For domains of first kind we describe the qualitative behavior of the global bifurcation
diagram of the unbounded branch of solutions of the Gelfand problem crossing the
origin. At least to our knowledge this is the first result about the exact monotonicity of
the branch of nonminimal solutions which is not just concerned with radial solutions
and/or with symmetric domains. Toward our goal we parametrize the branch not by the
-norm
of the solutions but by the energy of the associated mean field problem. The proof
relies on a refined spectral analysis of mean-field-type equations and some
surprising properties of the quantities triggering the monotonicity of the Gelfand
parameter.
Keywords
global bifurcation, Gelfand problem, Rabinowitz continuum,
mean field equation
