Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Rey, Olivier, and Wei, Juncheng. "Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity." Journal of the European Mathematical Society 007.4 (2005): 449-476. <http://eudml.org/doc/277359>.

@article{Rey2005,
abstract = {We show that the critical nonlinear elliptic Neumann problem $\Delta u−\mu u+u^\{7/3\}=0$ in $\Omega $, $u>0$ in $\Omega $, $\frac\{\partial u\}\{\partial \nu \}=0$ on $\partial \Omega $, where $\Omega $ is a bounded and smooth domain in $\mathbb \{R\}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists $\mu _K>0$ such that for $0<\mu <\mu _K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega $, as $\mu \rightarrow 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.},
author = {Rey, Olivier, Wei, Juncheng},
journal = {Journal of the European Mathematical Society},
keywords = {semilinear elliptic Neumann problems; critical Sobolev exponent; blow-up},
language = {eng},
number = {4},
pages = {449-476},
publisher = {European Mathematical Society Publishing House},
title = {Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity},
url = {http://eudml.org/doc/277359},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Rey, Olivier
AU - Wei, Juncheng
TI - Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 4
SP - 449
EP - 476
AB - We show that the critical nonlinear elliptic Neumann problem $\Delta u−\mu u+u^{7/3}=0$ in $\Omega $, $u>0$ in $\Omega $, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega $, where $\Omega $ is a bounded and smooth domain in $\mathbb {R}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists $\mu _K>0$ such that for $0<\mu <\mu _K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega $, as $\mu \rightarrow 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.
LA - eng
KW - semilinear elliptic Neumann problems; critical Sobolev exponent; blow-up
UR - http://eudml.org/doc/277359
ER -