elliptic equations in L 1

Albo Carlos Cavalheiro (2014)

Communications in Mathematics

In this article, we prove the existence of entropy solutions for the Dirichlet problem ( P ) - div [ ω ( x ) 𝒜 ( x , u , u ) ] = f ( x ) - div ( G ) , in Ω u ( x ) = 0 , on Ω where Ω is a bounded open set of N , N 2 , f L 1 ( Ω ) and G / ω [ L p ' ( Ω , ω ) ] N .

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ Δ ( | Δ u | p 2 Δ u ) d i v ( | u | p 2 u ) = λ f ( x , u ) + μ g ( x , u ) in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where Ω N (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

Gradient potential estimates

Giuseppe Mingione (2011)

Journal of the European Mathematical Society

Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.

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