Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds. (English) Zbl 1288.58013
Let \((M,g)\) be a smooth and compact Riemannian \(n\)-manifold. The paper deals with the equation
\[
\Delta_gu+hu=|u|^{2^*-2-\varepsilon}
\]
where \(\Delta_g=\text{div}_g\nabla\) is the Laplace–Beltrami operator on \(M,\) \(h\in C^1(M),\) \(2^*\) is the critical Sobolev exponent \({{2n}\over{n-2}}\) for the embedding \(H^2_1(M)\) into \(L^2(M)\) and \(\varepsilon\) is a small positive real parameter tending to \(0.\) The authors prove existence of blowing-up families of sign-changing solutions which develop bubble towers at some point where the function \(h\) is greater than the Yamabe potential \({{n-2}\over{4(n-1)}}\text{Scal}_g\) with \(\text{Scal}_g\) being the scalar curvature of \(M.\)
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
elliptic equations on manifolds; asymptotically critical equations; Riemannian manifold; Laplace-Beltrami operator; Yamabe potentialReferences:
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