Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth

Robert Černý

Displaying similar documents to “Generalized $n$ -Laplacian: semilinear Neumann problem with the critical growth”

Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

Robert Černý (2012)

Open Mathematics

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Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate...

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Robert Černý (2014)

Open Mathematics

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Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $u \in W_{0}^{1} L^{Φ} (Ω) a n d - d i v (Φ^{'} (|\nabla u|) \frac{\nabla u}{|\nabla u|}) + V (x) Φ^{'} (|u|) \frac{u}{|u|} = f (x, u) + μ h (x) i n Ω,$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the...

Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces

Robert Černý (2012)

Commentationes Mathematicae Universitatis Carolinae

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Let $n \geq 2$ and $Ω \subset ℝ^{n}$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_{0} L^{Φ} (Ω)$ , where the Young function $Φ$ behaves like $t^{n} {log}^{α} (t)$ , $α < n - 1$ , for $t$ large, into the Zygmund space $Z_{0}^{\frac{n - 1 - α}{n}} (Ω)$ . We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_{0}^{m} L^{\frac{n}{m}, q} {log}^{α} L (Ω)$ , $m < n$ , $q \in (1, \infty]$ , $α < \frac{1}{q^{'}}$ , embedded into the Zygmund space $Z_{0}^{\frac{1}{q^{'}} - α} (Ω)$ .

On generalized Moser-Trudinger inequalities without boundary condition

Robert Černý (2012)

Czechoslovak Mathematical Journal

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We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.

Displaying similar documents to “Generalized n -Laplacian: semilinear Neumann problem with the critical growth”

Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces

On generalized Moser-Trudinger inequalities without boundary condition

Displaying similar documents to “Generalized $n$ -Laplacian: semilinear Neumann problem with the critical growth”