Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type

The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x, u)\vert\nabla u\vert^{p_1} = G_u(x, u, v)&\hbox{ in $\Omega$,}\\[5pt] - {\rm div} (B(x, v)\vert\nabla v\vert^{p_2 -2} \nabla v) +\frac{1}{p_2}B_v(x, v)\vert\nabla v\vert^{p_2} = G_v\left(x, u, v\right)&\hbox{ in $\Omega$,}\\[5pt] u = v = 0&\hbox{ on $\partial\Omega$,} \end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $p_1$, $p_2>1$ and $A(x,u)$, $B(x,v)$ are $\mathcal{C}^1$-Carath\'eodory functions on $\Omega \times \mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$, $G_v(x,u,v)$ are given Carath\'eodory maps defined on $\Omega \times \mathbb{R}\times \mathbb{R}$ which are partial derivatives of a function $G(x,u,v)$. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional $\cal{J}$, related to problem $(P)$, admits at least one critical point in the ''right'' Banach space $X$. Moreover, if $\cal{J}$ is even, then $(P)$ has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ''good'' decomposition of the Banach space $X$ and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.


Introduction
In this paper we investigate the existence of one or more solutions for the coupled quasilinear elliptic system with homogeneous Dirichlet boundary conditions u, v) in Ω, where Ω is an open bounded domain in R N , N ≥ 2, p 1 , p 2 > 1, functions A, B : Ω × R → R admit partial derivatives for a.e. x ∈ Ω, for all u ∈ R, v ∈ R, (1.2) and a function G : Ω × R × R → R exists such that ∇G(x, u, v) = (G u (x, u, v), G v (x, u, v)), i.e.
On the other hand, a generalization of system (1.5) has been studied in [9] by using a cohomological local splitting while quasilinear elliptic systems similar to (1.1) are studied in [6] by means of an approximation approach and in [4,27] via nonsmooth techniques.
Here, we want to extend the existence result in [7, Theorem 3] to our quasilinear problem (1.1), but with stronger subcritical growth conditions which allow us to find bounded solutions. Moreover, we want also to state a multiplicity result in hypothesis of symmetry.
To this aim, following the approach developed in [10,11], we state enough conditions for recognizing the variational structure of problem (1.1) (see Proposition 3.7). Thus, investigating for solutions of (1.1) reduces to looking for critical points of the nonlinear functional in the Banach space X = X 1 × X 2 , where X i = W 1,pi 0 (Ω) ∩ L ∞ (Ω), for i = 1, 2. Unluckily, in our setting classical abstract theorems such as the statements in [2,5] cannot be applied, since functional J does not satisfy neither the Palais-Smale condition nor its Cerami's variant; in fact, already in the simpler case (1.4) a Palais-Smale sequence converging in the W 1,p 0 (Ω) norm but unbounded in L ∞ (Ω) can be found (see, for example, [13,Example 4.3]). Therefore, by exploiting the interaction between two different norms on X, we introduce the weak Cerami-Palais-Smale condition (see Definition 2.1) and apply the abstract theorems in [12] in order to prove the existence of at least one solution (see Theorem 5.1) or of infinitely many distinct ones if J is even (see Theorem 5.2 and Corollary 5.4). The multiplicity result is based on a "good" decomposition of the Sobolev spaces W 1,pi 0 (Ω) as given in [11,Section 5]. In this paper we assume that the coefficients A(x, u) and B(x, v) are C 1 -Carathéodory, bounded on bounded sets, greater of a strictly positive constant and interact properly with their partial derivatives (interactions trivially satisfied by A * (x) and B * (x) in problem (1.5)), while G(x, u, v) is a C 1 -Carathéodory function which has a suitable subcritical growth and satisfies an Ambrosetti-Rabinowitz type condition. Moreover, in order to state an existence result (see Theorem 5.1), we assume a condition on the behavior of G(x, u, v) at (0, 0) which imposes a growth of G(x, u, v) below a constant of the first straight lines associated to the −∆ p1 and −∆ p2 so that functional J is greater than a positive constant (and thus greater than J (0, 0)) in a sphere of small radius as required for applying the Mountain Pass Theorem (see Theorem 2.3). On the other hand, a multiplicity result can be stated (see Theorem 5.2) by using a "good" version of the Symmetric Mountain Pass Theorem (see Theorem 2.4) once J is even and G(x, u, v) grows fast enough at infinity. Anyway, in order to not weigh this introduction down with too many details, we prefer to specify each hypothesis when required and to state our main results at the beginning of Section 5 (see Theorems 5.1 and 5.2).
This work is organized as follows. In Section 2 we introduce an abstract setting and, in particular, the weak Cerami-Palais-Smale condition and some related existence and multiplicity results which generalize the Ambrosetti-Rabinowitz Mountain Pass Theorem (see [26,Theorem 2.2]) and its symmetric version (see [26,Theorem 9.12]). In Section 3, after introducing some first hypotheses for A(x, u), B(x, v) and G(x, u, v), we give the variational principle for problem (1.1). Then, in Section 4 we prove that J satisfies the weak Cerami-Palais-Smale condition. Finally, in Section 5, a "good" decomposition of X is given and the main results are stated and proved. Moreover, some explicit examples are pointed out (see Corollaries 5.5 and 5.6)

Abstract setting
We denote N = {1, 2, . . . } and, throughout this section, we assume that: • (X, · X ) is a Banach space with dual (X ′ , · X ′ ); • (W, · W ) is a Banach space such that X ֒→ W continuously, i.e. X ⊂ W and a constant σ 0 > 0 exists such that ξ W ≤ σ 0 ξ X for all ξ ∈ X; • J : D ⊂ W → R and J ∈ C 1 (X, R) with X ⊂ D; • K J = {ξ ∈ X : dJ(ξ) = 0} is the set of the critical points of J in X.
Furthermore, fixing β ∈ R, we denote as the set of the critical points of J in X at level β; • J β = {ξ ∈ X : J(ξ) ≤ β} the sublevel of J with respect to β.
Anyway, in order to avoid any ambiguity and simplify, when possible, the notation, from now on by X we denote the space equipped with its given norm · X while, if the norm · W is involved, we write it explicitly.
Moreover, β is a Cerami-Palais-Smale level, briefly (CP S)-level, if there exists a (CP S) βsequence. As (CP S) β -sequences may exist which are unbounded in · X but converge with respect to · W , we have to weaken the classical Cerami-Palais-Smale condition in a suitable way according to the ideas already developed in previous papers (see, e.g., [10,11,12]).
Definition 2.1. The functional J satisfies the weak Cerami-Palais-Smale condition at level β (β ∈ R), briefly (wCP S) β condition, if for every (CP S) β -sequence (ξ n ) n , a point ξ ∈ X exists, such that If J satisfies the (wCP S) β condition at each level β ∈ I, I real interval, we say that J satisfies the (wCP S) condition in I.
Since in [12] a Deformation Lemma has been proved if the functional J satisfies a weaker version of the (wCP S) β condition, namely any (CP S)-level is also a critical level, in particular we can state the following result.
Lemma 2.2 (Deformation Lemma). Let J ∈ C 1 (X, R) and consider β ∈ R such that • J satisfies the (wCP S) β condition, Then, fixing anyε > 0, there exist a constant ε > 0 and a homeomorphism h ε : X → X such that 2ε <ε and Moreover, if J is even on X, then h ε can be chosen odd.
Proof. It is enough reasoning as in [12,Lemma 2.3] with β 1 = β 2 = β and pointing out that the deformation h ε : X → X is a homeomorphism.
From Lemma 2.2 we obtain the following generalization of the Mountain Pass Theorem (compare it with [12,Theorem 1.7] and the classical statement in [26,Theorem 2.2]). Theorem 2.3. Let J ∈ C 1 (X, R) be such that J(0) = 0 and the (wCP S) condition holds in R + . Moreover, assume that there exist some constants R 0 , ̺ 0 > 0, and e ∈ X such that Then, J has a Mountain Pass critical point ξ ∈ X such that J(ξ) ≥ ̺ 0 .
Furthermore, with the stronger assumption that J is symmetric, the following generalization of the symmetric Mountain Pass Theorem can be stated (see [ Theorem 2.4. Let J ∈ C 1 (X, R) be an even functional such that J(0) = 0 and the (wCP S) condition holds in R + . Moreover, assume that ̺ > 0 exists so that: (H ̺ ) three closed subsets V ̺ , Y ̺ and M ̺ of X and a constant R ̺ > 0 exist which satisfy the following conditions: (ii) M ̺ = ∂N , where N ⊂ X is a neighborhood of the origin which is symmetric and bounded with respect to · W ; Then, if we put the functional J possesses at least a pair of symmetric critical points in X with corresponding critical level β ̺ which belongs to [̺, If we can apply infinitely many times Theorem 2.4, then the following multiplicity abstract result can be stated.
Corollary 2.5. Let J ∈ C 1 (X, R) be an even functional such that J(0) = 0, the (wCP S) condition holds in R + and a sequence (̺ n ) n ⊂ ]0, +∞[ exists such that ̺ n ր +∞ and assumption (H ̺n ) holds for all n ∈ N. Then, functional J possesses a sequence of critical points (u kn ) n ⊂ X such that J(u kn ) ր +∞ as n ր +∞.
For simplicity, here and in the following we denote by | · | the standard norm on any Euclidean space, as the dimension of the considered vector is clear and no ambiguity occurs, and by C any strictly positive constant which arises by computation.
In order to investigate the existence of weak solutions of the nonlinear problem (1.1), consider p 1 , p 2 > 1 and, for i ∈ {1, 2}, the related Sobolev space From the Sobolev Embedding Theorem, for any r ∈ [1, For simplicity, we put p * i = +∞ and ∈ R be two given functions and assume that G : Ω × R × R → R exists such that the following conditions hold: (h 0 ) A and B are C 1 -Carathéodory functions with partial derivatives as in (1.2); (h 1 ) for any ρ > 0 we have that and and Remark 3.2. We note that the subcritical growth assumptions (3.4) and (3.5) are required in order to prove the (wCP S) condition (see Proposition 4.9) but not the variational principle stated in Proposition 3.7. Moreover, conditions (g 0 ), (3.2) and (3.3) imply the existence of a constant σ 1 > 0 such that In fact, from (g 0 ), (3.2) and (3.3), for a.e. x ∈ Ω and for any (u, v) ∈ R 2 the Mean Value Theorem implies the existence of t ∈ ]0, 1[ such that and so (3.6) follows with σ 1 = max{σ, |G(·, 0, 0)| ∞ }.
Here, the notation introduced for the abstract setting at the beginning of Section 2 is referred to our problem with while the Banach space (X, · X ) is defined as where with the norms we have that X in (3.8) can also be written as and can be equipped with the norm By definition, for i ∈ {1, 2} we have X i ֒→ W i and X i ֒→ L ∞ (Ω) with continuous embeddings.
. Hence, if both p 1 > N and p 2 > N , then X = W 1 × W 2 and the classical Mountain Pass Theorems in [2] can be used, if required. (3.10) For simplicity, we put So, from (3.10) it follows that (3.12) and As useful in the following, we recall this technical lemma (see [21]).
From Lemma 3.5 and reasoning as in [16,Lemma 3.4], we have the following estimate.
Now, we can state a regularity result.
and, moreover, M > 0 exists such that Hence, J is a C 1 functional on X with Fréchet differential defined as in (3.10).
To this aim, we note that From (h 1 ), (3.16), (3.18) and Lemma 3.6 it follows that On the other hand, from Dominated Convergence Theorem we have that as (3.16) and (h 0 ) imply that Similarly, by assumptions (h 0 )-(h 1 ), (3.17), (3.18) and again Lemma 3.6, we obtain that Finally, (3.16), (3.17) and (g 0 ) imply that while, by conditions (3.6) and (3.18), a constant C > 0 exists such that thus, by Dominated Convergence Theorem we conclude that as from (3.13) it follows that To this aim, firstly let w ∈ X 1 be such that w X1 ≤ 1. So, from (3.11), condition |w| ∞ ≤ 1 implies that Moreover, with arguments which are similar to those ones used previously up for proving (3.20), respectively (3.21), we have that Furthermore, since w W1 ≤ 1, by using Hölder inequality we have that in Ω so Dominated Convergence Theorem applies. At last, hypothesis (h 1 ), (3.18) and Hölder inequality imply that We note that if p 1 > 2, then Lemma 3.5 and again Hölder inequality, with conjugate exponents as (3.16) implies that ( u n W1 ) n is bounded.
On the other hand, if 1 < p 1 ≤ 2, from Lemma 3.5 we have Hence, from (3.16) and (3.24)-(3.26) it follows that Thus, summing up, we have that and (3.22) holds. Finally, similar arguments allow us to prove that Hence, in the proof of Proposition 3.7 no sub-critical growth is required for G u (x, u, v) and G v (x, u, v).

The weak Cerami-Palais-Smale condition
In order to prove some more properties of functional J : X → R defined as in (1.7), we require that R ≥ 1 exists such that the following conditions hold: in Ω, for all u, v ∈ R; (h 3 ) there exists µ 1 > 0 such that in Ω if |v| ≥ R; (h 4 ) there exist θ 1 , θ 2 , µ 2 > 0 such that and in Ω, for all u ∈ R, in Ω, for all v ∈ R; (g 2 ) taking θ 1 , θ 2 as in (h 4 ), we have Remark 4.1. In (h 3 ), respectively (h 4 ), we can always assume that Hence, we have that in Ω if |v| ≥ R.
In order to prove the weak Cerami-Palais-Smale condition, we require the following boundedness result (for its proof, see [22, Theorem II.5.1]).
Up to now, no upper bound is required for the growth of the nonlinear term G(x, u, v). Anyway, subcritical assumptions need for proving the weak Cerami-Palais-Smale condition at any level, more precisely we will require that the whole assumption (g 1 ) is satisfied, i.e., the exponents in Remark 4.7. Consider 1 < p 1 < N and 1 < p 2 < N . If s 1 ∈ R is as in (3.5), then s 3 > 0 exists such that 1 < s 3 < p * 1 , 0 ≤ s 4 := s 1 s 3 In fact, (3.5) implies that Similarly, if s 2 ∈ R is as in (3.5), then s 5 > 0 exists such that 1 < s 5 < p * 2 , 0 ≤ s 6 := s 2 s 5 and Remark 4.8. Taking 1 < p 1 < N and 1 < p 2 < N , if (g 0 )-(g 1 ) hold, then from (3.6) and the notation and estimates in Remark 4.7 it follows that a.e. in Ω, for all (u, v) ∈ R 2 . Hence, putting q 1 = max{q 1 , s 3 , s 6 } and q 2 = max{q 2 , s 4 , s 5 }, (4.27) up to change the constant C > 0 we have that At last, if also (g 2 ) holds, then from (4.11), (4.24) and (4.26) it follows that Proof. Let β ∈ R be fixed. We have to prove that if ((u n , v n )) n ⊂ X is a sequence such that then (u, v) ∈ X exists such that To this aim, our proof is divided in several steps; more precisely, we will prove that: 1. ((u n , v n )) n is bounded in W ; hence, there exists (u, v) ∈ W such that, up to subsequences, we have that 3. for any k > 0, defining T k : R → R such that then, if k ≥ max{|(u, v)| L , R} + 1 (with R ≥ 1 as in our set of hypotheses), we have and For simplicity, here and in the following we use the notation (ε n ) n for any infinitesimal sequence depending only on ((u n , v n )) n , while (ε k,n ) n for any infinitesimal sequence depending also on some fixed integer k. Moreover, we assume that 1 < p i < N for both i ∈ {1, 2}, otherwise the proof is simpler.
Step 2. As we want to prove that (u, v) ∈ L, arguing by contradiction we assume that either u / ∈ L ∞ (Ω) or v / ∈ L ∞ (Ω). If u / ∈ L ∞ (Ω), either ess sup For example, suppose that (4.36) holds. Then, for any fixed k ∈ N, k > R (with R ≥ 1 as in the hypotheses), we have meas(Ω + k ) > 0, (4.38) with From (4.31), we have R + k u n ⇀ R + k u in W 1 . Thus, by the sequentially weakly lower semicontinuity of · W1 , it follows that i.e., with Ω + n,k := {x ∈ Ω | u n (x) > k}. On the other hand, by definition, we have R + k u n X1 ≤ u n X1 . Hence, (3.14) and (4.30) imply that thus, from (4.38) an integer n k ∈ N exists such that ∂J ∂u (u n , v n ) R + k u n < meas(Ω + k ) for all n ≥ n k . (4.40) Taking any k > R and n ∈ N, from (3.11), hypothesis (h 3 ) with µ 1 < 1 (see Remark 4.1) and condition (h 2 ) it follows that which, together with (4.40), implies that We note that, from (4.33) and (g 0 ), we have in Ω. Moreover, hypothesis (3.5) implies that suitable exponents can be choosen so that (4.24) holds. Thus, since |R + k u n (x)| ≤ |u n (x)| for a.e. x ∈ Ω, by using (4.25) in (3.2), from (3.4), (4.24) and (4.32) we have that a function h ∈ L 1 (Ω) exists such that x ∈ Ω, up to subsequences (see, e.g., [8,Theorem 4.9]). So, by using the Dominated Convergence Theorem, we have lim Hence, summing up, from (4.39), (4.41) and (4.42) we obtain which implies, by using again (3.2), that (4.44) Taking q 1 as in (4.27), in our setting of hypotheses the estimates in (4.29) hold, so and, since k ≥ 1, we have that Thus, (4.43) and (4.44) imply that with C = C( v W2 ) > 0. At last, from (4.29) and direct computations we obtain that while, as p 1 < N , from (4.24) it results Thus, summing up, from (4.45) we have that with C = C( u W1 , v W2 ) > 0; so, as k ≥ 1 implies 1 ≤ k p1 , Lemma 4.6 applies and yields a contradiction to (4.36). Now, suppose that (4.37) holds which implies that, fixing any k ∈ N, k ≥ R, it is In this case, by replacing function we can reason as above so to apply again Lemma 4.6 which yields a contradiction to (4.37). Then, it has to be u ∈ L ∞ (Ω). Similar arguments but considering ∂J ∂v (u n , v n ) and R + k v n , respectively R − k v n , and the related sets, allow us to prove that it has to be also v ∈ L ∞ (Ω).
By definition, we have that T k (u n , v n ) X ≤ (u n , v n ) X and R k (u n , v n ) X ≤ (u n , v n ) X for all n ∈ N; (4.46) furthermore, we note that Then, from (4.31)-(4.33) it follows that v) a.e. in Ω, (4.47) and also, again from (4.32) and (4.33), we have that meas(Ω u n,k ) −→ 0 and meas(Ω v n,k ) −→ 0 as n → +∞. Furthermore, (3.14), (4.30) and (4.46) imply that Now, by reasonig as in the proof of Step 2 but replacing R + k u n and R + k v n with R k u n , respectively R k v n , from (4.49) we have that as the same arguments used for proving (4.42) apply so that from (4.48) we obtain Whence, (4.50) and (4.51) imply not only that Now, in order to prove (4.34), from (3.15) it is enough to verify that To this aim, let w ∈ X 1 , z ∈ X 2 be such that w X1 = 1, z X2 = 1. Then, direct computations imply that (4.56) From (3.14) and (4.30) we have that Moreover, (4.2) and (4.53) imply that On the other hand, it results We note that from (g 0 ) and (4.33), respectively (4.47), we have a.e. in Ω, a.e. in Ω, while from (g 1 ) and Young inequality, direct computations imply that a.e. in Ω, with s 4 as in (4.24) and s 6 as in (4.26), and thus, (4.24), (4.26), (4.32) and [8,Theorem 4.9] allow us to apply the Dominated Convergence Theorem so that we obtain So, summing up, from (4.55) and (4.56), all the previous limits imply that where both (ε k,n ) n represent suitable infinitesimal sequences independent of w, respectively z.
Using the same notation introduced in Step 2, we evaluate the last integrals in (4.57) by reasoning as in the proof of Step 3 in [11, Proposition 4.6] but taking new test functions and passing to the limit in ∂J ∂u Hence, we are able to claim that (4.54) hold. At last, direct computations imply that where from (4.28), (4.29), (4.32), (4.33), (4.47), and again the Dominated Convergence Theorem we have Thus, (4.35) follows from (4.30) and (4.53).
Step 4. By following some ideas introduced in [3] and considering the real map ψ(t) = te ηt 2 , where η can be fixed in a suitable way, in particular by applying the same arguments developed in the proof of [14,Proposition 3.6] in order to estimate ∂J ∂u (T k (u n , v n ))[ψ(T k u n − u)], we prove that T k u n − u W1 → 0. (4.58) Moreover, reasonig in the same way but considering ∂J (4.59) Then, condition (i) follows from (4.52), (4.58) and (4.59).

Main results
In order to introduce a suitable decomposition of the space X, we recall that, if p > 1 but p = 2, the spectral properties of the operator −∆ p in W 1,p 0 (Ω) are still mostly unknown. In particular, with respect to the semilinear case, the drawback in using its known eigenvalues is that, for the Banach space W 1,p 0 (Ω), their use does not provide a decomposition having properties similar to that of the Hilbert space H 1 0 (Ω) by means of the eigenfunctions of −∆ on Ω with null homogeneous Dirichlet data.
Here, we consider the sequence of pseudo-eigenvalues of the operator −∆ p in W 1,p 0 (Ω) as introduced in [11, Section 5] as a suitable decomposition of the Sobolev space W 1,p 0 (Ω) occurs, so that it turns out to be the classical one for p = 2.
In order to present this definition, firstly let us recall that if V ⊆ X is a closed subspace of a Banach space X, a subspace Y ⊆ X is a topological complement of V , briefly X = V ⊕ Y , if Y is closed and every x ∈ X can be uniquely written as w + y, with w ∈ V and y ∈ Y ; furthermore, the projection operators onto V and Y are (linear and) continuous (see, e.g., [8, p. 38]). When X = V ⊕ Y and V has finite dimension, we say that Y has finite codimension, with codim Y = dim V . Now, as in [11,Section 5], for i ∈ {1, 2}, we start from λ i,1 , first eigenvalue of −∆ pi in W i , which is characterized as and is strictly positive, simple, isolated and has a unique eigenfunction ϕ i,1 such that ϕ i,1 > 0 a.e. in Ω, ϕ i,1 ∈ L ∞ (Ω) and |ϕ i,1 | pi = 1 (5.2) (see, e.g., [23]). Then, we have the existence of a sequence (λ i,m ) m such that with corresponding functions (ψ i,m ) m such that ψ i,1 ≡ ϕ i,1 and ψ i,m = ψ i,j if m = j. They generate the whole space W i , are also in L ∞ (Ω), hence in X i , and are such that where V i,m = span{ψ i,1 , . . . , ψ i,m } and its complement Y i,m in W i can be explicitely described.
Moreover, for all m ∈ N on the infinite dimensional subspace Y i,m the following inequality holds: and then, since V i,m is also a finite dimensional subspace of X i ,  ) and B(x, v) be two given real functions defined in Ω × R such that conditions (h 0 )-(h 4 ) hold. Moreover, suppose that the real function G(x, u, v), defined in Ω × R 2 , satisfies hypotheses (g 0 )-(g 2 ). In addition, we assume that with λ i,1 , i ∈ {1, 2}, as in (5.1). Then, functional J in (1.7) possesses at least one nontrivial critical point in X; hence, problem (1.1) admits a nontrivial weak bounded solution.
Then, the even functional J in (1.7) possesses a sequence of critical points ((u m , v m )) m ⊂ X such that J (u m , v m ) ր +∞; hence, problem (1.1) admits infinitely many distinct weak bounded solutions.
Remark 5.3. Subcritical growth conditions (3.5) are stronger than the "classical" ones required for Laplacian coupled systems in [7]. In our setting, we need it for proving that weak limits of (CP S)-sequences in the product Sobolev space W belong also to L, i.e. they are bounded functions (see Step 2 in the proof of Proposition 4.9).
Some corollaries to previous Theorem 5.2 can be stated, both when condition (g 4 ) is replaced by a stronger assumption, which is easier to verify, and when such a theorem is applied to the special cases obtained by coupling Example 4.2 with Example 4.4, respectively Example 4.5.
Then, the even functional J in (1.7) possesses a sequence of critical points ((u m , v m )) m ⊂ X such that J (u m , v m ) ր +∞; hence, problem (1.1) admits infinitely many distinct weak bounded solutions.