Continuation of Solutions of Constrained Extremum Problems and Nonlinear Eigenvalue Problems

In this paper we continue our investigations. begun in the previous paper, of describing the solution sets of constrained extremum problems inf Au). l&l-‘(p) (*) where fand f are twice continuously differentiable functionals on a reflexive Banach space V, and r-‘(p) denotes the level set of the functional f with value p E R. Considering p as a parameter in (*) we obtain results concerning the continuation of solutions of (*) and consequently also concerning specific solution branches of the nonlinear eigenvalue problem f’(u) = /.&f’(U). (**) The general results are applied to functionals which lead to nonlinear eigenvalue problems of a semilinear elliptic type and in particular we consider a specific example for which there occurs “bending” of a solution curve (II+) of (**). 1. PRELIMINARIES AND NOTATION Let I/ be a reflexive Banach space, I/* its dual and ( , ) the duality map. We consider two functionals f and t defined on V which we assume to satisfy cfl) fis weakly lower semicontinuous, andfis coercive on V [i.e., f(u) + m if 11 u 11 v + co], (tl) t is weakly continuous. cCf&) f,t E C’(V,R). We consider le\lel sets of the functional t: r-‘(p) := {u E v / f(U) = p} for p E t(V) where f(V) denotes the range of the functional t. The tangent space at a point fi E t-‘(t(k)) is defined if t’(h) # 0 as 71) := {v E v 1 (t’(ic).v) = O} c v.