On a Neumann problem involving two critical Sobolev exponents

Abstract

at one or more boundary points. By comparing it with the boundary conditionof the Yamabe Problem, ( ) appears as a natural condition in order to recovercompactness by breaking the conformal invariance.The proofs of the existence results of the above quoted papers rely on theminimization of a Sobolev–type quotient. By the geometrical assumption ( ) oneis able to overcome the lack of compactness due to the presence of the limitingexponent. Indeed, by means of energy estimates obtained by concentration ofsuitable test functions at some boundary points, one can prove that the infimumlies below the first energy level where the Palais–Smale condition fails. A similarperspective has been adopted by the authors in [14] where they solved problem(P) under the assumption that g is a linear function g(x;u)=l(x)uand ( ) holdsat one boundary point and (x) a given function. In [11], the same energy estimatewas obtained, using a more refined technique, when Ω is not conformally relatedto the unit ball (it has to be noticed that in the case of the ball one can exhibitthe explicit expression of the solution).However, this type of discussion can not be extended to the class of problems(P) when g has a superlinear rate of growth at infinity, even under assumption( ) on its linear part at zero, since, when one applies the blow–up technique, thecontribution of the superlinear term dominates that of the linear part. Further-more, when the opposite inequality holds in ( ) one can find examples where theminimization argument (or the application of the Mountain Pass lemma) fails.A natural question is then what is the real role of assumption ( ) (even in itsrelaxed form) in the solvability of (P).The purpose of the present work is to solve (P) under a condition weakerthan ( ) on the linearization l(x)ofgat zero; in addition, we allow a superlinear(but subcritical) rate of growth of g at infinity. As the reader will see, this studywill involve both topological and geometrical considerations. For this reason weshall restrict our analysis to the domain Ω having the simplest geometry andtopology: the unit ball B