JournalsjemsVol. 7 , No. 4DOI 10.4171/jems/35

Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity

  • Juncheng Wei

    University of British Columbia, Vancouver, Canada
  • Olivier Rey

    École Polytechnique, Palaiseau, France
Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity cover

Abstract

We show that the critical nonlinear elliptic Neumann problem

Δuμu+u7/3=0  \mboxin \Om,  u>0 \mboxin \Om \mboxand uν=0  \mboxon \Om\Delta u -\mu u + u^{7/3} = 0 \ \ \mbox{in} \ \Om, \ \ u >0 \ \mbox{in} \ \Om \ \mbox{and} \ \frac{ \partial u}{\partial \nu} = 0 \ \ \mbox{on} \ \partial \Om

where \Om\Om is a bounded and smooth domain in R5\R^5, has arbitrarily many solutions, provided that μ>0\mu>0 is small enough. More precisely, for any positive integer KK, there exists μK>0\mu_K >0 such that for 0<μ<μK0 <\mu < \mu_K , the above problem has a nontrivial solution which blows up at KK interior points in Ω\Omega, as μ0\mu \to 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.