Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth
Applications of Mathematics, Tome 58 (2013) no. 5, pp. 555-593
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
Let $\Omega \subset \mathbb R^n$, $n\geq 2$, 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 $$ \displaylines { u\in W^1 L^{\Phi }(\Omega ), \quad -\operatorname {div}\Big (\Phi '(|\nabla u|)\frac {\nabla u}{|\nabla u|}\Big ) +V(x)\Phi '(|u|)\frac {u}{|u|}=f(x,u)+\mu h(x)\quad \text {in} \Omega ,\cr \frac {\partial u}{\partial {\bf n}}=0\quad \text {on} \partial \Omega ,\cr } $$ 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, $\mu \geq 0$ is a small parameter and ${\bf n}$ denotes the outward unit normal to $\partial \Omega $.
Let $\Omega \subset \mathbb R^n$, $n\geq 2$, 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 $$ \displaylines { u\in W^1 L^{\Phi }(\Omega ), \quad -\operatorname {div}\Big (\Phi '(|\nabla u|)\frac {\nabla u}{|\nabla u|}\Big ) +V(x)\Phi '(|u|)\frac {u}{|u|}=f(x,u)+\mu h(x)\quad \text {in} \Omega ,\cr \frac {\partial u}{\partial {\bf n}}=0\quad \text {on} \partial \Omega ,\cr } $$ 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, $\mu \geq 0$ is a small parameter and ${\bf n}$ denotes the outward unit normal to $\partial \Omega $.
DOI :
10.1007/s10492-013-0028-0
Classification :
26D10, 46E30, 46E35
Keywords: Orlicz-Sobolev space; Mountain Pass Theorem; Palais-Smale sequence; Ekeland Variational Principle
Keywords: Orlicz-Sobolev space; Mountain Pass Theorem; Palais-Smale sequence; Ekeland Variational Principle
@article{10_1007_s10492_013_0028_0,
author = {\v{C}ern\'y, Robert},
title = {Generalized $n${-Laplacian:} semilinear {Neumann} problem with the critical growth},
journal = {Applications of Mathematics},
pages = {555--593},
year = {2013},
volume = {58},
number = {5},
doi = {10.1007/s10492-013-0028-0},
mrnumber = {3104618},
zbl = {06282096},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0028-0/}
}
TY - JOUR AU - Černý, Robert TI - Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth JO - Applications of Mathematics PY - 2013 SP - 555 EP - 593 VL - 58 IS - 5 UR - http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0028-0/ DO - 10.1007/s10492-013-0028-0 LA - en ID - 10_1007_s10492_013_0028_0 ER -
%0 Journal Article %A Černý, Robert %T Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth %J Applications of Mathematics %D 2013 %P 555-593 %V 58 %N 5 %U http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0028-0/ %R 10.1007/s10492-013-0028-0 %G en %F 10_1007_s10492_013_0028_0
Černý, Robert. Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth. Applications of Mathematics, Tome 58 (2013) no. 5, pp. 555-593. doi: 10.1007/s10492-013-0028-0
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