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

Robert Černý

Applications of Mathematics (2013)

  • Volume: 58, Issue: 5, page 555-593
  • ISSN: 0862-7940

Abstract

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Let Ω ⊂ ℝ n , n ≥ 2 , be a bounded connected domain of the class C 1 , θ for some θ ∈ ( 0 , 1 ] . Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem u ∈ W 1 L Φ ( Ω ) , - div Φ ' ( | ∇ u | ) ∇ u | ∇ u | + V ( x ) Φ ' ( | u | ) u | u | = f ( x , u ) + μ h ( x ) in Ω , ∂ u ∂ 𝐧 = 0 on ∂ Ω , where Φ is a Young function such that the space W 1 L Φ ( Ω ) is embedded into 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 nontrivial continuous function, μ ≥ 0 is a small parameter and 𝐧 denotes the outward unit normal to ∂ Ω .

How to cite

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Černý, Robert. "Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth." Applications of Mathematics 58.5 (2013): 555-593. <http://eudml.org/doc/260656>.

@article{Černý2013,
abstract = {Let $\Omega \subset \mathbb \{R\}^n$, $n\ge 2$, be a bounded connected domain of the class $C^\{1,\theta \}$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem \[ \{ u\in W^1 L^\{\Phi \}(\Omega ), \quad -\operatorname\{div\}\Big (\Phi ^\{\prime \}(|\nabla u|)\frac\{\nabla u\}\{|\nabla u|\}\Big ) +V(x)\Phi ^\{\prime \}(|u|)\frac\{u\}\{|u|\}=f(x,u)+\mu h(x)\quad \text\{in\} \Omega ,\cr \frac\{\partial u\}\{\partial \{\bf n\}\}=0\quad \text\{on\} \partial \Omega ,\cr \} \] where $\Phi $ is a Young function such that the space $W^1 L^\{\Phi \}(\Omega )$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V(x)$ is a continuous potential, $h\in (L^\{\Phi \}(\Omega ))^*$ is a nontrivial continuous function, $\mu \ge 0$ is a small parameter and $\{\bf n\}$ denotes the outward unit normal to $\partial \Omega $.},
author = {Černý, Robert},
journal = {Applications of Mathematics},
keywords = {Orlicz-Sobolev space; Mountain Pass Theorem; Palais-Smale sequence; Ekeland Variational Principle; Orlicz-Sobolev space; mountain-pass theorem; Palais-Smale sequence; Ekeland variational principle},
language = {eng},
number = {5},
pages = {555-593},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth},
url = {http://eudml.org/doc/260656},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Černý, Robert
TI - Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 5
SP - 555
EP - 593
AB - Let $\Omega \subset \mathbb {R}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem \[ { u\in W^1 L^{\Phi }(\Omega ), \quad -\operatorname{div}\Big (\Phi ^{\prime }(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big ) +V(x)\Phi ^{\prime }(|u|)\frac{u}{|u|}=f(x,u)+\mu h(x)\quad \text{in} \Omega ,\cr \frac{\partial u}{\partial {\bf n}}=0\quad \text{on} \partial \Omega ,\cr } \] where $\Phi $ is a Young function such that the space $W^1 L^{\Phi }(\Omega )$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V(x)$ is a continuous potential, $h\in (L^{\Phi }(\Omega ))^*$ is a nontrivial continuous function, $\mu \ge 0$ is a small parameter and ${\bf n}$ denotes the outward unit normal to $\partial \Omega $.
LA - eng
KW - Orlicz-Sobolev space; Mountain Pass Theorem; Palais-Smale sequence; Ekeland Variational Principle; Orlicz-Sobolev space; mountain-pass theorem; Palais-Smale sequence; Ekeland variational principle
UR - http://eudml.org/doc/260656
ER -

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