JournalsjemsVol. 16, No. 8pp. 1687–1748

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

  • Manuel del Pino

    Universidad de Chile, Santiago, Chile
  • Fethi Mahmoudi

    Universidad de Chile, Santiago, Chile
  • Monica Musso

    Ponificia Universidad Catolica de Chile, Santiago, Chile
Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents cover
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Abstract

Let Ω\Omega be a bounded domain in Rn\mathbb R^n with smooth boundary Ω\partial \Omega. We consider the equation d2Δuu+unk+2nk2=0 in Ωd^2\Delta u - u+u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{ in }\,\Omega , under zero Neumann boundary conditions, where Ω\Omega is open, smooth and bounded and dd is a small positive parameter. We assume that there is a kk-dimensional closed, embedded minimal submanifold KK of Ω\partial\Omega, which is non-degenerate, and certain weighted average of sectional curvatures of Ω\partial\Omega is positive along KK. Then we prove the existence of a sequence d=dj0d=d_j\to 0 and a positive solution udu_d such that

d2ud2SδKasd0d^2 |\nabla u_{d} |^2\,\rightharpoonup \, S\,\delta_K \: \mathrm {as} \: d \to 0

in the sense of measures, where δK\delta_K stands for the Dirac measure supported on KK and SS is a positive constant.

Cite this article

Manuel del Pino, Fethi Mahmoudi, Monica Musso, Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents. J. Eur. Math. Soc. 16 (2014), no. 8, pp. 1687–1748

DOI 10.4171/JEMS/473