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

Robert Černý

Open Mathematics (2014)

  • Volume: 12, Issue: 1, page 114-127
  • ISSN: 2391-5455

Abstract

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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 ∈ W 0 1 L Φ Ω a n d - d i v Φ ' ∇ u ∇ u ∇ u + V x Φ ' u 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 case of Ω = ℝn.

How to cite

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Robert Černý. "Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range." Open Mathematics 12.1 (2014): 114-127. <http://eudml.org/doc/269606>.

@article{RobertČerný2014,
abstract = {Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem \[u \in W\_0^1 L^\Phi \left( \Omega \right) and - div\left( \{\Phi ^\{\prime \}\left( \{\left| \{\nabla u\} \right|\} \right)\frac\{\{\nabla u\}\}\{\{\left| \{\nabla u\} \right|\}\}\} \right) + V\left( x \right)\Phi ^\{\prime \}\left( \{\left| u \right|\} \right)\frac\{u\}\{\{\left| u \right|\}\} = f\left( \{x,u\} \right) + \mu h\left( x \right) in \Omega ,\] 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 case of Ω = ℝn.},
author = {Robert Černý},
journal = {Open Mathematics},
keywords = {Orlicz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; Moser iteration; generalized -Laplacian; boundedness of weak solutions; Dirichlet problem},
language = {eng},
number = {1},
pages = {114-127},
title = {Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range},
url = {http://eudml.org/doc/269606},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Robert Černý
TI - Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range
JO - Open Mathematics
PY - 2014
VL - 12
IS - 1
SP - 114
EP - 127
AB - Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem \[u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi ^{\prime }\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi ^{\prime }\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,\] 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 case of Ω = ℝn.
LA - eng
KW - Orlicz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; Moser iteration; generalized -Laplacian; boundedness of weak solutions; Dirichlet problem
UR - http://eudml.org/doc/269606
ER -

References

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