Displaying similar documents to “On weighted estimates of solutions of nonlinear elliptic problems”

On a higher-order Hardy inequality

David Eric Edmunds, Jiří Rákosník (1999)

Mathematica Bohemica

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The Hardy inequality Ω | u ( x ) | p d ( x ) - p x ¨ c Ω | u ( x ) | p x ¨ with d ( x ) = dist ( x , Ω ) holds for u C 0 ( Ω ) if Ω n is an open set with a sufficiently smooth boundary and if 1 < p < . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for p = 1 .

Nonlinear elliptic problems with jumping nonlinearities near the first eigenvalue

Pavel Drábek (1981)

Aplikace matematiky

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In this paper existence and multiplicity of solutions of the elliptic problem u + λ 1 u + μ u + v u - + g ( x , u ) = f in Ω B u = 0 on Ω , are discussed provided the parameters μ and v are close to the first eigenvalue 1 . The sufficient conditions presented here are more general than those in given by S. Fučík in his aerlier paper.

Transition from decay to blow-up in a parabolic system

Pavol Quittner (1998)

Archivum Mathematicum

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We show a locally uniform bound for global nonnegative solutions of the system u t = Δ u + u v - b u , v t = Δ v + a u in ( 0 , + ) × Ω , u = v = 0 on ( 0 , + ) × Ω , where a > 0 , b 0 and Ω is a bounded domain in n , n 2 . In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded.

The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems

Pavel Drábek (1995)

Mathematica Bohemica

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We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem - div ( a ( x , u ) | | p - 2 u ) = λ b ( x , u ) | u | p - 2 u in Ω , u = 0 on Ω , where Ω is a bounded domain, p > 1 is a real number and a ( x , u ) , b ( x , u ) satisfy appropriate growth conditions. Moreover, the coefficient a ( x , u ) contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in L ( Ω ) . The main tool is the investigation of the associated homogeneous eigenvalue problem...

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

Ryuji Kajikiya (2014)

Mathematica Bohemica

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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 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...