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

Open Mathematics (2014)

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

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topRobert 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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