Nonlinear Fredholm Operators with Noncompact Fibers and Applications to Elliptic Problems on RN

Under suitable conditions, an equation F(x)=y between Banach spaces involving a nonlinear Fredholm mapping F of nonnegative index is shown to have a noncompact and hence infinite set of solutions for almost every y for which the equation is solvable. The proof of this nonuniqueness (but not existence) result relies on an entirely new line of arguments in which the concept of generalized critical value plays a central role. When F: W2, p(RN)→Lp(RN) is associated with a quasilinear elliptic PDE on RN with “constant coefficient,” it often happens that a crucial denseness hypothesis in the abstract theorem is equivalent to the existence of a nontrivial solution to the equation F(x)=F(0) and hence can be verified in practice. Generalizations exist for some classes of problems with nonconstant coefficients and for problems on exterior domains.