Displaying similar documents to “Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data”

Generic existence result for an eigenvalue problem with rapidly growing principal operator

Vy Khoi Le (2004)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the eigenvalue problem - div ( a ( | ∇ u | ) ∇ u ) = λ g ( x , u ) in Ω u = 0 on ∂ Ω , in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.

On a class of nonlinear problems involving a p ( x ) -Laplace type operator

Mihai Mihăilescu (2008)

Czechoslovak Mathematical Journal

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We study the boundary value problem - d i v ( ( | ∇ u | p 1 ( x ) - 2 + | ∇ u | p 2 ( x ) - 2 ) ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in ℝ N . Our attention is focused on two cases when f ( x , u ) = ± ( - λ | u | m ( x ) - 2 u + | u | q ( x ) - 2 u ) , where m ( x ) = max { p 1 ( x ) , p 2 ( x ) } for any x ∈ Ω ¯ or m ( x ) < q ( x ) < N · m ( x ) ( N - m ( x ) ) for any x ∈ Ω ¯ . In the former case we show the existence of infinitely many weak solutions for any λ > 0 . In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ 2 -symmetric version for even...