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

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 4, page 449-476
- ISSN: 1435-9855

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topRey, 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 -

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