Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper we consider the following coupled gradient-type quasilinear elliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v)&\hbox{ in $\Omega$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left(x, u, v\right)&\hbox{ in $\Omega$,}\\[10pt] u = v = 0&\hbox{ on $\partial\Omega$,} \end{array} \right. \end{equation*} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N\ge 2$. We suppose that some $\mathcal{C}^{1}$-Carath\'eodory functions $A, B:\Omega\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ exist such that $a(x,t,\xi) = \nabla_{\xi} A(x,t,\xi)$, $A_t(x,t,\xi) = \frac{\partial A}{\partial t} (x,t,\xi)$, $b(x,t,\xi) = \nabla_{\xi} B(x,t,\xi)$, $B_t(x,t,\xi) =\frac{\partial B}{\partial t}(x,t,\xi)$, and that $G_u(x, u, v)$, $G_v(x, u, v)$ are the partial derivatives of a $\mathcal{C}^{1}$-Carath\'eodory nonlinearity $G:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$. Roughly speaking, we assume that $A(x,t,\xi)$ grows at least as $(1+|t|^{s_1p_1})|\xi|^{p_1}$, $p_1>1$, $s_1 \ge 0$, while $B(x,t,\xi)$ grows as $(1+|t|^{s_2p_2})|\xi|^{p_2}$, $p_2>1$, $s_2 \ge 0$, and that $G(x, u, v)$ can also have a supercritical growth related to $s_1$ and $s_2$. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.

Since the coefficients depend on the solution and its grandient themselves, the study of the interaction of two different norms in a suitable Banach space is needed.
In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.

Introduction
The study of partial differential equations involving nonlinearities with critical or supercritical growths, is a very complex matter and for many critical and supercritical problems some basic issues are mostly unknown or undiscovered. For example, let us consider the quasilinear elliptic problem −∆ p u = λ|u| p−2 u + |u| q−2 u in Ω, u = 0 on ∂Ω, (1.1) where Ω is an open bounded domain in R N , N ≥ 3, and 1 < p < N . In spite of the simple looking structure of the problem, if we ask q to be critical or supercritical from the viewpoint of the Sobolev Embedding Theorem, namely q ≥ p * = N p N −p , some significant difficulties arise. Among other problems, in general the lack of compactness which occurs, does not guarantee even the existence of solutions, which has been derived only in few cases and frequently under assumptions on the shape of the domain Ω (for the classical nonexistence result due to the Pohožaev's identity see [36], or also [37,Theorem III.1.3]).
The existence of positive solutions of (1.1) has been successfully addressed either by adding some lower order term to the critical nonlinearity (see [9]) or by considering domains which are not starshaped (see, e.g., [6,22]) if p = 2, q = 2 * and λ = 0, while the existence of sign-changing solutions of (1.1) have been obtained if p = 2, q = 2 * but λ = 0 (see, e.g., [3,20]). To our knowledge, all the results carried over so far, are built on the key assertion that the functional associated to the critical problem (1.1) satisfies the Palais-Smale condition even if only in certain ranges of energy.
On the other hand, taking p = 2, due to the hardship in handling a quasilinear operator, very few results of existence have been derived so far, not even under assumptions of symmetry on the domain (we refer to [31] for a wider discussion). We limit ourselves to point out that, as derived in [34], a Pohožaev type nonexistence result is not yet available for sign-changing solutions of (1.1), as the unique continuation principle for the p-Laplacian is not known, while it has been proved for nonnegative solutions (see [23]). However, the existence of a positive solution in a domain with a sufficiently small hole has been shown for 2N N +2 ≤ p ≤ 2, as well as an existence and multiplicity result has been proved under further assumptions of symmetry (see [21,31,32,34,35] and references therein).
In spite of the mentioned difficulties, in recent years there has been a marked increase of research in critical and supercritical problems. The interest in these problems is related to their similarity to some variational problems which arise in Geometry and Physics where the lack of compactness also occurs. In this sense, one of the best known challenges is the so called Yamabe's problem, but also some examples related to the existence of extremal functions for isoperimetric inequalities, Hardy-Littlewood-Sobolev inequalities and trace inequalities can be addressed (see, e.g., [25,27,29]). However, in general, also in the "simplest" cases some problems are still open and some classical variational tools, largely used in the subcritical case, do not work in the critical and supercritical ones.
x ∈ Ω (see [14] and also, for other approaches, [4,30]). One of the most remarkable feature of this work is that, unlike the results mentioned above, both an existence and a multiplicity result have been provided in the supercritical case for a more general problem without taking any symmetry assumption on the domain Ω.
Here, following the ideas introduced in [14], we look for solutions of the family of coupled gradient-type quasilinear elliptic systems in Ω, where Ω is an open bounded domain in R N , N ≥ 2, and A, B : Ω× R× R N → R are given functions with partial derivatives Roughly speaking, here we assume that A(x, u, ∇u) grows at least as (1+|u| s1p1 )|∇u| p1 , p 1 > 1, s 1 ≥ 0, while B(x, v, ∇v) grows at least as (1 + |v| s2p2 )|∇v| p2 , p 2 > 1, s 2 ≥ 0 (see Remark 3.2 and assumption (h 7 )), and that G(x, u, v) can also have a supercritical growth depending on s 1 and s 2 (see hypothesis (g 2 )).
While subcritical quasilinear systems have been handled through several techniques (see, e.g., [5,8,10,16,18]), as far as we know very few existence results have been determined for supercritical quasilinear elliptic systems (see, for example, [19,24] and references therein), even though no result occurs for supercritical systems with coefficients depending on the solution and its gradient themselves, as that one in (1.2). Moreover, as in [14], even if a supercritical growth occurs, the domain Ω is only open and bounded as we consider homogeneous Dirichlet boundary condition and the solutions we are looking for, are weak.
Thus, following the same approach used in [16] and [18], but carefully adapting the ideas in [14] to our supercritical setting, we give some sufficient conditions for recognizing the variational structure of problem (1.2), so that investigating solutions of (1.2) reduces to find critical points of functional 2}. Moreover, since in the Banach space X our functional J does not satisfy the Palais-Smale condition, or one of its standard variants, we are not allowed to use directly existence and multiplicity results as the classical Ambrosetti-Rabinowitz theorems stated in [2] or in [7]. Hence, we have to submit a weaker definition of the Cerami's variant of Palais-Smale condition, the so-called weak Cerami-Palais-Smale condition (see Definition 2.1). We believe that the use of this definition, introduced in the pioneering paper [11] and employed in the framework of a quasilinear supercritical system, represents another major improvement of the work in this field. In fact, here Definition 2.1 is used for stating an extended Mountain Pass Theorem and also its symmetric version of which we avail to gain our existence and multiplicity results (see Theorems 2.2 and 2.3), but we do not exclude the chance that this feature may be also employed to recover other kind of problems (see, e.g., [33]). In fact, we highlight that this technique has been adapted to address problems placed over unbounded domains both in radial and in non-radial setting (see [15], respectively [17]) but so far only in subcritical growth assumptions (in [1] the existence of solutions for some critical and supercritical problems have been proved by using a different (radial) approach, which is not applicable for non-autonomous equations).
On the other hand, this enhancement imposes to pay the price that some technical assumptions on the involved functions are needed. Namely, if we just assume the Carathéodory functions A(x, t, ξ), B(x, t, ξ), G(x, u, v) and their partial derivatives fit some proper polynomial growths to show the C 1 regularity of the functional J in (1.6), on the other hand the proof of the weak Cerami-Palais-Smale condition passes through some fine requirements on the involved functions (see Section 3) and a very remarkable result (see Lemma 3.7) which has interest own self and can be employed regardless of this scenario to fix a problem of common trouble in this field. Now, in order to draw the attention to the enhancement of our main results, we state them here in a "streamlined" version but we refer the reader to Section 4 for all the needed hypotheses on the involved functions and the precise statement of the results (see Theorems 4.1 and 4.2). Theorem 1.1. Suppose that A(x, t, ξ) grows at least as (1 + |t| s1p1 )|ξ| p1 , with p 1 > 1, s 1 ≥ 0, while B(x, t, ξ) grows at least as (1 + |t| s2p2 )|ξ| p2 , with p 2 > 1, s 2 ≥ 0. Moreover, assume that the C 1 -Carathéodory function A(x, t, ξ), respectively B(x, t, ξ), and its partial derivatives fits some suitable interaction properties among themselves, while the C 1 -Carathéodory nonlinear term G(x, u, v) satisfies the Ambrosetti-Rabinowitz condition for systems with coefficients θ 1 , θ 2 > 0 such that θ i < 1 pi , i ∈ {1, 2}, and has a proper polynomial growth which can also be supercritical depending on s 1 and s 2 . If lim sup (u,v)→(0,0) G(x, u, v) |u| p1 + |v| p2 < α 2 min{λ 1,1 , λ 2,1 } uniformly a.e. in Ω, , then problem (1.2) admits a nontrivial weak bounded solution.
Finally, in order to better explain the required hypotheses, we consider the particular setting and with c * ≥ 0 and some positive exponents and, in a suitable set of assumptions, system (1.2) turns into the model problem (1.9) Hence, the previous results can be reworded in this way.
Our paper is organized as follows. In Section 2 we introduce the abstract setting needed to recognize the variational structure of our problem (1.2), as well as some extended versions of the Mountain Pass Theorems are shown up. Furthermore, a regularity result for the functional J in (1.6) is provided, too. Then, in Section 3 some further assumptions on A(x, t, ξ), B(x, t, ξ) and G(x, u, v) are addressed in order to show that the functional J verifies the weak Cerami-Palais-Smale condition. Lastly, in Section 4 our main results are stated and proved.

Abstract tools and variational setting
We denote N = {1, 2, . . . } and, as long as we introduce our abstract setting, we employ the following notations: • (W, · W ) is a Banach space such that X ֒→ W continuously, i.e. X ⊂ W and a constant σ 0 > 0 exists such that y W ≤ σ 0 y X for all y ∈ X, 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. Now, taking β ∈ R, we say that a sequence (y n ) n ⊂ X is a Cerami-Palais-Smale sequence at level β, briefly (CP S) β -sequence, if lim n→+∞ J(y n ) = β and lim n→+∞ dJ (y n ) X ′ (1 + y n X ) = 0.
As pointed out in [13, Example 4.3], a (CP S) β sequence can be constructed so that it is unbounded in · X but converges with respect to · W . Thus, as in [14], we introduce the following definition.
We say that J satisfies (wCP S) in I, I real interval, if J satisfies the (wCP S) β condition in X at each level β ∈ I.
Anyway, even if we deal with a weaker version of the Cerami's variant of the Palais-Smale condition, some classical abstract results can be extended so to fit to our purposes. Actually, as in [14, Lemma 2.2] (see also [12, Lemma 2.3]) a Deformation Lemma can be stated which provides the following extended version of the Mountain Pass Theorem given in [2] (see [14,Theorem 2.3] for a detailed proof).
Theorem 2.2. 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 a continuous map ℓ : X → R, some constants r 0 , ̺ 0 > 0, and e ∈ X such that Then, J has a Mountain Pass critical point y * ∈ X such that J(y * ) ≥ ̺ 0 .
If, in addition, we require that J is symmetric, then a more general version of the Symmetric Mountain Pass Theorem in [2] can be stated, too (for the proof, see [14,Theorem 2.4]). Theorem 2.3. 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 ̺ , Z ̺ and M ̺ of X and a constant R ̺ > 0 exist which satisfy the following conditions: 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 [̺, Finally, if we can apply Theorem 2.3 infinitely many times, then the following multiplicity abstract result can be stated, too.
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 assumption (H ̺ ) holds for all ̺ > 0. Then, the functional J possesses a sequence of critical points (y n ) n ⊂ X such that J(y n ) ր +∞ as n ր +∞. Now, we proceed introducing the notations related to our specific issue. If Ω ⊂ R N is an open bounded domain, N ≥ 2, we denote by: • L ∞ (Ω) the space of Lebesgue-measurable and essentially bounded functions y : Ω → R with norm |y| ∞ = ess sup Ω |y|; • W 1,p 0 (Ω) the Sobolev space equipped with the norm y W 1,p 0 = |∇y| p if 1 ≤ p < +∞; • meas(D) the usual Lebesgue measure of a measurable set D in R N ; • | · | the standard norm on any Euclidean space, as the dimension of the vector taken into account is clear and no ambiguity occurs.
Moreover, for any m ∈ N, we say that h : Let A, B : Ω × R × R N → R be such that the following conditions hold: Furthermore, let G : Ω × R × R → R be a map which satisfies the following hypotheses: is a C 1 -Caratheodory function with partial derivatives as in (1.5), such that G(·, 0, 0) ∈ L ∞ (Ω) and G u (x, 0, 0) = G v (x, 0, 0) = 0 for a.e. x ∈ Ω; (g 1 ) a constant σ > 0 and some exponents Remark 2.6. Hypotheses (g 0 )-(g 1 ), the Mean Value Theorem and direct computations ensure the existence of a positive constant σ 1 > 0 such that Now, taking any couple of real numbers t 3 , t 5 > 1, from Young inequality we obtain where, for simplicity, we set Thus, from (2.6) and (2.7) we infer that for a suitable constant σ 2 > 0.
In order to recall some features shared by the subcritical systems in [16] and [18], if needed, here we introduce similar notations.
For each i ∈ {1, 2} let p i > 1 be as in assumption (h 1 ) and let us consider the related Sobolev space From the Sobolev Embedding Theorem, for any r ∈ [1, Here, the notation (W, · W ), introduced for the abstract setting at the beginning of this section, is referred to our problem with Moreover, we consider the Banach space (X, · X ) defined as where are endowed with the norms we have that X in (2.13) can also be written as and its norm is such that Clearly, from (2.14), for both i ∈ {1, 2} we have that the continuous embeddings X i ֒→ W i and X i ֒→ L ∞ (Ω) hold.
Remark 2.7. If p i > N for both i ∈ {1, 2}, then the embedding W i ֒→ L ∞ (Ω) means that X i = W i . Thus, X = W and the classical Mountain Pass Theorems in [2] may be applied.
Firstly, we note that if conditions (h 0 )-(h 1 ), (g 0 )-(g 1 ) hold, then direct computations imply that J (u, v) in (1.6) is well defined for all (u, v) ∈ X. Moreover, taking any (u, v), (w, z) ∈ X, the Gâteaux differential of functional J in (u, v) along the direction (w, z) is given by For simplicity, we set hence, from (2.16), it follows that Furthermore, above remarks and direct computations give not only the estimates but also At last, from (2.19) we infer that Finally, we can state the regularity of functional J defined in (1.6) (for the proof, see [18, Proposition 3.5]). Proposition 2.8. Assume that conditions (h 0 )-(h 1 ), (g 0 )-(g 1 ) hold. Let ((u n , v n )) n ⊂ X and (u, v) ∈ X be such that If M > 0 exists such that |u n | ∞ ≤ M and |v n | ∞ ≤ M for all n ∈ N, Hence, J is a C 1 functional on X with Fréchet differential defined as in (2.16).

The set up for the weak Cerami-Palais-Smale condition
In order to prove some more properties of functional J in (1.6), let p 1 > 1 and p 2 > 1 as in the earlier hypothesis (h 1 ). Then, assume that R ≥ 1 exists such that the following conditions hold: a(x, t, ξ) · ξ ≥ µ 0 (1 + |t| s1p1 )|ξ| p1 a.e. in Ω, for all (t, ξ) ∈ R × R N , b(x, t, ξ) · ξ ≥ µ 0 (1 + |t| s2p2 )|ξ| p2 a.e. in Ω, for all (t, ξ) ∈ R × R N ; (h 4 ) a constant µ 1 > 0 exists such that (h 5 ) some constants θ 1 , θ 2 , µ 2 > 0 exist such that and in Ω, for all t ∈ R; (g 2 ) for i ∈ {1, 2}, taking p i as in hypothesis (h 1 ), q i , t i as in assumption (g 1 ) and s i as in (h 3 ), we assume that and Remark 3.1. Assumption (3.5) shows up the supercritical nature of our problem which vanishes if s 1 = s 2 = 0 as it reduces exactly to the subcritical condition (g 1 ) in [16,18]. However, in general, if one or both s 1 > 0, s 2 > 0 hold, then a supercritical growth on the nonlinear term G(x, u, v) is allowed. Moreover, we emphasize that the growth hypothesis (g 2 ) is needed to prove that the functional J satisfies the (wCP S) condition, but has not been required for the variational principle stated in Proposition 2.8.
We note that, if then we can always choose η 1 in (h 2 ) large enough so that (3.19) is satisfied.
Remark 3.4. Conditions in (3.20) not only relate the exponents s 1 , s 2 provided in assumption (h 3 ) with the powers p 1 , p 2 > 1 used in the growth conditions (h 1 ) and θ 1 , θ 2 claimed in (3.4), but also they tell us how far we can take it. In particular, it implies that in our set of hypotheses, a supercritical growth is allowed as long as s 1 , s 2 cover the whole range stated in (3.20).
Then, y ∈ W 1,p 0 (Ω) exists such that |y| s y ∈ W 1,p 0 (Ω), too, and, up to subsequences, we have that y n ⇀ y weakly in W 1,p 0 (Ω), |y n | s y n ⇀ |y| s y weakly in W 1,p 0 (Ω), y n → y strongly in L r (Ω) for each r ∈ [1, p * (s + 1)[, y n → y a.e. in Ω.  Let Ω be an open bounded subset of R N and consider y ∈ W 1,p 0 (Ω) with p ≤ N . Suppose that γ > 0 and k 0 ∈ N exist such that with Ω + k = {x ∈ Ω : y(x) > k} and r, m, α j , ε j positive constants such that Then, ess sup Ω y is bounded from above by a positive constant which can be chosen so that it depends only on meas(Ω), N , p, γ, k 0 , r, m, ε j , α j , |y| p * (eventually, |y| l for some l > r if p * = +∞).
As pointed out in Remark 3.1, the upper bounds in (g 2 ) were not required so far, but will be essential in the incoming results. Therefore, some consequences of the estimates in (3.5) and (3.6) are needed.

(ii) is satisfied.
For simplicity, here and in the following we will use the notation (ε n ) n for any infinitesimal sequence depending only on ((u n , v n )) n . Moreover, we denote by c i every positive constant which arises during our computations.
Step 2. Due to the Sobolev Embedding Theorem, this step requires a proof only if either p 1 ≤ N or p 2 ≤ N . So, if p 1 < N (when p 1 = N the arguments can be simplified) we want to prove that u ∈ L ∞ (Ω). Arguing by contradiction, we assume that u / ∈ L ∞ (Ω) as either ess sup If (3.34) holds, then for any k ∈ N we have that and for an integerk > 0 we consider the function R + k : t ∈ R → R + k t ∈ R defined as Now, we consider condition (3.36) for a fixed integer k > R (with R ≥ 1 as in our setting of hypotheses) and, takingk = k s1+1 , for simplicity we put and, as |t| s1 t >k ⇐⇒ t > k, we have that Thus, from condition (3.30) and the sequentially weakly lower semicontinuity of · W1 we have that with Ω + n,k = {x ∈ Ω : u n (x) > k} = {x ∈ Ω : w n (x) >k}. On the other hand, by definition (3.37) withk replaced with k, it is R + k u n X1 ≤ u n X1 , so from (2.20), (3.27) and (3.36), an integer n k ∈ N exists such that ∂J ∂u (u n , v n ) R + k u n < meas(Ω + u,k ) for all n ≥ n k .
We claim that In fact, from (3.32) and (g 0 ) we have that while, thanks to assumption (g 2 ), formulae (2.5), (2.7), (3.24), (3.25) and (3.31) ensure the existence of h ∈ L 1 (Ω) such that or better, from (2.7) but according to the choises in Remark 3.8, by taking q 1 > 1 as in (2.10) and being u > 1 in Ω + u,k , definition (3.38) implies that We claim that In fact, if t 4 = 0 then (3.45) reduces to > 1 and the Hölder inequality with such an exponent, together with (2.11), implies that On the other hand, if, for simplicity we put r = q 1 s1+1 , from (3.26), direct computations and, again, (2.11) we have that which, together with (3.45), allows us to reduce (3.44) to the estimate At last, as p 1 < N , from (3.25) we have that so, from (3.26) and (3.39), since (3.46) holds for allk large enough, we have that Lemma 3.7 applies and ess sup Ω w < +∞ in contradiction to (3.34).
Similar arguments, but modified in a suitable way, ensures that even (3.35) cannot occur, then it has to be u ∈ L ∞ (Ω), and also that it has to be v ∈ L ∞ (Ω).
Step 3 The proof can be obtained by reasoning as in the proof of [18, Step 3 in Proposition 4.8] but with m = 2 and by replacing the estimates in [18,Remark 4.5] with those ones in Remark 3.8 together with (3.26), and also by using (3.31) at the place of [18, (4.19)].

Existence and multiplicity results
Now, we can state our leading results. To this aim, we refer to the decomposition of X already introduced in [16, Section 5]. For the sake of convenience, here we recall the main issues. For i ∈ {1, 2}, the first eigenvalue of −∆ pi in W i is given by Such an eigenvalue is simple, positive, 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 (4.2) (see, e.g., [28]). Furthermore, a sequence of positive real numbers exists such that with corresponding pseudo-eigenfunctions (ψ i,m ) m which not only generate the whole space W i , but are in L ∞ (Ω), too. Thus, (ψ i,m ) m ⊂ X i , and, for any fixed m ∈ N, we consider is satisfied (cf. [11,Proposition 5.4]).
Thus, for any m ∈ N definition (2.12) implies that while from (2.15) it follows that Now, we are ready to provide our existence and multiplicity results.
Thus, functional J in (1.6) possesses at least one nontrivial critical point in X; hence, problem (1.2) admits a nontrivial weak bounded solution.
Before turning to the proof of our main results, we observe that if assumption (h 3 ), and then (h 7 ), holds with s 1 = s 2 = 0, then Theorem 4.1 reduces to [18, Theorem 5.1] while Theorem 4.2 reduces to [18, Theorem 5.2] but with m = 2. Actually, the same holds true if both p 1 ≥ N and p 2 ≥ N . Thus, in order to improve such previous results, here we assume that either s 1 > 0 or s 2 > 0 and we define and then From definitions (4.5) and (4.6) we have that and Moreover, takingp = min{p 1 , p 2 }, direct computations imply that Remark 4.4. For both i ∈ {1, 2} definition (2.14) and identity (3.23) imply that the function y → |y| si y Wi is continuous and well defined in (X i , · Xi ) and so ℓ i : X i → R is continuous, too. Thus, from (2.15) we have that (u, v) → (|u| s1 u, |v| s2 v) W is continuous and well defined in (X, · X ), then also ℓ : X → R is continuous with respect · X and definition (4.6) implies that ℓ(u, v) ≥ (u, v) W for all (u, v) ∈ X with ℓ(0, 0) = 0.
Throughout the remaining part of this section, for simplicity we assume that since they share the same differential on X and so the same critical points. Moreover, we denote by c i every positive constant which arises during computations. Now, we can prove our existence result.
In order to prove our multiplicity theorem, some geometric conditions are needed. In particular, if assumptions (h 0 )-(h 6 ) and (g 0 )-(g 3 ) hold, we are able to state the following results. and, since from (4.3) it follows that R i,m ր +∞ as m → +∞, we have that while, by choosing m > m ̺ , the m-dimensional space V m is such that codim Y m̺ < dim V m , and from Proposition 4.6 a radius R Vm > 0 exists so that J (u, v) ≤ 0 for all (u, v) ∈ V m such that (u, v) X ≥ R Vm .
Hence, assumption (H ̺ ) in Theorem 2.3 is verified. Then, the arbitrariness of ̺ > 0 so that (H ̺ ) holds, together with Propositions 2.8 and 3.10, allows us to apply Corollary 2.5 and the existence of a sequence of diverging critical levels for the functional J in X is provided.
Proof of Theorem 1.3. Taking A(x, t, ξ) and B(x, t, ξ) as in (1.7), from (1.10) it follows that conditions (h 0 )-(h 4 ) and (h 6 ) hold. Moreover, if G(x, u, v) is as in (1.8), assumptions (1.10)-(1.12) and Young inequality imply that (g 0 )-(g 2 ) are satisfied with On the other hand, again from (1.10), direct computations allow us to prove that hypotheses (h 5 ) and (g 3 ) are verified, too. At last, also condition (g 5 ) holds as (1.10) and direct computations allow us to prove that for any R ≥ 2 it is G(x, u, v) Then, since the symmetric assumptions (h 8 ) and (g 6 ) are trivially satisfied, the thesis follows from Theorem 4.2.