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

Olivier Rey, Juncheng 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 ℝ 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...

Area integral estimates for higher order elliptic equations and systems

Björn E. J. Dahlberg, Carlos E. Kenig, Jill Pipher, G. C. Verchota (1997)

Annales de l'institut Fourier

Let L be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in L p norm between the maximal function and the square function of solutions to L in Lipschitz domains. Several applications of this result are discussed.

Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary

Vladimir Kozlov, Vladimir Maz'ya (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive an asymptotic formula of a new type for variational solutions of the Dirichlet problem for elliptic equations of arbitrary order. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.

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