| Title:
|
Generalized $n$-Laplacian: semilinear Neumann problem with the critical growth (English) |
| Author:
|
Černý, Robert |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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58 |
| Issue:
|
5 |
| Year:
|
2013 |
| Pages:
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555-593 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $. (English) |
| Keyword:
|
Orlicz-Sobolev space |
| Keyword:
|
Mountain Pass Theorem |
| Keyword:
|
Palais-Smale sequence |
| Keyword:
|
Ekeland Variational Principle |
| MSC:
|
26D10 |
| MSC:
|
46E30 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 06282096 |
| idMR:
|
MR3104618 |
| DOI:
|
10.1007/s10492-013-0028-0 |
| . |
| Date available:
|
2013-09-14T11:43:43Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143432 |
| . |
| Reference:
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