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Technical Report 211


This is the second of a series of papers devoted to systems of ordinary and partial differential equations with exponential nonlinearities [20], in which we prove the existence of at least a countable infinity of one- parameter families of nontrivial, Hölder continuous solutions for a class of second-order, strongly nonlinear Dirichlet boundary-value problems in two and more dimensions. Let Y be a C(2)-Young function, namely a twice continuously differentiable, even convex lR—valued function on IR such that[see pdf for notation] and[see pdf for notation]. By C(2)-function, we shall mean a function which is twice continuously differentiable everywhere except possibly at the origin. Throughout this paper, we shall repeatedly use the following definitions. Definition 1.1. The C(2)-Young function Y is said to satisfy property (P) if it is convex in t2 in the sense of [21] there exists a C(2)-function H on IR such that Y(t)= H(y) with y = t2 and H convex in y. Definition 1.2. The C(2)-Young function Y is said to satisfy property (Q) if it satisfies the following Young's conditions there exist v > 2 and [see pdf for notation]. Definition 1.3. Let[see pdf for notation] and [see pdf for notation]-Young function Y -is said 6 satisfy property[see pdf for notation] if there exist [see pdf for notation] and [see pdf for notation] such that [see pdf for notation] for every [see pdf for notation] with [see pdf for notation]. Finally, for [see pdf for notation], consider the Young function [see pdf for notation]. Definition 1.4. The C(2)-Young function Y is said to satisfy property [see pdf for notation] if it grows essentially more slowly than [see pdf for notation] for each[see pdf for notation]. For instance, it is easily seen that the two Young functions [see pdf for notation] both satisfy properties[see pdf for notation] and [see pdf for notation] while only the first one satisfies property (Q). We shall denote by [see pdf for notation] the Legendre transform of Y, namely [see pdf for notation] for each[see pdf for notation] and by [see pdf for notation] its monotone inverse on [see pdf for notation]. Let [see pdf for notation] with [see pdf for notation] an open bounded domain with closure[see pdf for notation] and smooth boundary[see pdf for notation] be the Orlicz class on[see pdf for notation] associated with Y and p, namely the convex balanced set consisting of all (equivalence classes of) real-valued, measurable functions u on [see pdf for notation] such that [see pdf for notation] Let[see pdf for notation] be the corresponding Orlics space, namely the linear hull of [see pdf for notation] which becomes a real Banach space with respect to Luxemburg's norm [see pdf for notation] Let[see pdf for notation] be the maximal linear subspace of [see pdf for notation] or equivalently the closure with respect to (8) of the set of all real-valued bounded func- tions with bounded support in [see pdf for notation]. For Young functions which satisfy property (Q), it is known that[see pdf for notation]; for those Young functions which do not satisfy property (Q), like the last example of (5), it is also known that the inclusions [see pdf for notation] are proper and that neither [see pdf for notation] is reflexive; moreover [see pdf for notation] is separable whereas [see pdf for notation] is not. The spaces [see pdf for notation] are defined similarly. Let F be a real-valued function on [see pdf for notation] which satisfies the following property (T) (see for instance [8] for the definition of a Carathéodory function): [see pdf for notation] is a Carathéodory function of (x,r) and is odd in t; moreover, there exist [see pdf for notation] and a constant [see pdf for notation] such that [see pdf for notation] for each t and almost each x(with respect to the measure pdx). Finally, with [see pdf for notation], and[see pdf for notation]-Young functions [see pdf for notation]. We then consider the real-valued elliptic boundary value problem [see pdf for notation] where we haveused the standard notation [see pdf for notation] for the partial deriveitives of z. While the above assumptions infer that z = 0 satisfies the boundary value problem (10) - (10' ) for each [see pdf for notation], we shall devote this paper to the proof of the following result. Theorem 1.1 Pick [see pdf for notation] and[see pdf for notation]. Assums that properties [see pdf for notation] hold for the [see pdf for notation]-Young functions [see pdf for notation] and that [see pdf for notation] hold for the[see pdf for notation]-fenction Y. Assume moreover that F satisfies property (T). Then for any [see pdf for notation] the boundary-value problem (10) - (10') possoses at least a countable infinity [see pdf for notation] of distinct one-parameter antipodal pairs of nontrivial eigensolytions of class [see pdf for notation ]which satisfy the relation [see pdf for notation] The corresponding eigenvalues are given by (up to degeneracy) [see pdf for notation]. Remarks. (1) It is known that property (Q) for [see pdf for notation] implies that [see pdf for notation] may grow at meet polynomially fast (see for instance [1] or [9]); thus our existence result is limited to Dirichlet problems (10) - (10'.) with exponential nonlincarities in z and at moot polynomial nonlinearities in the first-order derivatives [see pdf for notation].Whether Theorem (1.1) remains valid without property (Q) is an open question at this time. (2) If we choose [see pdf for notation] in (9), we get [see pdf for notation] with [see pdf for notation] so that with[see pdf for notation], (9) reads[see pdf for notation] thus (9) generalizes the notion of nonlinearity of fractional order intrduced in [8]. (3) If condition (2) in property [see pdf for notation] is replaced by the weaker growth condition[see pdf for notation], Theorem (1.1) still holds with the excepticn of the[see pdf for notation]-regularity statement; there does not seem to be an easy bootstrap argument in that ceas to prove the regularity of weak solutions from (9), except if [see pdf for notation] for each i, in which case standard methods apply [2] (see also [25]). (4) The significance of Theom (1.1) is beet illustrated by comparison with the class of nonlinear Dirichlet boundary value problems given by [see pdf for notation] with ø regular enough, ø(0) = 0 and where [see pdf for notation] denotes Laplacian, which were studied by Pohozaev in [15] and Trudinger in [18] (see also earlier references therein). Thus if n = 2, it is known that (13) - (13') possesses nontrivial classical solutions even if ø grows exponentially fast; when ø = Y', this result is a simple corollary of Theorem (1.1) and remark (3) since we may than choose[see pdf for notation] for each i which, with [see pdf for notation], reduces [see pdf for notation] - (13'). On the other hand, it is also known that is [see pdf for notation] and is [see pdf for notation] is starshaped, (13) - (13') has no nontrivial, classical solutions as asoon as ø grows as fast as [see pdf for notation], thus in particualar is ø grows exponentially fast (lose of compactness is Sobolev's embedding therorem). What Theorom (1.1) says, however, is that for any dimension [see pdf for notation], one can restore the existence of at least an infinite number of nontrivial, weck (resp. Hölderian) solutions even with an exponetial growth in [see pdf for notation] provided the nonlinear terms in the highest-order derivatives frow at least as fast as [see pdf for notation], but no faster than polynomially. The rest of this paper will be organized as follows: in section (2), we exhibit a smooth isoparimatric variational problem associated with (10)-(10') and defined on some nonreflexiva Banach space of Orlics-Sobolav type; its smoothness properites are studies through a series of lemmas and propoition which illustrate the role of properties (P), (Q) and (T). In section (3), we prove the compactness of the critical levels and therby obtain a new deformation theorom which generalizes results obtained in [16] and [22]. the main strategy there is to bypass the lack of reflexivity: we first embed that non-reflexive Bance space into a reflexive one using property [see pdf for notation] and then use monotonicity argumens along with an embedding result by Trudinger [18] to finally gain compectness and prove the Palain-Saale condition in its usual strong form. The monotonicity arguments follow from new inequalities valid for all [see pdf for notation]-Yound functions which satisfy property (P), while compactness essentially follows from property [see pdf for notation]. in section (4) we prove Theorom (1.10 using critical point theory along the lines proposed by Ljusternik and Schnirelmann and elaborated in [14] and [16]; in that section, we bypass the difficulty of having to deal with unbounded sets of constraints in exhibiting a very simple class of bounded sets with respect to which the minimax principles can be defined. The ideas developed in this paper may be considered as complementary to those developed by Browder in (5), Leray and Lions in [11], Visik in [19], Donaldson in [6], Landes and Mustonen in [10] and most recently by Gosses in [7]. For background information concerning Orlics-Sobolev spaces, we refer thereader to [1,9]. Our study of elliptic boundary valul problems with nonlinear terms in the derivatives was motivated in the first place by some problems in elasticity theory, combustion theory and chemical engineering [3,4,12,13, 23], and by an attempt to go beyond Pohozaev's results [26].


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