On a Class of Strongly Nonlinear Dirichlet Boundary-Value Problems: Beyond Pohozaev's Results

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Vuillermot, Pierre A.
Department of Mathematics
Nonlinear boundary value problems; Dirichlet problem; Isoperimetric inequalities
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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 [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 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 [3] and [4] for complete details.