Nonlinear elliptic functional equations in nonreflexive Banach spaces

Abstract

If the coefficient functions Aa have at most polynomial growth in u and its derivatives, we have shown in [ l ] how existence theorems for solutions of variational boundary value problems for the equation A (u) —f may be obtained from monotonicity assumptions on the nonlinear Dirichlet form of A, with extensions to weaker ellipticity assumptions in [3], [4], [9], and [15]. These results extend earlier theorems of Visik [18] obtained by other arguments, but (despite an apparent remark to the contrary in [l5]) do not extend automatically to the case of rapidly increasing coefficients treated by Visik in [ l9] . The crucial point for this more general case is that the Banach spaces in which the problems are appropriately formulated are derived from nonreflexive Orlicz spaces [14] and are themselves nonreflexive and nonseparable. On the other hand, the basic treatments of monotone and semimonotone operator equations have been carried out in reflexive Banach spaces (and in [15], it is even essential to consider only separable spaces). I t is our object in the present note to outline a treatment of these more general problems in nonreflexive spaces, using the remark already made by the writer in the final section of [4] on the useful properties of the weak* topology on a conjugate space. We formulate this treatment in a very general context of functional rather than simply differential equations, and apply to the resulting operator equations a new theorem on perturbation of semimonotone operators (Theorem 2 below) which is particularly efficacious in this general context. This result is extended and applied in more detail in [ l l ] to include all the elliptic results of [ l ] , [3], [4], [9], [IS], [18], and [ l9] , and a corresponding extension to unbounded operators and to monotone operators on convex sets gives a significant extension of