Geometry and topology of the boundary in the critical Neumann problem. (English) Zbl 0804.35036

Adimurthi; Mancini, G.

J. Reine Angew. Math. 456, 1-18 (1994).

The paper deals with existence and multiplicity results for semilinear elliptic equations with critical nonlinearity and Neumann (or mixed) boundary conditions. In dimension two, the number of (positive) solutions is shown to be bounded below by the category of the boundary, while in dimension greater than two a lower bound is given in terms of the topology of the positively curved part of the boundary. As a special case it is proved that the (homogeneous) Neumann problem has at least two positive solutions provided the mean curvature of the boundary is positive everywhere. An existence result without curvature assumptions is also given.

Reviewer: G.Mancini (Bologna)i

MSC:

35J65

Nonlinear boundary value problems for linear elliptic equations

Keywords:

Neumann or mixed boundary conditions; existence and multiplicity results; semilinear elliptic equations; critical nonlinearity