Displaying similar documents to “Nonlinear elliptic problems with jumping nonlinearities near the first eigenvalue”

On weighted estimates of solutions of nonlinear elliptic problems

Igor V. Skrypnik, Dmitry V. Larin (1999)

Mathematica Bohemica

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The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, 2 p < n for a solution of a degenerate nonlinear elliptic equation in a domain B ( x 0 , 1 ) F , F B ( x 0 , d ) = { x n | x 0 - x | < d } , d < 1 2 , under the boundary-value conditions u ( x , k ) = k for x F , u ( x , k ) = 0 for x B ( x 0 , 1 ) and where 0 < ρ d i s t ( x , F ) , w ( x ) is a weighted function from some Muckenhoupt class, and c a p p , w ( F ) , w ( B ( x , ρ ) ) are weighted capacity and measure of the corresponding sets.

A population biological model with a singular nonlinearity

Sayyed Hashem Rasouli (2014)

Applications of Mathematics

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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...

Korn's First Inequality with variable coefficients and its generalization

Waldemar Pompe (2003)

Commentationes Mathematicae Universitatis Carolinae

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If Ω n is a bounded domain with Lipschitz boundary Ω and Γ is an open subset of Ω , we prove that the following inequality Ω | A ( x ) u ( x ) | p d x 1 / p + Γ | u ( x ) | p d n - 1 ( x ) 1 / p c u W 1 , p ( Ω ) holds for all u W 1 , p ( Ω ; m ) and 1 < p < , where ( A ( x ) u ( x ) ) k = i = 1 m j = 1 n a k i j ( x ) u i x j ( x ) ( k = 1 , 2 , ... , r ; r m ) defines an elliptic differential operator of first order with continuous coefficients on Ω ¯ . As a special case we obtain Ω u ( x ) F ( x ) + ( u ( x ) F ( x ) ) T p d x c Ω | u ( x ) | p d x , ( * ) for all u W 1 , p ( Ω ; n ) vanishing on Γ , where F : Ω ¯ M n × n ( ) is a continuous mapping with det F ( x ) μ > 0 . Next we show that ( * ) is not valid if n 3 , F L ( Ω ) and det F ( x ) = 1 , but does hold if p = 2 , Γ = Ω and F ( x ) is symmetric and positive definite in Ω .

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in k at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.