Multiple Critical Points for Symmetric Functionals without upper Growth Condition on the Principal Part

This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For this purpose, nonsmooth variational methods are applied.


Introduction
Let Ω be a bounded and open subset of R n . We aim to prove the existence of multiple critical points, in a suitable sense, for a homogeneous Dirichlet problem associated with a functional of the form under assumptions that do not guarantee any upper growth condition on the principal part Ψ(x, ·).
If Ψ and G are smooth and subjected to suitable growth estimates, the functional f is of class C 1 on some Sobolev space W 1,p 0 (Ω), and standard variational methods apply (see e.g., [1,2]).
The case in which the growth conditions on G are relaxed, meaning that f is only continuous or even lower semicontinuous, has been already considered in [3,4], but standard growth conditions on the principal part Ψ are still imposed.
However, situations in which there is no upper growth condition on the principal part appear, for instance, in continuum mechanics, and a case in which Ω is one-dimensional has already been treated in [5]. On the other hand, to the best of our knowledge, in the multi-dimensional case, only the existence of minima has been proved thus far.
Let us also point out that the fact that each minimum satisfies the associated Euler-Lagrange equation can be not at all obvious. See, e.g., the survey paper [6]. This problem has also been addressed in [7], and the assumptions we will impose on Ψ are related to those required in [7].
In order to prove the existence of minima, the case in which the functional f is coercive is usually considered. As a first step in the direction of existence results for critical points, we will also consider a coercive case. When standard growth conditions on Ψ and G are satisfied, the existence of multiple critical points in the coercive case has been obtained, for instance, in [1,8]. We will prove a result in the line of ([1] Theorem 9.10) adapted to our setting.

Remark 2.
By standard results, the functional is convex and lower semicontinuous on L n/(n−1) (Ω) (see also the next Corollary cor:lsc), while the functional is continuous on L n/(n−1) (Ω). However, it is not locally Lipschitz, unless n = 1, as we do not have a convenient estimate of |g(x, s)|.
Let us point out that we need to consider the functional f λ on a Lebesgue space such as L n/(n−1) (Ω) and not, e.g., on W 1,1 0 (Ω), because Ψ(x, ·) is not assumed to be strictly convex and, consequently, it is impossible to prove a Palais-Smale condition related to a norm which requires the strong convergence of ∇u.

Remark 3.
According to Remark 1, we have that g(x, u) u ∈ L 1 (Ω) implies g(x, u) ∈ L 1 (Ω). Therefore, we have u ∈ V u and W 1,1 Let us state our main result.
Since we are mainly interested in the principal part of the functional, in the next examples, we propose the same lower-order term, even if other choices are possible.
Concerning the principal part, since there is no upper bound on Ψ(x, ·), one can consider, in particular, cases with nonstandard growth conditions. Example 1. The assumptions of Theorem 1 are satisfied by a functional of the form for a.a. x ∈ Ω and ϑ : R n → R is convex, even and satisfies Let us point out that, in the case n = 2, a possible choice is with a very different behavior in the the two variables ξ 1 and ξ 2 .
Let us also point out that, if n ≥ 2, the functional is continuous on L n/(n−1) (Ω), but not locally Lipschitz, unless further summability conditions on a 1 are imposed.

Example 2.
The assumptions of Theorem 1 are satisfied by the functionals with a 1 as in the previous example and Principal parts of this form appear, for instance, in the study of strongly nonhomogeneous materials and non-Newtonian fluids (see, e.g., [10][11][12] and references therein).
Concerning the first case, let us recall that, by Young's inequality, one has and assumption (ΨG 3 ) follows.
Under a smoothness assumption on Ψ(x, ·), we have that each energy critical point is also a weak solution of the associated Euler-Lagrange equation. Proposition 1. Let λ > 0 and let u ∈ W 1,1 0 (Ω) be an energy critical point of f λ . Assume also that, for a.e. x ∈ Ω, the function Ψ(x, ·) is of class C 1 . Then and we have Moreover, we also have In Section 2, we recall the tools of nonsmooth critical point theory we need. In Section 3, we adapt some basic results from [13] to our setting. The main technical results are contained in Section 4, where we show how the nonsmooth critical point theory can be applied to a functional such as f λ . Since we believe that the approach can be useful also when the functional is not coercive, in Section 4, assumption (G 4 ) is replaced by more general hypotheses. Finally, in Section 5, we prove the results stated in the Introduction.
In the following, we will denote by · p the usual norm in L p . For every s ∈ R, we also set s ± = max{±s, 0} , T k (s) = min{max{s, −k}, k} .
Let X be a metric space endowed with the distance d. We denote by B δ (u) the open ball of center u and radius δ. We will also consider the set X × R endowed with the distance When f is real-valued and continuous, the next notion has been independently introduced in [15,16] and in [18], while a variant has been developed in [17]. By means of a suitable device, also the general case was considered in [15,16]. Here, we follow the equivalent approach of [14].

Definition 2.
For every u ∈ X with f (u) ∈ R, we denote by |d f |(u) the supremum of the σ's in [0, +∞[ such that there exist δ > 0 and a continuous map The extended real number |d f |(u) is called the weak slope of f at u.
When f is real-valued and continuous, the next result provides a simple estimate.

Remark 4.
Let X be an open subset of a normed space and let f : X → R be of class C 1 . Then, we have |d f |(u) = f (u) for all u ∈ X.
We say that f satisfies the Palais-Smale condition at level c ((PS) c , for short), if every (PS) c -sequence for f admits a convergent subsequence in X.
Now, let us see, following [15,16], how the case of a general f can be reduced, to some extent, to the continuous case, taking advantage of the function G f introduced in [19].
Define a function

Proof. See ([14] Proposition 2.3).
We aim to reduce the study of a general f to that of the continuous function G f . In view of the natural correspondence u ↔ (u, f (u)), a key point is to have a control on pairs Several results of critical point theory can be extended to the nonsmooth case, by means of the previous concepts. In view of our purposes, let us mention an extension of D.C. Clark's theorem (see [1,8] when f is smooth). When dealing with the weak slope |d f |(u), an auxiliary concept is sometimes useful. From now on in this section, we assume that X is a normed space over R and f : X → [−∞, +∞] is a function.
The next notion has been introduced in [14].

Definition 6.
For every u ∈ X with f (u) ∈ R, v ∈ X and ε > 0, let f 0 ε (u; v) be the infimum of r's in R such that there exist δ > 0 and a continuous map Let us recall that the function f 0 (u; ·) : X → [−∞, +∞] is convex, lower semicontinuous and positively homogeneous of degree 1.

Definition 7.
For every u ∈ X with f (u) ∈ R, we set Remark 6. If f is convex, then ∂ f agrees with the subdifferential of convex analysis. If f is locally Lipschitz, then f 0 and ∂ f agree with Clarke's notions [20], while, in general, The subdifferential we have recalled is suitably related to the weak slope because of the next result.

Compactness and Lower Semicontinuity
Throughout this section, we consider the more general situation in which has the form We assume that L : Ω × R × R n → R satisfies (L 1 ) for every (s, ξ) ∈ R × R n , the function {x → L(x, s, ξ)} is measurable and, for a.e.
In both cases, n ≥ 2 and n = 1, we infer that (u k ) is bounded in W 1,1 0 (Ω), hence convergent, up to a subsequence we still denote by (u k ), to some u in L 1 (Ω), and that L(x, u k , ∇u k ) ≥ 0 a.e. in Ω.
In the case n = 1, we extend u k , u with value 0 outside Ω, so that u k , u ∈ W 1,1 (R), and fix τ ∈ R \ Ω. If (t k ) is convergent to t in Ω, then According to the previous step, it follows that Let us point out two obvious consequences.

Corollary 1. The functional f is lower semicontinuous.
Corollary 2. Let c ∈ R and let (u k ) be a (PS) c -sequence for f such that sup k u k n n−1 < +∞ .
Then, there exist u ∈ W 1,1 0 (Ω) with f (u) ≤ c and a subsequence (u k j ) such that lim j u k j − u n n−1 = 0 .
On the other hand, if n = 1, it follows from (ΨG 2 ) that, for every R > 0, there existŝ for a.a. x ∈ Ω and all s ∈ R with |s| ≤ R .
Proof. Assume, first, that n ≥ 2. From Lemma 1, we infer that [g(x, u)(u − v)] + ∈ L 1 (Ω). Assume now, for a contradiction, that there exist t k → 0 + and z k → u in L n/(n−1) (Ω) satisfying Then, from Lemma 1 and the (generalized) Fatou lemma, we infer that and a contradiction follows. In the case n = 1, the proof is similar, taking into account assumption (ΨG 2 ).
To show assertion (b) and complete the proof of assertion (a), consider µ ∈ ∂ f (u).
Since Ψ(x, 0) = 0, we have that Therefore, Ψ(x, ∇T k (v)) ∈ L 1 (Ω) and we can choose T k (v) as the test function in (2), obtaining We also infer that On the other hand, we have and (T k (v)) is convergent to v in L n/(n−1) (Ω). Combining the lower semicontinuity of f 0 (u; ·) with Fatou's lemma, we infer that and the proof of assertion (a) is complete.
Taking into account Definition 7, we deduce that and assertion (b) also follows.
Then, u is a minimum of the convex functional on the linear space V u if and only if u is a minimum of the same functional on the convex set v ∈ W 1,1 0 (Ω) : Ψ(x, ∇v) ∈ L 1 (Ω) , [g(x, u) v] − ∈ L 1 (Ω) .
Proof. Similar to before, we also have g(x, u) ∈ L 1 (Ω). Assume now that u is a minimum of the convex functional on the linear space V u and let v ∈ W 1,1 0 (Ω) with Ψ(x, ∇v) ∈ L 1 (Ω) and [g(x, u) v] − ∈ L 1 (Ω). Since Similar to before, we also have that The converse is easily seen.
It follows that f λ is bounded from below and that, for every c ∈ R, the set u ∈ W 1,1 0 (Ω) : f λ (u) ≤ c is bounded in W 1,1 0 (Ω), hence in L n/(n−1) (Ω). Combining this fact with Corollary 2, we infer that f λ satisfies (PS) c for all c ∈ R. Moreover, from Theorem 6, we deduce that f λ satisfies (epi) c for all c ∈ R.
By Theorem 3, we have that 0 ∈ ∂ f λ (u j ) for all j = 1, . . . , m. From Theorem 5 and Proposition 4, we deduce that each u j is an energy critical point of f λ .
Proof of Proposition 1. It is a particular case of Proposition 5 with g replaced by λg.
Author Contributions: All the authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.