Existence And Multiplicity Results For Super-linear And Sub-linear Operator Equations

Let X be a real infinite-dimensional Hilbert space with norm ‖·‖X and inner product (·,·)x.Let A:D(A) (?) Xâ†'X be an unbounded self-adjoint operator satisfying σ(A)=σd(A).Assume Φ satisfies R is differentiable and for any x ∈ Z there exists M>0 such that|Φ’(x)y|≤M‖y‖x,(?)y∈Z.It is easy to see (Φ0)implies that for any x∈Z there exists a unique element in X denoted by ▽Φ(x) such that Φ’(x)y=(▽Φ(x),y)x for all y∈Z. We consider the following operator equation: Ax-▽Φ(x)=0.Applying variationsl methods, minimax methods, index theories etc, we get the following results:1. If Φ satisfies the super-linear condition, under some conditions, both the first and second kind operator equations have one solution. Moreover, if Φ satisfies Φ(-x)=Φ(x),the first kind operator equations have infinitely many solutions.2. If Φ is unbounded on ker(A),under some conditions, there is one solution for both the first and second kind operator equations;if further Φ(-x)=Φ(x),there exist dim ker(A) pairs of distinct solutions for the first kind operator equations.3. If Φ satisfies the sub-linear condition,under some conditions, there exists one solution for both the first and second kind operator equations.Here the first kind operators are the ones with σ(A)=σd(A) bounded from below and the second with σ(A)=σd(A) unbounded from both above and below.