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Positive solutions of the p -Laplace Emden-Fowler equation in hollow thin symmetric domains

Ryuji Kajikiya (2014)

Mathematica Bohemica

We study the existence of positive solutions for the p -Laplace Emden-Fowler equation. Let H and G be closed subgroups of the orthogonal group O ( N ) such that H G ⊂ O ( N ) . We denote the orbit of G through x ∈ ℝ N by G ( x ) , i.e., G ( x ) : = { g x : g ∈ G } . We prove that if H ( x ) G ( x ) for all x ∈ Ω ¯ and the first eigenvalue of the p -Laplacian is large enough, then no H invariant least energy solution is G invariant. Here an H invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all H invariant functions. Therefore...

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

We consider linear elliptic equations - Δ u + q ( x ) u = λ u + f in bounded Lipschitz domains D ⊂ ℝ N with mixed boundary conditions ∂ u / ∂ n = σ ( x ) λ u + g on ∂ D . The main feature of this boundary value problem is the appearance of λ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient σ ( x ) . We study positivity principles and anti-maximum principles. One of our main results states that if σ is somewhere negative, q ≥ 0 and ∫ D q ( x ) d x > 0 then there exist two eigenvalues λ - 1 , λ 1 such the positivity principle...

Post-buckling range of plates in axial compression with uncertain initial geometric imperfections

Ivan Hlaváček (2002)

Applications of Mathematics

The method of reliable solutions alias the worst scenario method is applied to the problem of von Kármán equations with uncertain initial deflection. Assuming two-mode initial and total deflections and using Galerkin approximations, the analysis leads to a system of two nonlinear algebraic equations with one or two uncertain parameters-amplitudes of initial deflections. Numerical examples involve (i) minimization of lower buckling loads and (ii) maximization of the maximal mean reduced stress.

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