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Multiplicity of solutions for doubly resonant Neumann problems.

1 Introduction

Let Z [subset or equal to] [R.sup.N] be a bounded domain with a [C.sup.2]-boundary [partial derivative]Z. In this paper, we consider the following Neumann elliptic problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

Suppose f(z,x) = [[LAMBDA].sub.k]x + g(z,x) With [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] uniformly for a.e. z [member of] Z

and [[LAMBDA].sub.k] is an eigenvalue of the negative Neumann Laplacian. Then problem (1.1) is said to be resonant at infinity with respect to [[LAMBDA].sub.k]. If this happens for two successive distinct eigenvalues [[lambda].sub.k] < [[lambda].sub.k+1], then we say that the problem is "doubly resonant".

The goal of this paper is to prove multiplicity results under conditions of double resonance between two successive eigenvalues of the negative Neumann Laplacian. The doubly resonant situation was investigated in the past only in the context of the Dirichlet problem. In this direction, we mention the works of Berestycki-de Figueiredo [4] (who coined the term double resonance), Cac [6], Robinson [24], Su [25] and Zou [30]. To the best of our knowledge, there is no analogous study for the Neumann problem. Certain resonant Neumann problems, were studied by Iannacci-Nkashama [13], [14], Kuo [15], Mawhin-WardWillem [19], Rabinowitz [23]. Iannacci-Nkashama [14] and Kuo [15] used variants of the well-known Landesman-Lazer conditions (LL-conditions for short), which were first introduced in the pioneering "resonant" work of LandesmanLazer [16]. Iannacci-Nkashama [13] used a sign condition. Mawhin-Ward-Willem [19] used a monotonicity condition and finally Rabinowitz [23] employed a periodicity condition. With the exception of Iannacci-Nkashama [14], all the aforementioned Neuamnn works, treat problems resonant with respect to the principal eigenvalues [[lambda].sub.0] = 0 and none of them deals with the doubly resonant case. Moreover, all of them prove existence theorems, but do not address the question of existence of multiple nontrivial solutions. Multiplicity results for resonant Neumann problems, were obtained by Filippakis-Papageorgiou [9], Tang [27] and Tang-Wu [28]. However, their hypotheses do not allow for double resonance (neither at zero nor at infinity).

In this paper, we consider the case of double resonance at infinity, with respect to two successive eigenvalues of the negative Neumann Laplacian. Our approach combines variational techniques based on the critical point theory, together with Morse theory. We prove two multiplicity theorems.

The two multiplicity results are the following (for hypotheses [H.sub.1] (resp. [H.sub.2]), we infer to the beginning of Section 3 (resp. Section 4)).

Theorem 1.1. If hypotheses [H.sub.21] hold, then problem (1.1) has at least two nontrivial solutions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Theorem 1.2. If hypotheses [H.sub.2] hold, then problem (1.1) has at least three nontrivial solutions [x.sub.0], [v.sub.0] [u.sub.0] [member of ] [C.sup.1.sub.n] ([bar.Z]) with [x.sub.0] (z) > 0 > [v.sub.0] (z) for all z [member of] [bar.Z].

2 Mathematical background

We start by recalling some basic elements of critical point theory and of Morse theory, which we shall need in the sequel.

So, let X be a Banach space and [X.sup.*] its dual. By (*, *) we denote the duality brackets for the pair ([X.sup.*], X). Let [phi] [member of] [member of] [C.sup.1] (X). We say the cp satisfies the Cerami condition (the C-condition for short), if every sequence {[x.sub.n]}n [greater than or equal to] 1 [subset or equal to] X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is boundedin R and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

has a strongly convergent subsequence.

The next theorem is the well-known "mountain pass theorem" and gives a minimax characterization of certain critical values of a [C.sup.1]-functional.

Theorem 2.1. If X is a Banach space, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[phi] satisfies the C-condition and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a critical value of [phi].

Given [phi] [member of] [C.sup.1] (X) and c [member of] R, we use the following notation:

[[phi].sup.c] = {x [member of] X : [phi] (x) [less than or equal to] c} (the sub level set of [phi] at c),

K = {x [member of] X : [phi]'(x) = 0} (the critical set of [phi])

and [K.sub.c] = {x [member of] K : [phi] (x) = c} (the critical set of [phi] at the level c).

Suppose ([Y.sub.1], [Y.sub.2]) is a topological pair with [Y.sub.2] [sunset or equal to] [Y.sub.1] [subset or equal to] X. For every integer k [greater than or equal to] 0, by [H.sub.k] ([Y.sub.1], [Y.sub.2]) we denote the ftth-relative singular homology group of the pair ([Y.sub.i], [Y.sub.2]) with integer coefficients. The critical groups of [phi] at an isolated critical point x [member of] X with [phi](x) = c are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where U is a neighborhood of x such that K [intersection] [[phi].sup.c] [intersection] U = {x} (see Chang [8] and Mawhin-Willem [20]). The excision property of singular homology theory, implies that this definition of critical groups, is independent of the particular choice of the neighborhood U.

Now, suppose that [phi] satisfies the C-condition and -[infinity] < inf [phi] (K). Let c < inf [phi](K). Then the critical groups of [phi] at infinity, are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(see Bartsch-Li [3]). The deformation theorem, which is valid since by hypothesis [phi] satisfies the C -condition (see Bartolo-Benci-Fortunato [2]), implies that the above definition of critical groups at infinity, is independent of the particular level c < inf [phi](K) used.

If K is finite, then the Morse-type numbers of [phi] are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The Betti-type numbers of [phi], are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By Morse theory (see Bartsch-Li [3], Chang [8] and Mawhin-Willem [20]), the "Poincare-Hopf formula" holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

if all [M.sub.k], [[beta].sub.k] are finite and the two series converge.

Recall that if A and B are homotopy equivalent (in particular, if A and B are homeomorphic), then [H.sub.k] (X, A) = [H.sub.k](X, B) for all k [greater than or equal to] 0.

In the study of (1.1), we shall use the following two spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([parallel] * [parallel] denotes the usual norm of [H.sup.1] (Z)).

The space [C.sup.1.sub.n] ([bar.Z]) is an ordered Banach space, with order cone

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We know that this cone has a nonempty interior given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This space seems to be more natural for Neumann problems with homogeneous boundary conditions. However, the main reason for working with this new Sobolev space is Proposition 2.2 below. In general [H.sup.1.sub.n](Z) [not equal to] [pH.sup.1](Z).

Let [f.sub.0]: Z x R [right arrow] R be Caratheodory function (i.e., measurable in z [member of] Z and continuous in x [member of] R), with subcritical growth, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with [a.sub.0] [member of] [L.sup.[infinity]] (Z) +, [c.sub.0] > 0 and 1 < r < [2.sup.*] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and consider the (C.sup.1]-functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] R defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proposition 2.2. If [x.sub.0] [member of] [H.sup.1.sub.n] (Z) is a local [C.sup.1.sub.n] (Z)-minimizer of [[phi].sub.0], i.e., fZzere exists [r.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then [x.sub.0] [member of] [C.sub.1.sub.n] (Z) and it is also local [H.sup.1.sub.n] (Z)-minimizer of [[phi].sub.0], i.e., there exists [r.sub.1] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Remark 2.3. This result for Dirichlet spaces was first proved by Brezis-Nirenberg [5] for p = 2 and later generalized to all 1 < p < [infinity] (i.e., to the spaces [W.sup.1.p.sub.0] (Z)) by Garcia Azorero-Manfredi-Peral Alonso [10]. The corresponding result for Neumann spaces ============ (i.e., for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], was proved by Barletta-Papageorgiou [1] (for 2 < p < oo) and by Motreanu-Motreanu-Papageorgiou [21] (for 1 < p < [infinity]).

Let X = H be a Hilbert space, x [member of] H, U a neighborhood of x and [delta] [member of] [C.sup.2](U). If x [member of] H is [delta] critical point of cp, its "Morse index" is defined as the supremum of the dimensions of the vector subspaces of H on which [delta]p"{x) is negative definite.

Finally, let us recall some basic facts about the spectrum of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]). We shall do this in the more general context of weighted eigenvalue problems. So, let m [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (the weight function) and consider the following weighted linear eigenvalue problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

It is easy to see that [??] [greater than or equal to] 0 is a necessary condition for problem (2.2) to have a nontrivial solution. In fact [[lambda].sub.0] = [[lambda].sub.0] (m) = 0 is an eigenvalue of (2.2) with corresponding eigenspace 1R (the space of constant functions). Moreover, (2.2) has a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of distinct eigenvalues such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. If m = 1, then we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

For every integer k > 0, let E([??](m)) be the eigenspace corresponding to the eigenvalue [??](m) of (2.2). We know that E([??](m)) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (regularity theory) and it has the unique continuation property, namely, if u G E([??](m)) vanishes on a set of positive measure, then u(z) =0 for all z [member of] [bar].Z. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Then we have the following variational characterizations of the eigenvalues:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.4)

The minimum in (2.3) is attained on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = 1R. The maximum and the minimum in (2.4) are realized on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], k > 1. Then (2.3), (2.4) and the unique continuation property imply the following monotonicity property of the eigenvalues with respect to the weight function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Note that Ao(ra) = 0 is the only eigenvalue with eigenfunctions of constant sign. All other eigenvalues have nodal (i.e., sign changing) eigenfunctions.

Finally in what follows, for every x G 1R, we use the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3 Existence of two solutions

In this section we establish the existence of two nontrivial smooth solutions for problem (1.1) under double resonance conditions:

The hypotheses on the nonlinearity f(z, x), are the following:

[H.sub.1]: f : Z x M [right arrow] M is a function such that f(z, 0) = 0 a.e. on Z and

(i) for all x [membere of] R, z [right arrow] > f(z, x)[C.sup.1] is measurable;

(ii) for almost all z G Z, x - > x f(z,x) is C1;

(iii) for almost all z - Z and all i [member of] R, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z)+, c > 0 and 0 < p < 2/N-2 = [2.sup.*] - 2 if N [greater than or equal to] 3 and 0 < p < [infinity] if N = 1,2;

(iv) there exists an integer k [greater than or equal to] 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] uniformly for a.a. z [member of]G Z;

(v) suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[i] if [x.sub.n] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then there exist 70 > 0 and no > 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.];

[ii] if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] 1, then there exist [[gamma].sub.1] > 0 and [[eta].sub.1] > 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(vi) there exist [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Remark 3.1. Hypothesis [H.sub.1]{iv) implies that at [[+ or -][infinity], we have double resonance with respect to the successive eigenvalues [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Hypothesis [H.sub.1](v), is a generalization of the well-known LL-sufficiency conditions for the solvability of resonant problems, first introduced in the work of Landesman-Lazer [16]. Analogous conditions can be found in the study of resonant Dirichlet problems, see Landesman-Robinson-Rumbos [17], Robinson [24] and Su [25]. Also we note that in this case the growth hypothesis H\{iii) is stated in terms of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](z, *) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], because we want to corresponding eu;er functional of the problem to be [C.sup.2]. Indeed, hypotheses [H.sub.1](ii), (Hi) imply that the integral functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] F(z,u(z))dz is [C.sup.2]. Finally, we should point out that our hypotheses near the origin (see [H.sub.1]{vi)) are minimal and particular they do not necessarily dictate a linear growth there for f(z, *) as in [24], [25].

Example 3.2. Consider the following nonlinearity f(x) (for the sake of simplicity, we drop the z-dependence)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Suppose that near the origin, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and for [absolute valeu of x] large, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Then such a nonlinearity f(*) satisfies hypotheses [H.sub.1]. The generalized LL-condition (hypothesis [H.sub.1] (v)), can be verified using Lemma 2.1 ofSu-Tang [26].

We consider the Euler functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] IR for problem (1.1), defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hypotheses H\ imply that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. In the sequel, by (*, *) we denote the duality brackets for the pair ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since we are dealing with the Neumann problem, Poincare's inequality is not valid (hence [parallel]Du[parallel]p is not equivalent to the Sobolev norm) and this makes the verification of the C-condition more difficult. In fact the possibility of resonance at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] adds to the above difficulties.

Proposition 3.3. If hypotheses H\ hold, then <p satisfies the C-condition.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be a sequence such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

We shall show that the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is bounded. We argue indirectly. So, suppose that the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is unbounded. We may assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and so we may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is embedded compactly in [L.sup.2](Z)). By virtue of hypothesis H\(iii), (iv), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.2)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. From (3.2) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is bounded.

So, by passing to a suitable subsequence if necessary, we may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

For every [epsilon] > 0 and ft [greater than or equal to] 1, we introduce the sets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So, by virtue of hypothesis H\ (iv), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then the dominated convergence theorem implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

From the definition of the sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We pass to the limit as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], use (3.4) together with Mazur's lemma and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

Moreover, it is clear from (3.2) that

h(z) = 0 a.e. on y = 0}.

Combining (3.5), (3.6), (3.7), we infer that

h(z) = g(z)y(z) a.e. on Z, (3.8)

with g [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] a.e. on Z.

Let A [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] be defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Clearly A is monotone, hence maximal monotone. Also let N : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [L.sup.2](Z) be defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.9)

From (3.1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.10)

for all n [greater than or equal to] 1 (see (3.9)).

In (3.10), we choose v = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So, from (3.10) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.11)

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. So, from (3.11), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

On the other hand, we also have D[y.sub.n] Dy in [L.sup.2](Z,[R.sup.N]). Hence, from the Kadec-Klee property of Hilbert spaces, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore, [parallel]y[parallel] = 1. Passing to the limit as n ->* oo in (3.10), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (3.8)),

From this equation, using Green's identity, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12)

Standard regularity theory, implies that y [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We consider three distinct cases, depending on the position of g in the spectral interval [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]]. Case 1: g(z) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] a.e. on Z.

Then, from (3.12) we infer that y G [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]\(0}. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.13)

where [x.sub.n] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], n > 1. From (3.1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.14)

Note that in the last two implications we have used the orthogonality of the spaces E([[lambda].sub.k]) and [V.sub.k]. But then, because of (3.13), we see that (3.14) contradicts hypothesis [H.sub.1] (v) [i].

Case 2: g(z) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] a.e. on Z.

This case is treated similarly as Case 1, using this time hypothesis [H.sub.1] (v) [ii].

Case 3: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] a.e. on Z and the two inequalities are strict on sets (not necessarily the same) of positive measure.

From the monotone dependence of the eigenvalues [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] on the weight function g [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]+, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.15)

From (3.15), it follows that 1 is not an eigenvalue of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) and so in (3.12) we must have y = 0, a contradiction to the fact that [parallel]y[parallel] = 1.

So, in all three cases we have reached a contradiction. This means that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) is bounded. Therefore, we may assume that

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (3.16)).

From this as before, using the Kadec-Klee property of Hilbert spaces, we conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z). This proves that [delta] satisfies the C-condition.

Proposition 3.4. If hypotheses Hi hold, then the origin is a local minimizer of [psi].

Proof Let [[delta].sub.0] > 0 be as in hypothesis [H.sub.1] (vi) and consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then by virtue of hypothesis [H.sub.1] (vi), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.17)

So, for any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (3.17)).

Therefore the origin is a local [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]-minimizer of [delta]. Invoking Proposition 2.2, we conclude that the origin is a local [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]-minimizer of 95.

We may assume that x = 0 is an isolated critical point and local minimizer of cp, or otherwise we have a whole sequence of distinct nontrivial solutions of (1.1) and we are done. Then because of Proposition 3.4, we have (see Chang [8], p.33 and Mawhin-Willem [20], p.175).

Proposition 3.5. If hypotheses [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([[delta].sub.m] denotes the Kronecker function).

Since we have assumed without any loss of generality that x = 0 is an isolated critical point and local minimizer of cp, we can find p > 0 small, [rho] < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]/ such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.18)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proposition 3.6. If hypotheses [H.sub.1] hold, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], with [rho] > 0 as in (3.18).

Proof. We proceed by contradiction. So, suppose we can find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.19)

We may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.20)

Exploiting the compact embedding of [H.sup.1.sub.n](Z) into [L.sup.2](Z) and the sequential weak lower semicontinuity of the norm functional in a Banach space, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (3.19))

and [parallel]y[parallel] < p.

Because of (3.18), we must have y = 0. From the mean value theorem, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.21)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.22)

We may assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (3.19)) and 0 = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], from (3.21) and (3.22), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

a contradiction to the fact that [parallel][y.sub.n] [parallel] = p for all n [greater than or equal to] 1. ?

Now we are ready to produce the first nontrivial solution for problem (1.1).

Proposition 3.7. If hypotheses Hi hold, then problem (1.1) has a nontrivial solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. From Proposition 3.6, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.23)

Also, by hypothesis [H.sub.1] (vi), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.24)

From (3.23),(3.24) and Proposition 3.3, we see that we can apply Theorem 2.1 (the mountain pass theorem) and obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

an

So, [x.sub.0] [member of] [H.sup.1.sub.n](Z) is a nontrivial solution of problem (1.1) and the regularity theory implies [x.sub.0] [member of] [C.sup.1.sub.n]([bar].Z).

Proposition 3.8. If hypotheses [H.sub.1] hold and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], is the nontrivial solution of problem (1.1) obtained in Proposition 3.7, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],.

Proof. We shall show that we can apply Corollary 8.5, p.195 of Mawhin-Willem [20]. To this end, it suffices to check that, if the Morse index of xq is equal to 0, then its nullity is less than 2. So, we may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.25)

(i.e., the Morse index of xq is equal to 0). Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],, (3.26)

where [??](z) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

If [??]+ = 0, then clearly the only solution of (3.26) is u = 0 and so we are done. If [??]+ [empty set], then we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],. (3.27)

From (3.25), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],. (3.28)

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], 0, the from Proposition 2.2 of Godoy-Gossez-Paczka [12], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], = 0, which contradicts (3.28).

So, we must have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], < 0. Then according in Proposition 2.7 of GodayGossez-Paczka [12], we have dimker[delta]([x.sub.0]) [less than or equal to]< 1. Therefore, we can apply Corollary 8.5, p.195 of Mawhin-Willem [20] and infer that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for all m [greater than or equal to] 0.

To compute the critical groups at cp at infinity, we shall need the following slight modification of Lemma 2.4 of Perera-Schechter [22]. The new formulation is suitable for problems in which the Euler functional satisfies the C-condition.

Lemma 3.9. IfH is a Hilbert space, (t, x) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (x) is a function belonging in C( [0,1] x X) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],(x) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],(x) are both locally Lipschitz on H and there exists R > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],, (3.30)

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for all m [greater than or equal to] 0.

Proof. We choose [eta] < inf[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],] such that

If no such [eta] [member of] R can be found, then [C.sub.m]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],) = [C.sub.m]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], for all m [greater than or equal to] 0.

For definiteness, we assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (the argument is similar if we assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],). Take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and consider the following Cauchy problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (331)

Since by hypothesis both [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], are locally Lipschitz, then from the local existence theorem (see, for example, Gasinski-Papageorgiou [11], p.618), we know that (3.31) admits a local flow denoted b h(t) (or h(t, u) to emphasize the initial point u). If by (*, -)h we denote the inner product of h, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (see (3.31))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (3.32)

Therefore, the flow h(*) is global (i.e., on the whole [R.sub.+]) and we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], is homeomorphic to a subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (see (3.32)).

Similarly, if we consider the family ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), then an analogous argument gives us that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], is homeomorphic to a subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Therefore we conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is homotopic to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Using this lemma we can compute the critical groups of [delta] at infinity.

Proposition 3.10. If hypotheses [H.sub.1] hold, then [C.sub.m](cp, 00) = 5m^kZ for all m > 0, wz'f/z

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. We consider the following one-parameter family of functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Claim: We can find R > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We proceed by contradiction. So, suppose that the Claim is not true. Then we can find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.33)

for all v [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) with en [down arrow] 0.

We set [y.sub.n] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] 1. Then [parallel][y.sub.n] [parallel] = 1 for all n [greater than or equal to] 1 and so, we may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We multiply (3.33) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.34)

for all v [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z).

We choose v = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) and then pass to the limit as n [right arrow] [infinity] in (3.34). Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) is bounded (see the proof of Proposition 3.3), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Also, arguing as in the proof of Proposition 3.3, we can show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.35)

As in the proof of Proposition 3.3, we consider three distinct cases depending on the position of the weight function gt = tg + (1 - t)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. in the spectral interval [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Case 1: t = 1 and g = [[lambda].sub.k].

In this case (3.35) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Then arguing as in Case 1 in the proof of Proposition 3.3, we reach a contradiction to the generalized LL-condition (see hypothesis Hi(v) [i]). Case 2: t = 1 and g = [[lambda].sub.k+1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3.36]

This is treated as Case 1 using this time hypothesis [H.sub.1] (v) [ii]. Case 3: 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].) In this case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3-36)

Combining (3.35) and (3.36), we infer that y = 0, a contradiction to the fact that [parallel]y[parallel] = 1. This proves the Claim. Clearly, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]..

Therefore, we can apply Lemma 3.9 and obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].. (3.37)

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], x = 0 is the only critical point of [[delta].sub.0]. It is a nondegenerate critical point with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Morse index u = [d.sub.k], where [d.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].. So, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now we are ready for the first multiplicity theorem for doubly resonant semilinear Neumann problems.

Theorem 3.11. If hypotheses [H.sub.1] hold, then problem (1.1) has at least two nontrivial solutions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z).

Proof. From Proposition 3.7, we already have one nontrivial solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z).

Suppose that (0,[X.sub.0]} are the only critical points of [delta]. Then from Propositions 3.5, 3.8,3.10 and the Poincare-Hopf formula (see (2.1)), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

a contradiction. This means that there is one more critical point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) of [phi], distinct from 0 and [X.sub.0]. Then [y.sub.0] is a solution of (1.1) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the regularity theory.

4 Existence of three solutions

In this section we strengthen hypotheses [H.sub.1] and we produce three nontrivial solutions for problem (1.1).

The new hypotheses on the nonlinearity f(z, x) are the following:

[H.sub.2] f : Z x R [right arrow] R is a function such that f(z,0) = 0 a.e. on Z and hypotheses [H.sub.2](z), (ii), (Hi), (v), (vi) are the same as the corresponding hypotheses [H.sub.1](i),(ii),(iii),(v),(vi)

(iv) there exists an integer k [greater than or equal to] 1 such that [d.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is even and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.];

(vii) there exists [??] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Remark 4.1. In hypothesis [H.sub.2](iv) since [d.sub.k] is assumed to be even, k can not be zero. That is why we assume k[greater than or equal to]1.

We introduce the positive and negative truncations of f(z, *) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]ds and then for e G (0,1) we introduce the functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]R defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z)).

Proposition 4.2. If hypotheses [H.sub.2] hold, then the functionals [psi] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] satisfy the C-condition.

Proof. That [psi] satisfies the C-condition follows from Proposition 3.3.

Next we prove that <pe+ satisfies the C-condition. The proof for [[psi].sup.[epsilon][+ or -] is similar. So, we consider a sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

From (4.1), we easily see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then [parallel][y.sub.n][parallel] = 1 for all n [greater than or equal to] 1 and so we may assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Reasoning as in the proof of Proposition 3.3, we show that:

"If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. on Z and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Moreover, in the limit as n [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.2)

Since [d.sub.k] is even, k [greater than or equal to] > 1 and so from (4.2), we have that y is nodal, a contradiction to the fact that y [greater than or equal to] 0, y [empty set] 0. This proves that ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) is bounded, hence ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) is bounded. From this, as before, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] satisfies the C-condition.

Similarly for [[psi].sup.[psilon]].

Proposition 4.3. If hypotheses [H.sub.2] hold, then u = 0 is a local minimizer for the Junctionals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof Let [[delta].sub.0] > 0 be as in hypothesis H(vi). Then for every u [member of] [C.sup.1.sub.n](Z) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[F.sub.+](z,u(z)) [less than or equal to] < 0 a.e. on Z

(recall that [F.sub.+](z,x) = 0 for a.a. z [member of] Z, all x [less than or equal to] 0). Hence for u [member of] [C.sup.1.sub.n](Z) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see Proposition 2.2).

The proof for [[psi].sup.[epsilon]]-, is similar.

As before we may assume that u = 0 is an isolated critical point and local minimizer of [[psi].sup.[epsilon]][+ or -] (otherwise we have a whole sequence of distinct constant sign solutions). Then as in the proof of Proposition 3.6, we obtain:

Proposition 4.4. If hypotheses [H.sub.2] hold, then there exists [rho] > 0 small such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now we are ready for the second multiplicity result for problem (1.1).

Theorem 4.5. If hypotheses [H.sub.2] hold, then problem (1.1) has at least three nontrivial solutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. By virtue of hypothesis [H.sub.2] (iv), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since k [greater than or equal to] 1 (recall [d.sub.k] is even, see Remark), [[lambda].sub.k] > 0 and so for large x > 0, F(z, x) > 0. Hence, if [theta] > 0 is large (such that [parallel][theta][parallel] > [rho], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hence Propositions 4.2 and 4.4 permit the application of Theorem 2.1. So, we obtain [x.sub.0] [member of] [H.sup.1.sub.n](Z) a critical point of [[psi].sup.[epsilon].sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Also, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

We act with the test function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) and obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So, (4.3) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

using Green's identity. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.](Z) is a solution of problem (1.1). Moreover, regularity theory implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then, by virtue of hypothesis [H.sub.2](vii), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (see (4.4))

using the strong maximum principle (see Vazquez [29]). As in the proof of Proposition 3.8, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.5)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and using a result of Liu-Wu [18] (see also Chang [7]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.6)

Similarly, working this time with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain [v.sub.0] [member if] -int[C.sub.+] a solution of (1.1) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.7)

Finally, from Propositions 3.5 and 3.10, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4.9)

Suppose ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]} are the only critical points of cp. Then from (4.6), (4.7), (4.8), (4.9) and the Poincare-Hopf formula (see (2.1)), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.11)

a contradiction, since by hypothesis [d.sub.k] is even. Therefore, there must be a third nontrivial critical point [U.sub.0] of [psi], distinct from ([x.sub.0],[v.sub.0]}. This is a solution of (1.1) and regularity theory implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Remark 4.6. If N = 1 (i.e., ordinary differential equation problem), then [d.sub.k] =even means that k [greater than or equal to] 0 is odd. Recall that in this case dim [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = lfor all k [greater than or equal to] 0 and so dim[bar].H.sub.k] = k + 1.

AKCNOWLEDGEMENT: The authors wish to thank the referee for pointing out a mistake in the proof of Theorem 4.5 and for other correction and constructive remarks.

Received by the editors July 2009--In revised version in December 2009.

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Communicated by P. Godin.

Michael E. Filippakis * Nikolaos S. Papageorgiou

Department of Mathematics,

Hellenic Military Academy

Vari 16673 Athens, Greece,

email:mfilip@math.ntua.gr

Department of Mathematics,

National Technical University,

Zografou Campus, Athens 15780, Greece

email:npapg@math.ntua.gr

* Researcher supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)
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