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- Nonlinear boundary value problems; Dirichlet problem; Isoperimetric inequalities
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 boundary-value 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 boundary-value 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  and ). 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 first-order 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 boundary-value 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  and  for complete details.