Abstract:

Let [see pdf for notation] be an open bounded domain with closure [see pdf for notation] and smooth boundary [see pdf for notation]. Consider the
class of nonlinear boundaryvalue problems defined by [see pdf for notation] with [see pdf for notation] regular enough,[see pdf for notation] and
where [see pdf for notation] denotes Laplacian. If n = 2, it is known that the boundaryvalue problem (1.1) possesses nontrivial, classical
eigensolutions (with appropriate eigenvalues ^) even if ø grows exponentially fasts if n ≥ 3, it is also known that problem (1.1) has, with ^
starshaped, no nontrivial, classical solutions as soon as ø grows as fast as [see pdf for notation], thus in particular if 0 grows exponentially
fast (loss of compactness in Sobolevis embedding Theorem, see [1] and [2]). On the basis of simple energy considerations, it is however natural to
expect that, if ^z in (1.1) is replaced by some sufficiently strong nonlinear term in the firstorder partial derivatives, [see pdf for notation]
say [see pdf for notation] for some suitably chosen [see pdf for notation], one can restore the existence of nontrivial eigenfunctions in the
boundaryvalue problem (1.1) for any dimension n ≥ 3, even with an exponential growth in ø (for a certain appropriate class of ø's, see below).
In this paper, we announce new results which precisely go in that direction. The proofs are ommitted and we refer the reader to [3] and [4] for
complete details. 