Infinitely many solutions for elliptic equations with non-symmetric nonlinearities

We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.


Introduction
Let us consider the problem − u = |u| p−1 u + w in , where is a smooth bounded domain of R n , with n ≥ 1, w ∈ L 2 ( ), p > 1 and p < n+2 n−2 when n ≥ 3. If w ≡ 0 in , the corresponding energy functional E : H 1 0 ( ) → R, defined by is not even, so the equivariant Lusternik-Schnirelmann theory for Z 2 -symmetric sets cannot be applied to find infinitely many solutions as in the case w ≡ 0 (see for instance [1,3,18,19,27,28,32,34] and also [9,17] for a more general framework).
In the case w ≡ 0 in , a natural question (which goes back to the beginning of the eighties) is wether the infinite number of solutions still persists under perturbation. A detailed analysis was originally carried on in [2, 3, 5-8, 25, 29, 30, 33, 35, 39] by Ambrosetti, Bahri, Berestycki, Ekeland, Ghoussoub, Krasnoselskii, Lions, Marino, Prodi, Rabinowitz, Struwe and Tanaka by introducing new perturbation methods. In particular, this question was raised to the attention by Rabinowitz also in his monograph on minimax methods (see [34,Remark 10.58]). In [4] Bahri proved that, if n ≥ 3 and 1 < p < n+2 n−2 , then there exists an open dense set of w in L 2 ( ) such that problem (1.1) admits infinitely many solutions. In [8] Bahri and Lions proved that, if n ≥ 3 and 1 < p < n n−2 , then problem (1.1) admits infinitely many solutions for every w ∈ L 2 ( ).
These results suggest the following conjecture, proposed by Bahri and Lions in [8]: the multiplicity result obtained in [8] holds also under the more general assumption 1 < p < n+2 n−2 . More recently, a new approach to tackle the break of symmetry in elliptic problems has been developed by Bolle, Chambers, Ghoussoub and Tehrani (see [10,11,15], which include also applications to more general nonlinear problems). However that approach did not allow to solve the Bahri-Lions conjecture.
In the present paper we describe a new possible method to approach this problem. The idea is to trying to piece together solutions of Dirichlet problems in subdomains of chosen in a suitable way. This idea has been first used by Struwe in earlier papers (see [35][36][37] and references therein). In the present paper we consider as nodal regions subdomains of that are deformations of cubes by suitable Lipschitz maps (we say that nodal solutions of this type have a check structure).
It is interesting to observe that such a class of Lipshitz maps also appeared in some recent works by Rabinowitz and Byeon (see [13,14] and the references therein) concerning a rather different problem: construct solutions having certain prescribed patterns for an Allen-Cahn model equation. Also in that papers, as in the present one, Lipschitz condition is combined with the structure of Z n and the covering of R n by cubes with vertices in Z n .
The main multiplicity result of this paper is stated in Theorem 2.7 and says that if satisfies a suitable geometric condition (condition (2.53)) then problem (1.1) admits infinitely many nodal solutions having as nodal structure suitable partitions of in subdomains that are Lipschitz deformations of arbitrarily small cubes.
In order to explain the meaning of the geometrical condition (2.53), which plays a crucial role in this paper, we need some more detail on the method we use in the proof. Let P k denotes the union of all the cubes with sides of length 1 k and vertices in 1 k Z n , that are enclosed in and, for L > 1, consider the class C L (P k , ) consisting of the bilipschitz maps between P k and with Lipshitz constants in 1 L , L . For all T in C L (P k , ) and k in N, we construct by a mini-max argument a nodal function u T k in H 1 0 ( ), whose nodal regions are the deformations of these cubes by the map T . For k large enough, u T k satisfies equation (1.1) in each nodal region (by Proposition 2.4) and it is solution of the Dirichlet problem (1.1) when, in addition, it satisfies the assumptions of Proposition 2.5 (a kind of stationarity property). Then, for all k, we minimize the energy functional E in the set {u T k : T ∈ C L (P k , )} and show that, if the minimum is achieved by a bilipschitz map T L k with Lipschitz constants in the interior of 1 L , L , then the corresponding function u T L k k satisfies the stationarity condition of Proposition 2.5, so it is a solution of Problem (1.1) for k large enough.
Condition (2.53) requires just that, for a suitable choice of L > 1, there existsk in N and L in ]1, L[ such that the minimum is achieved by a map T L k in CL (T k , ) for all positive integer k ≥k.
Thus, if satisfies condition (2.53), we obtain infinitely many nodal solutions u T L k k having check structure. Moreover, the number of nodal regions of u T L k k and its energy E(u T L k k ) tend to infinity as k → ∞, while the size of the nodal regions tends to zero. Lemma 2.9 and Corollary 2.10 show that condition (2.53) holds true, for example, when n = 1 (the proof may be also adapted to deal with radial solutions in domains having radial symmetry).
Indeed, in dimension n = 1, a more general result was obtained by Ehrmann in [23] (see also [24,26] for related results). Here it is proved that the ordinary differential equation has infinitely many distinct solutions when f is a function with superlinear growth satisfying quite general assumptions. However, the method here used relies on a shooting argument, typical of ordinary differential equations, combined with counting the oscillations of the solutions in the interval (0, 1). Therefore, this method, which gives the existence of solutions having a sufficiently large number of zeroes in dimension n = 1, cannot be extended to higher dimensions. On the contrary, in the present paper we use a method which is more similar to the one introduced by Nehari in [31], that can be in a natural way extended to the case n > 1. In fact, for example, Nehari's work was followed up by Coffman who studied an analogous problem for partial differential equations (see [18,19]). Independently, this problem was also studied by Hempel (see [27,28]).
More recently, the method introduced by Nehari has been also used by Conti, Terracini and Verzini to study optimal partition problems in n-dimensional domains and related problems: in particular, existence of minimal partitions and extremality conditions, behaviour of competing species systems with large interactions, existence of changing sign solutions for superlinear elliptic equations, etc. (see [20][21][22]40]).
Notice that Nehari's work deals with an odd differential operator, so the corresponding energy functional is even. Moreover, Nehari proves that for every positive integer k there exists a solution having exactly k zeroes. On the contrary, in the present paper (as Ehrmann in [23]) we find only solutions with a large number of zeroes; moreover, we prove that, for all w in L 2 ( ), the zeroes tend to be uniformly distributed in all of the domain as their number tends to infinity (see Lemmas 2.9 and 3.2). The reason is that, when w ≡ 0, the Nehari type argument we use in the proof works only when the sizes of all the nodal regions are small enough, so their number is sufficiently large.
In order to show that our existence result is sharp, we prove also that the term w in problem (1.1) can be chosen in such a way that the problem does not have solutions with a small number of nodal regions. More precisely, in the case n = 1 we show that for all positive integer h there exists w h in L 2 ( ) such that every solution of problem (1.1) with w = w h has at least h zeroes (see Corollary 3.6). Indeed, we show that for all n ≥ 1 and for every eigenfunction e k of the Laplace operator − in H 1 0 ( ) there existsw k in L 2 ( ) such that every solution u of problem (1.1) with w =w k must have the sign related to the sign of e k in the sense that every nodal region of e k has a subset where u and e k have the same sign (see Proposition 3.5).
In the case n > 1, condition (2.53) seems to be more difficult to be verified because the class C L (P k , ) of the admissible deformations of the nodal structure might result too large. In fact, as we point out in Remark 3.1, the minimality of the map T L k implies that as k → ∞ the nodal regions of u T L k k tend to have all the same size. In the case n = 1, this property is sufficient to control the Lipschitz constant of T L k , for k large enough, in order to prove condition (2.53). On the contrary, if n > 1 the Lipschitz constant of T L k might be very large, even if the nodal regions of u T L k k tend as k → ∞ to have all the same size, because their shape might be very different from the cubes of R n . Therefore, a natural idea is to restrict the class of admissible deformations C L (P k , ), taking also into account that our method to construct nodal solutions with many small nodal regions having check structure can be easily adapted to deal with other classes of bilipschitz maps, different from C L (P k , ).
For example, we can fix a bilipschitz map T 0 : → and consider as a class of admissible deformations a suitable neighbourhood of T 0 in C L (P k , ), that is the class of all the maps in C L (P k , ) that are close to T 0 in a suitable sense (see Remark 3.1). Then, the geometrical condition (2.53) has to be replaced by a similar condition that holds or fails depending on the choice of T 0 and of its neighborhood (condition (3.7)) whose meaning is again that, for k large enough, the minimization problem settled in this new class of admissible deformations is achieved by a map which is, in some sense, in the interior of this new class. Thus, if n > 1, the problem is to choose carefully a suitable class of admissible deformations.
In a similar way, for example, we can prove that if is a cube of R n with n > 1, p > 1, p < n+2 n−2 if n > 2, for all w in L 2 ( ) there exist infinitely many solutions u k (x) of problem (1.1) such that the nodal regions of the function u k x k , after translations, tend to the cube as k → ∞ (the proof will be reported in a paper in preparation).
Notice that, in particular, this result shows that the Bahri-Lions conjecture is true when is a cube of R n with n ≥ 3.
Let us point out that our method does not require techniques of deformation from the symmetry and may be applied to more general problems: for example, when the nonlinear term |u| p−1 u is replaced by c + (u + ) p − c − (u − ) p with c + and c − two positive constants (see Lemma 3.2), in case of different, nonhomogeneous boundary conditions and even in case of nonlinear elliptic equations involving critical Sobolev exponents.

Existence of infinitely many nodal solutions
In order to find infinitely many solutions with an arbitrarily large number of nodal regions, we proceed as follows. Let us set Notice that there exists k in N such that Z k = ∅ ∀k ≥ k . For all subsets P, Q of R n and for all L ≥ 1, let us denote by C L (P, Q) the set of all the functions T : P → Q such that For all k ≥ k , z ∈ Z k , L ≥ 1, T ∈ C L (P k , ) let us set Since p < n+2 n−2 when n ≥ 3, one can easily verify that the infimum in (2.4) is achieved. Moreover, for all L ≥ 1 and k ≥ k , also the infimum is achieved (as one can prove by standard arguments using Ascoli-Arzelà Theorem) and the following lemma holds. (2.7) We say that In fact, arguing by contradiction, assume that lim inf It follows that (up to a subsequence) (ū k ) k is bounded in H 1 0 ( ) and there exists a functionū ∈ H 1 0 ( ) such thatū k →ū, as k → ∞, weakly in H 1 0 ( ), in L p+1 ( ), and almost everywhere in (hereū k is extended by the value 0 in \ T k ( 1 k C z k )). Since meas T k ( 1 k C z k ) → 0 as k → ∞, from the almost everywhere convergence we obtainū ≡ 0 in , which is a contradiction becauseū k →ū in L p+1 ( ) and (2.7) holds for all k ≥ k . Thus (2.8) is proved.
Notice that (2.11) As a consequence, for all k ≥ k we obtain (2.12) and, as k → ∞, which completes the proof.
is achieved by a unique minimizing functionũ T k,z . Moreover, we have On the other hand, n−2 when n ≥ 3, one can prove by standard arguments that (up to a subsequence) it converges to a functionũ T k,z ∈ H 1 0 T 1 k C z such that In order to prove (2.15) we argue by contradiction and assume that lim sup Then, for all k ≥ k(L) there exist z k ∈ Z k and T k ∈ C L (P k , ) such that (up to a subsequence) Since E(ũ T k k,z k ) ≤ 0 and the sequenceũ T k k,z k (extended by the value zero outside 1 k C z k ) is bounded in L p+1 ( ), we infer that it is bounded also in H 1 0 ( ). We say that, as a consequence, . Thus, we can conclude that (2.15) holds. Finally, notice thatũ T k,z is the unique minimizing function for (2.14) because the functional E is strictly convex in a suitable neighborhood of zero. So the proof is complete.
Taking into account Corollary 2.2, for all k ≥ k(L), z ∈ Z k and T ∈ C L (P k , ) we can consider a minimizing functionũ T k,z for the minimum (2.14). Moreover, since p > 1, for all u ∈ H 1 0 1 k C z there exists the maximum Proof Let us consider a minimizing sequence (u i ) i∈N for the minimum (2.23). Whitout any loss of generality, we can assume that It follows that this sequence is bounded in H 1 0 T 1 k C z . Therefore, since p < n+2 n−2 when n ≥ 3, up to a subsequence it converges weakly in H 1 0 , in L p+1 and a.e. to a function Notice that the L p+1 convergence and (2.24) imply We say that, indeed, the convergence is strong in H 1 0 T 1 k C z . In fact, arguing by contradiction, assume that (up to a subsequence) Therefore, we can conclude that u i →û in Proof It is clear that the functionũ T k,z (local minimum of the functional E) is a solution of the Dirichlet problem (2.28). In order to prove that, for k large enough, also u T k,z is solution of the same problem, let us consider the function G : Let us assume, for example, σ (z) = 1 (in a similar way one can argue when σ (z) = −1). One can verify by direct computation that for all u ≡ũ T k,z there exists a unique t u > 0 such that Taking into account thatũ T k,z is solution of problem (2.28), we obtain by direct computation Notice that and, by (2.15), Moreover, we have where, for all k ∈ N, As a consequence, since p < n+2 n−2 when n ≥ 3, there existsψ in H 1 0 ( ) such that (up to a subsequence) ψ i →ψ as i → ∞ weakly in H 1 0 ( ), in L p+1 ( ) and a.e. in . Moreover, since lim i→∞ meas T i 1 k i C z i = 0, the a.e. convergence implies thatψ ≡ 0 in , which is in contradiction with the convergence in L p+1 ( ) because |ψ| p+1 dx = 1 ∀i ∈ N.
Thus, we can conclude that lim k→∞ λ k = ∞. It follows that, for k large enough, As a consequence, if we denote by the set defined by Therefore, there exists a Lagrange multiplier μ ∈ R such that On the other hand, since u T k,z −ũ T k,z ≥ 0 in T 1 k C z , we have so u T k,z is a solution of problem (2.28).
When the function u T k = z∈Z k u T k,z satisfies a suitable stationarity property, then it is solution of problem (1.1) (here the function u T k,z is extended by the value zero outside T 1 k C z ). In fact, the following proposition holds.

Proposition 2.5
Assume that k ≥ k 1 (L) and T ∈ C L (P k , ). Moreover, assume that the function u T k = z∈Z k u T k,z satisfies the following condition: where ν k,z denotes the outward normal on ∂ T 1 k C z . Thus, in order to obtain E (u T k ) [ϕ] = 0, we have to prove that if z 1 , z 2 ∈ Z k and |z 1 − z 2 | = 1 (that is T 1 k C z 1 and T 1 k C z 2 are adjacent subdomains of ) then Taking into account that u T k,z satisfies problem (2.28) for all z ∈ Z k , for all vector field In order to obtain a function u T k which is stationary in the sense of Proposition 2.5, we can, for example, minimize E(u T k ) with respect to T for k large enough. First notice that, since is a smooth bounded domain, there exist k ≥ k and L ≥ 1 such that, for all k ≥ k and L ≥ L , we have (2.49) Moreover, using Ascoli-Arzelà Theorem, one can show the following lemma.
For all L ≥ 1 and T ∈ C L (P k , ), let us set (2.51) Using again Ascoli-Arzelà Theorem, we infer that, for all L ≥ L and k ≥ k , there exists (2.52) Notice that T L k depends only on the geometrical properties of the subdomains T L k 1 k C z with z ∈ Z k . A large L(T L k ) means that there are large differences in the sizes and in the shape of these subdomains. We can now state the following multiplicity result.
Thus, taking into account Proposition 2.5 we have to prove that E (u k )[v · Du k ] = 0 for all vector field v ∈ C 1 ( , R n ) such that v · ν = 0 on ∂ . Therefore, for all vector field v ∈ C 1 ( , R n ) such that v · ν = 0 on ∂ and for all τ ∈ R, let us consider the function D τ : → defined by the Cauchy problem (2.60) Thus, we have to prove that For the proof, we argue by contradiction and assume that (2.61) does not hold. For example, we assume that . As a consequence, there exists a sequence of positive numbers From Corollary 2.2 we infer that, if we choosek large enough, for all k ≥k, z ∈ Z k and i ∈ N there exists a unique minimizing functionũ

As in the proof of Proposition 2.4, let us consider the functions
for i large enough. In fact, arguing by contradiction, assume that (up to a subsequence still denoted by (τ i ) i∈N ) the inequality (2.66) does not hold. Then, for all i ∈ N and z ∈ Z k , there exists t z,i ≥ 0 such that It follows that lim i→∞ t z,i = 1 ∀z ∈ Z k and in contradiction with (2.62). Thus, (2.66) holds. Notice that, ifk is chosen large enough, and tends to −∞ as t → ∞. As a consequence, there exists t i k,z ∈ R such that Therefore, from (2.66) and (2.71) we obtain for i large enough, in contradiction with (2.59). Thus, we can conclude that d Let us point out that, if n = 1, condition (2.53) in Theorem 2.7 is satisfied. In fact, it is a consequence of the following lemma (see also Remark 3.1 concerning the case n > 1).

Lemma 2.9
Assume n = 1, p > 1 and w ∈ L 2 ( ). Then, for all L > 1 there existsk(L) ∈ N such that , which implies (2.74) as one can easily verify. Also, notice that in this case we have Taking into account that for all w ∈ L 2 ( ) we obtain lim sup and, as a consequence, Then, taking into account that and that lim i→∞ J (k i ) As one can easily verify, for all i ∈ N, there exists a function T i ∈ C L (P k i , ) such that Taking into account that E(ū On the other hand, which is a contradiction because When the case (2.85) occurs, we argue in analogous way. In this case, for all i ∈ N we choosẽ z in Z k i such that Moreover, we can consider a function T i ∈ C L (P k i , ) satisfying all the properties of T i with z i andz i instead of z i andz i . Then, we can repeat for T i , z i andz i the same arguments as before. In particular, the property (analogous to (2.94)) now follows from (2.85) and (2.98). Thus, also in this case we obtain again a contradiction with the minimality property of . So the proof is complete.
As a direct consequence of Theorem 2.7, Proposition 2.8 and Lemma 2.9 we obtain the following corollary.

Final remarks
Notice that the method we used in Sect. 2 to find infinitely many solutions of problem (1.1) with a large number of nodal regions having a prescribed structure (a check structure) may be used also in other elliptic problems as we show in this section. It is clear that in this method condition (2.53) plays a crucial role. In Sect. 2 this condition is proved only in the case n = 1. In next remark, we discuss about the case n > 1. As a consequence, we can construct a sequence (k i ) i∈N such that Notice that L(T L i k i ) is large, for example, when there are large differences in the sizes or in the shapes of the subdomains T L i k i 1 k i C z with z ∈ Z k i . For k i large enough, too large differences seem to be incompatible with the minimality property This fact explains why condition (2.53) holds in the case n = 1. In the case n > 1, on the contrary, even if the subdomains T L i k i 1 k i C z with z ∈ Z k i have all the same shape and the same size, we cannot exclude that L T L i k i is large as a consequence of the fact that the shape of these subdomains is very different from the cubes of R n . This explains why it is difficult to prove that condition (2.53) holds also for n > 1.
Therefore, in the case n > 1, the natural idea is to restrict the class of the admissible deformations of the nodal regions.
For example, we can fixL ≥ L , T 0 ∈ CL ( , ), r > 0 and consider the set of deformations (analogous to condition (2.53)) is satisfied. It is clear that condition (3.7) holds or fails depending on the choice ofL, T 0 and r that have to be chosen in a suitable way. For example, in the case n = 1, if we choose T 0 (x) = x ∀x ∈ , (3.7) holds for allL > 1 and r > 0 as follows from Lemma 2.9.
In the case n > 1, condition (3.7) seems to have more chances than condition (2.53) to be satisfied. In fact, as we show in a paper in preparation, a variant of this method works for example when is a cube of R n with n > 1, p > 1, p < n+2 n−2 if n > 2 and, for all w ∈ L 2 ( ), allows us to find infinitely many solutions u k (x) such that the nodal regions of u k x k , after translations, tend to the cube as k → ∞. Therefore, it seems quite natural to expect that, by a suitable choice ofL, T 0 and r , for every bounded domain in R n with n > 1 and for all w ∈ L 2 ( ) one can find infinitely many nodal solutions of problem (1.1) with p > 1 and p < n+2 n−2 if n > 2.
Notice that this method to construct solutions with nodal regions having this check structure works for more general nonlinearities, even when they are not perturbations of symmetric nonlinearities: for example when in problem (1.1) the term |u| p−1 u + w is replaced by In fact, this method does not require any technique of deformation from the symmetry. For example, let us show how Lemma 2.9 has to be modified in this case. In this case the energy functional is We denote by F 0 the functional F when w = 0. Now, consider the numberL ≥ 1 defined byL = 1 min{t,2−t} wheret ∈]0, 2[ is the unique number such that Notice thatt = 1 (and soL = 1) if and only if c + = c − . Then, we have the following lemma which extends Lemma 2.9. cannot happen. In fact, for all i ∈ N we can chooseẑ i andẑ i + 1 in Z k i such that (up to a subsequence) Taking into account the minimality property As a consequence, we obtain In order to prove that lim i→∞ L(T L k i ) =L, arguing by contradiction, assume that lim i→∞ L(T L k i ) >L. As a consequence, since there exists a sequence (z i ) i∈N such that z i ∈ Z k i ∀i ∈ N and 18) or there exists a sequence (z i ) i∈N such that z i ∈ Z k i ∀i ∈ N and Assume, for example, thatt ≤ 1 (otherwise we argue in a similar way but witht replaced by 2 −t). Then,L = 1 t and, ift = 1, Lemma 2.9 applies. Thus, it remains to consider the casê t ∈]0, 1[. Consider first the case where (3.18) holds. Notice that there exists a sequence (ζ i ) i∈N such that ζ i ∈ Z k i and |z i − ζ i | = 1 ∀i ∈ N. Then, the minimality property (3.13) implies and, arguing as in the proof of Lemma 2.9 but with meas T L As a consequence of (3.20) and (3.21), we obtain which is a contradiction because with 1 t > 2 −t fort ∈]0, 1[. Thus, we can conclude that the case (3.18) cannot happen. In a similar way we argue in order to obtain a contradiction in the case (3.19). In fact, assume that (3.19) holds. Notice that there exists a sequence (ζ i ) i∈N such that ζ i ∈ Z k i and |z i − ζ i | = 1 ∀i ∈ N. As before, the minimality property (3.13) implies that and lim i→∞ k i meas T L As a consequence, we infer that Thus, we can conclude that lim i→∞ L(T L k i ) =L so the proof is complete.

Remark 3.3
The results we present in this paper concern the existence of solutions with a large number of nodal regions. In particular, when ⊂ R n with n = 1, these solutions must have, as a consequence, a large number of zeroes. In next propositions we show that the term w can be chosen in such a way that the sign of the solutions is related to the nodal regions of the eigenfunctions of the Laplace operator − in H 1 0 ( ). In particular, if n = 1 we show that for suitable terms w in L 2 ( ), problem (1.1) does not have solutions with a small number of zeroes: more precisely, we show that for all positive integer h there exists w h ∈ L 2 ( ) such that every solution of problem (1.1) has at least h zeroes (it follows from Corollary 3.6). (3.28) Let u ∈ H 1 (D) be a weak solution of the equation then sup D u > 0.
Proof Let e 1 be a positive eigenfunction corresponding to the eigenvalue λ 1 , that is e 1 + λ 1 e 1 = 0, e 1 > 0 in D, e 1 ∈ H 1 0 (D). (3.31) Arguing by contradiction, assume that (3.28) holds and u ≥ 0 in D. Then, from (3.29) we infer that where ν denotes the outward normal on ∂ D, so that and g(x, t) ≥ λ 1 t + c ∀x ∈ D, ∀t ∈ R for a suitable constant c > 0. It follows that which implies c D e 1 dx ≤ 0, that is a contradiction. Thus, the function u cannot be a.e. nonnegative in D.
In a similar way one can show that we cannot have u ≤ 0 a.e. in D when (3.30) holds, so the proof is complete.
In particular, Lemma 3.4 may be used to obtain informations on the effect of the term w on the sign changes of the solutions of problem (1.1), as we describe in the following proposition.
Proposition 3.5 Let ⊂ R n with n ≥ 1 and e k ∈ H 1 0 ( ) be an eigenfunction of the Laplace operator − with eigenvalue λ k , that is e k + λ k e k = 0 in . Assume that w ∈ L 2 ( ) satisfies Proof Notice that λ k is the first eigenvalue of the Laplace operator − in H 1 0 ( k ) and |e k | is a corresponding positive eigenfunction. Moreover, if we set g(x, t) = |t| p−1 t + w(x), we infer from (3.36) that, if w(x) > 0, g(x, t) ≥ λ k t +c ∀t ≥ 0 (3.39) and, if w(x) < 0, g(x, t) ≤ λ k t −c ∀t ≤ 0 (3.40) wherec = inf |w| − max{λ k t − t p : t ≥ 0} > 0. (3.41) Since ue k ≥ 0 and we k ≥ 0 in k and e k has constant sign in k , we have u ≥ 0 and w > 0 in k if e k > 0 in k and u ≤ 0, w < 0 in k in the opposite case. Thus, our assertion follows from Lemma 3.4. In fact, for example, in the case e k > 0 in k we cannot have − u = |u| p−1 u + w in k (3.42) otherwise inf k u < 0, because of Lemma 3.4, while u ≥ 0 in k . In the opposite case, when e k < 0 in k , one can argue in a similar way, so the proof is complete. Assume, for example, that e k > 0 on I 1 (in a similar way one can argue if e k < 0 in I 1 ). Then, from Proposition 3.5 we infer that for every solution u of problem (1.1) we have inf I i u < 0 for i odd and sup I i u > 0 for i even.

Remark 3.7
Notice that all the assertions in Proposition 3.5 and Corollary 3.6 still hold when the nonlinear term |u| p−1 u is replaced by c + (u + ) p − c − (u − ) p where c + and c − are two positive constants. In this case we have only to replace max{λ k t − t p : t ≥ 0} by max{λ k t −ct p : t ≥ 0}, wherec = min{c + , c − } > 0.
Notice that this method to construct solutions with nodal regions having a check structure may be used for nonlinear elliptic problems with different boundary conditions, for systems and also when the nonlinear term has critical growth. For example, for all λ ∈ R consider the Dirichlet problem − u = |u|  Using this method, if the functional F satisfies condition (2.53), one can prove that for n ≥ 4 and λ > 0 the functional F has an unbounded sequence of critical levels. More precisely, the following theorem can be proved.
Theorem 3.8 Let n ≥ 4, λ > 0, w ∈ L 2 ( ) and assume that condition (2.53) holds for the functional F . Then, there existsk ≥ k such that for all k ≥k there exists TL k ∈ CL (P k , ) and a solution u k of problem (3.44) such that TL k (P k ) = and, if for all z ∈ Z k we set u z k (x) = u k (x) when x ∈ TL k Let us point out that Theorem 3.8 gives a new result also when w ≡ 0 in . In fact, in this case the functional F is even but well known results (see [12,16,38]) guarantee only the existence of a finite number of solutions (because some compactness conditions hold only at suitable levels of F ). On the contrary our method, combined with some estimates as in [12] and in [16], allows us to construct infinitely many solutions with many nodal regions and arbitrarily large energy level.