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Dead cores of singular Dirichlet boundary value problems with φ -Laplacian

Ravi P. Agarwal, Donal O'Regan, Staněk, Svatoslav (2008)

Applications of Mathematics

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem ( φ ( u ' ) ) ' = λ f ( t , u , u ' ) , u ( 0 ) = u ( T ) = A . Here λ is the positive parameter, A > 0 , f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p -Laplacian operator

De-xiang Ma, Weigao Ge, Zhan-Ji Gui (2007)

Czechoslovak Mathematical Journal

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ( φ p ( u ' ) ) ' + q ( t ) f ( u ) = 0 , 0 < t < 1 , where φ p ( s ) : = | s | p - 2 s , p > 1 , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q ( t ) may be singular at t = 0 , 1 .

Existence and L∞ estimates of some Mountain-Pass type solutions

José Maria Gomes (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We prove the existence of a positive solution to the BVP ( Φ ( t ) u ' ( t ) ) ' = f ( t , u ( t ) ) , u ' ( 0 ) = u ( 1 ) = 0 , imposing some conditions on Φ and f. In particular, we assume Φ ( t ) f ( t , u ) to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An L bound for the solution is provided by the L norm of any test function with negative energy.

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