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

Olivier ReyJuncheng Wei — 2005

Journal of the European Mathematical Society

We show that the critical nonlinear elliptic Neumann problem Δ u μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , u ν = 0 on Ω , where Ω is a bounded and smooth domain in 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ 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...

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