A variant of Clark’s theorem and its applications for nonsmooth functionals without the global symmetric condition


 We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].


Introduction
The Clark's theorem is a important result in critical point theory (see [4,8]). Using this theorem for the even coercive functional, the existence of a sequence of negative critical values tending to is obtained. Speci cally, in [8], Heinz obtained a variant of the Clark theorem as follows: Clark Theorem (see [21]). Let X be a Banach space and assume J ∈ C (X) satis es, J( ) = , the (P-S) conditions, is bounded from below and even. For any positive integer k, there exists a k-dimensional subspace X k of X and β k > such that sup X k ∩S β k J < , here S β = {u ∈ X : u = β}, then there exists a sequence of negative critical values for J tending to . This Clark's theorem was improved by Kajikiya in [9] and Liu-Wang in [18], under the same conditions as in the above Clark's theorem, they showed the critical points of J also tend to in X. We remark that Liu-Wang also studied the existence of periodic solutions for sub-linear Hamiltonian systems and showed a new version of the Clark's theorem for non-smooth functionals. Very recently, in [3] Chen-Liu-Wang showed a version of the Clark's theorem without the Palais-Smale conditions ((P-S) conditions). And then they studied the existence of in nitely many solutions for a degenerate quasi-linear elliptic operator and a second-order Hamiltonian system via their abstract theory.
However, all those versions of Clark's theorem references to above rely on the symmetric condition about the Euler-Lagrange functional. In [10], Kajikiya established the existence of in nitely many critical points about C functional without the global symmetric condition. As applications, they obtained the existence of in nitely many solutions of the sub-linear elliptic equation with a small perturbation. We note that since the *Corresponding Author: Chen Huang, College of Mathematics and Informatics, FJKLMAA, Fujian Normal University, Fuzhou, 350117, PR China, E-mail: chenhuangmath111@163.com perturbation term only satis es continuity, the Euler-Lagrange functional corresponding to the sub-linear elliptic equation may not be even.
But for some quasi-linear elliptic problems with continuous perturbations, here the problems do not have a C variational formulation and do not satisfy the global symmetric condition. For these reasons, both the classical Clark's Theorem in [8] and the abstract result in [10] cannot be applied directly. In this situation, we need develop a new non-smooth variational method based on the Clark's theorem.
In order to state the new variant of Clark's theorem, we rstly give the following assumption: Condition (I). Let X be an in nite dimensional Banach space and E be dense subspace of X. For any ε ∈ [ , ], let Iε be a continuous functional de ned on X which is E-di erentiable. Iε satis es (I ) − (I ) below.
( (I ): For any u ∈ X \ { } there exists a unique t(u) > such that I (tu) < if < |t| < t(u) and I (tu) ≥ if |t| ≥ t(u). In order to explain some concepts in Condition (I), we recall some de nitions as follows: (1) for all u ∈ X and φ ∈ E the derivative of J in the direction φ at u exists and will be denoted by D E J(u), φ :

De nition 1.2. The slope of an E-di erentiable functional J at u denoted by |D E J(u)| is an extended number in
A point u ∈ X is said to be a critical point of J at the level c if |D E J(u)| = and J(u) = c. Secondly, we give an another direct proof about Corollary 1.1 in Section 3. Our method is based on the approach developed by Degiovanni-Lancelotti [6]. By this approach, Li-Liu [11] considered a similar perturbation for semilinear elliptic equation in a bounded domain. Since our problems do not have a C variational formulation, the method in [11] cannot be applied directly.
Finally, as an application of Theorem 1.1 and Corollary 1.1, we consider the following quasi-linear problem Besides, a ij (x, u) satis es (a): a ij ∈ C (R N × [−a, a], R), a ij is even, a ij = a ji , for all x ∈ R N and |s| ≤ a, Ds a ij (x, s)s ≥ and there Quasi-linear elliptic equation of the form (1.1) contains the quasi-linear Schrödinger equation which corresponds to the special case a ij (x, t) = ( + t )δ ij . The problem (1.2) arose in several models of physical phenomena, such as super uid lms in plasma physics (see e.g. [1,2,19]). And it has received considerable attention in mathematical analysis in the last twenty years (see [5,12,20,22,23]).
Since the variational functional of the quasi-linear problem (1.1) is merely continuous. Even though ε = , limited work has been done in the general form of the quasi-linear problem (1.1). In [13], a least energy signchanging solution of (1.1) with ε = is obtained via the Nehari manifold method. Multiple solutions for (1.1) with ε = was rst proved in [16], where a 4-Laplacian perturbation term is added to (1.1) so that the associated functionals are well-de ned on W , (Ω). Then in [15], they obtained multiplicity of sign-changing solutions for general form of the quasi-linear problem (1.1) with ε = . This idea is further developed in [17], where treated the critical exponent case giving new existence results. The previous mentioned four results yield only with the power range f (x, u) = |u| s− u ( < s ≤ · * ). The case < s < is investigated in [18], by using the variants of Clark's theorem, the quasi-linear problem (1.1) with ε = has a sequence of solutions with L ∞ -norms tending to zero. For < s < , less results are known, by using the perturbation approach and the invariant sets approach, in [7] Jing-Liu-Wang showed the problem (1.1) with ε = has at least six solutions.
When ε ≠ , under our assumptions on g, the Euler-Lagrange functional corresponding to (1.1) may be not even with respect to u. In Section 4, for |ε| small enough, implying Theorem 1.1, we prove that the generalized quasi-linear elliptic problem with small perturbations (1.1) has in nitely many solutions.
Next, we give our second main result.   This paper is organized as follows. In Section 2, we give the proofs of the abstract result Theorem 1.1 and Corollary 1.1. Then we give the another direct proof about Corollary 1.1 in Section 3. As an application of Theorem 1.1, we prove Theorem 1.2 in Sections 4.
In what follows, C denotes positive generic constants.

Proof of Theorem 1.1
In this section, we shall prove Theorem 1.1. To this end, we use the following topological lemma, which is an analogue of a result of [10]. Throughout this section, X is an in nite dimensional Banach space and E be dense subspace of X. For any ε ∈ [ , ], functionals Iε are an E-di erentiable functional de ned on X and satis es (I ) − (I ). (2) Let v be mentioned in Lemma 2.1-(1), there exists a δ > such that

Proof. Notice that condition (I ) means that
which plays a key role in the proof of this lemma. The details in the proof see also [10].
Recall the de nitions in Section 1, as follows: Using these de nitions and the condition (I ), we have the following lemma.

Lemma 2.2.
For any k ∈ N, there exists an α k ∈ A k such that Proof. Let k be a xed positive integer. For any α ∈ A k , by the condition (I ), there exists t(α) such that Then we can take α k = t(α) α, which satis es For any k ∈ N, it follows from Lemma 2.2 and the de nition of d k that Next we suppose that there exists a positive integer k such that d k < d k+ < . De ne S k+ Using the above de nitions and Lemma 2.1, we show the next fundamental lemma holds. We use the same method as in Kajikiya [10]. For the completeness of the article, we give a detailed proof as follows.

Lemma 2.3. There exists an f k+
Proof. Let d k and r > be de ned in (H ). From the de nition of d k , we can choose an α ∈ A k satisfying Then take M = α(S k ). It follows from (2.2) that M is compact and ∉ M. From Lemma 2.1-(2), there exist v ∈ X and δ > such that Next, we denote Then for x ∈ S k+ + , we take We only need extend the continuous function f k+ (x) onto S k+ as an odd mapping f k+ (x). Then f k+ ∈ A k+ ∩ H k+ and (2.3) imply that I ( f k+ (x)) = I (f k+ (x)) < , for x ∈ S k+ + .

Lemma 2.4. Each d k is a critical value of I (u) and
Before completing the proof of Lemma 2.4, we need the following non-smooth deformation lemma and a notion of genus (see [14,21]): Lemma 2.5 (The First Deformation Lemma). [14] Assume J is an E-di erentiable functional de ned on X and satis es the (P-S) conditions. For some c ∈ R, let N be a neighborhood of Kc = {u ∈ X : Then there exists a deformation map σ : With the help of lemma 2.5, and using the standard method as in Rabinowitz's argument [21] (see Proposition 9.33), we show the following lemma holds.
Lemma 2.6. For all k ∈ N, e k is a critical value of I (u) and e k → as k → ∞.
Proof. Under the condition (I ), we know that e k > −∞.
Remark 2.1 means e k ≤ d k < . Since Γ k+ ⊂ Γ k , it follows that e k ≤ e k+ . Then from the above facts, we have lim k→∞ e k := e∞ ≤ .
In contradiction to Lemma 2.6, we suppose that Next we use the following notation: By (P-S) conditions on I , we have K is compact. It is clear that K is symmetric. Due to (2.4), we have ∉ K.
Then by the properties of genus, there exists a δ ′ > small enough such that By Lemma 2.5 with c = e∞, there exists δ > such that Now we x an integer j ∈ N such that e∞ − δ < e j . (2.6) By the de nition of e i+j , there exists P ∈ Γ i+j such that Let Q = P \ N δ ′ (K), then from (2.5) and (2.7), we have From (2.6), (2.8) and the above fact, we get This is a contradiction.
The proof of Lemma 2.4. From the de nition of d k , we know that for any ϵ > , there exists an α k ∈ A k such that sup Here we x ϵ = δ which is mentioned in Lemma 2.5. Assume to the contrary that the conclusions are false. On the other hand, it is straightforward to show that σ( , α k ) ∈ A k . Then by the de nition of d k , it implies sup x∈S k I (σ( , α k (x))) ≥ d k , which contradicts (2.9). Hence d k is a critical value of I . Next we shall prove that d k → as k → ∞. Due to e k ≤ d k < , it is enough to show the convergence of e k to zero. This fact follows from Lemma 2.6. The proof is complete. Now, we are ready to prove the variant of Clark's theorem. The proof of Theorem 1.1. Fixed a positive integer k, such that d k < d k+ < and d k + r < d k+ , for some r > From Lemma 2.3 and I (u) is even on u, we have Choose ε k+ ∈ ( , ] so small that For all ε ∈ [ , ε k+ ], de ne b k+ (ε) := inf On one hand, from condition (I ), it implies b k+ (ε) ≤ max S k+ On the other hand, for any h ∈ H k+ xed, denote the odd extension of h on S k+ by h, then h ∈ A k+ . Since I (u) is even, it holds that max S k+ Then max Taking the in mum on h ∈ H k+ in the above inequality, we have Then, from (2.10) and ( . ), it implies Next we shall prove b k+ (ε) is critical value of Iε. Assume to the contrary that the conclusions are false. b k+ (ε) is a regular value. Then by Lemma 2.5 with c = b k+ (ε) and c − δ = d k + r + ψ(ε), we have an δ ∈ ( , δ) and σ : [ , ] × X → X satisfying the conditions below: By the de nition of b k+ (ε), there exists an h ∈ H k+ such that max S k+ By the deformation property (i), we have (2.12) Since h ∈ H k+ , we get From this, we have Thus σ( , h (x)) satis es (H ) and (H ) and then Then, by the de nition of b k+ (ε), we obtain max S k+ For any δ > and k ∈ N xed. Let n(i) (i ∈ { , , · · · , k}) be a increasing positive integers sequence, such that −δ < d n( ) and d n(i) ≤ c n(i) < d n(i+ ) for i ∈ { , , · · · , k}.

The another proof of Corollary 1.1
In this section, we shall give the another proof of Corollary 1.1. To this end, let us recall some notions and facts from Degiovanni-Lancelotti (see [6]). And set I b = {u ∈ X : I(u) ≤ b}.

and such that A is a strong deformation retract of B.
Let I be a merely continuous functional in X. The next lemma shows the main property of essential values. In what follows, let I be a E-di erentiable functional de ned on X, then by the following lemma, we show the relationship between essential values and critical values of I.

Lemma 3.2. Let functional I be E-di erentiable functional de ned on X and c be an essential value of I. If (P-S) conditions hold for I, then c is a critical value of I.
Proof. By contradiction, let us assume that c is not a critical value of I. By (P-S) conditions of I, there exist positive constants ε and d such that Then let a, b ∈ (c − ε, c + ε) with a < b. By Lemma 2.5, there exists a deformation map σ : [ , ] × X → X such that σ( , u) = u, I(σ(t, u)) ≤ I(u), if u ∈ I b , then σ( , u) ∈ I a , and if u ∈ I a , then σ(t, u) = u.
This means I a is a strong deformation retract of I b , so that the pair (I b , I a ) is trivial, which is a contradiction since c is an essential value of I.
The idea of the proof of Corollary 1.1 is taken from [11]. But our abstract result extends Theorem 1.2 in [11] by relaxing the C assumption of I. Moreover, our abstract result is powerful in application such as quasi-linear elliptic problems (see Section 4).
The proof of Corollary 1.1. Set S ∞ = {u ∈ X : u = } and O = {tu : < t < t(u), u ∈ S ∞ }, where for u ∈ S ∞ , t(u) is mentioned in condition (I ). From the above de nitions, we rst show O is contractible, which will be used in the sequel. De ne G = {t(u)u/ : u ∈ S ∞ }. Consider g : S ∞ → G as g(u) = t(u)u/ . Then the inverse of g is given by g − : G → S ∞ , g − (u) = u/ u and both g and g − are continuous. It implies G is homeomorphic to S ∞ . On the other hand, G is a strong deformation retract of O. Thus O is contractible since S ∞ is contractible.
Next, for k ∈ N, let S k− be the unit sphere in R k and de ne It is easy to see p ≤ p ≤ · · · ≤ p k . For any k ∈ N, by Lemma 3.1 and 3.2, there exists ε k > such that if ≤ ε ≤ ε k then p k ≤ max Recall the de nition of e k in De nition 2.2. By {h(S k− ) : h ∈ H k } ⊂ γ k , it implies that p k ≥ e k . From Lemma 2.6, we see that e k → as k → ∞. So p k → as k → ∞. Finally, we set E = {c < : c is an essential value of I }. It su ces to prove E ≠ ∅ and sup E = . By contradiction, there exists k ∈ N such that p k < p k+ and [p k , ) ∩ E = ∅. Then there exist constants α ′ , a, α ′′ such that p k < α ′ < a < α ′′ < p k+ .
By the de nition of p k , we can choose h ∈ H k satisfying max , we know β < . Then there exist constants b and β ′ such that β < b < β ′ < .
Due to [p k , ) ∩ E = ∅, the pair (I b , I a ) is trivial. This means that we can choose two closed subsets A and B of X satisfying and there exists a deformation map σ : Then we consider h ′′ (x) = σ( , h ′ (x)) which satis es Denote the odd extension of h ′′ on S k by h * , which satis es I (h * (S k )) ⊂ I α ′′ . However this contradicts the de nition of p k+ , i.e. p k+ ≤ max Consequently, there exists {c k } ⊂ E such that c < c < c < · · · < c k < , and c k → as k → ∞.
For any k ∈ N, by Lemma 3.1 and 3.2, there exists ε k > such that if |ε| ≤ ε k then Iε has at least k distinct critical points with negative critical values.

Proof of Theorem 1.2
In this section, we shall prove Theorem 1.2. For this purpose, we rstly make some modi es. Motivated by the similar modi es in [10,18], we make use of the following approach: From condition (f ), decreasing a if necessary, we may assume For xed a > . Let η ∈ C ∞ (R, [ , ]) be a cut-o even function such that Using this cut-o function η, we consider the following modi ed functions: where C > is a positive constant mentioned in condition (a). Next, we give some properties for f , g and a ij . Proof. From the de nition of f (x, u) and g(u), it is easy to show that (f ′ ), (f ′ ), (f ′ ) and (f ′ ) hold. To obtain (f ′ ) a little manipulation is needed. For < u < a and x ∈ R N , from the de nition of η, we consider On the other hand, ∂ ∂u (u − F(x, u)) can also be denoted by F(x, u)).
Combing the above two equations and the fact that u f (x, u)− F(x, u) is even on u, we see that f (x, u) satis es condition (f ′ ).
Lemma 4.2. a ij (x, u) satis es (a ′ ): a ij ∈ C (R N × R, R), a ij is even, a ij = a ji and there exist two constants C and C such that C |ξ | ≤ a ij ξ i ξ j ≤ C |ξ | ; (a ′ ): there exists some C > , such that Proof. It is clear that a ij satis es (a ′ ) and (a ′ ). We now verify (a ′ ). Observe the relation, Using condition (a) and the de nition of η, we get We set X := {W , (R N ) : R N V(x)u dx < ∞} in which the norm is given by and E := X ∩ L ∞ (R N ). For u ∈ X, we consider the following modi ed functional corresponding to (1.1), Now by Lemma 4.1, 4.2, Iε is well de ned in X. And it is easy to say functionals Iε are E-di erentiable in the directions φ ∈ E. Next, we verify the (P-S) conditions for functional Iε. Proof. For any xed u ∈ X and ε ∈ [ , ], we have Thus, and then Iε(u) is coercive and bounded below. Let (εn , un) ∈ [ , ] × X be any sequence such that Iε n (un) → c and |D E Iε n (un)| → .
Then {un} and {εn} are bounded. Therefore, there exists a subsequence of {εn} converges to ε and a subsequence of {un} converges to u weakly in X and a.e. on R N . Next, we shall show this convergence becomes a strong one.
Step 1. u is critical point of Iε. Take T > a, and de ne Choose φ ∈ E, φ ≥ and ψn = φ exp(−Hu T n ) where H > large enough such that −H a ij + Ds a ij is negatively de nite. Obviously, ψn can be seen as a test function in D E Iε(un), ψn → , that is Ds a ij (x, un)D i un D j un ψn dx where we used the Fatou's Lemma and the lower semi-continuity. Thus, for all φ ∈ E, φ ≥ , we have We can choose φ = ϕ exp(Hu T ) in (4.2) for ϕ ∈ E and ϕ ≥ we obtain Ds a ij (x, u)D i uD j uϕdx Similarly, by choosing ψ = φ exp(Hu T n ), we can get an opposite inequality. Hence, u is a critical point of Iε.
Step 2. We shall show the facts that From {un} converges to u weakly in X and a.e. on R N , we know For any R > , by the Young inequality and the Hölder inequality, we have Recall that for |u| ≥ a it implies g(u) = . From this and Lebesgue Dominated Convergence Theorem, we have Step 3. un → u strongly in X. Due to u T n X ≤ C with C independent of T and n, we get Taking T → +∞, we obtain that Similarly, we have | D E Iε(u), u | = . From the de nition of a ij and condition (a ′ ), we obtain Then which implies that un − u X → as n → ∞. Proof. For any u ∈ X \ { } xed. For t > , we consider and Then observe the relation, I (tu) = t (P(t) + Q(t)).
De ne J(t) := P(t) + Q(t). By condition (f ′ ) and (a ′ ), we have and Ds a ij (x, tu)uD i uD j udx > for t > .
Take ϵ > small enough such that µ(Dϵ) > and Dϵ := {x ∈ R N : ϵ < |u(x)| < /ϵ}, where µ denotes the Lebesgue measure of R N . Due to F(x, u) ≥ , we can estimate the function J(t) as follows: By condition (f ′ ) in Lemma 4.1, it implies that lim t→ + J(t) = −∞. Hence, J(t) < for t > small enough. On the other hand, for t > large, by ≤ r < we have Accordingly, for xed u ∈ X \ { }, J(t) has a unique zero t(u) such that Thus the above result holds for I (tu).
Before proving the multiplicity of the critical points for Iε, by the Moser's iteration (see Lemma 3.7 in [7]) and condition (f ′ ), we give some priori estimates. Proof. Notice that from conditions (VW), (K) and (f ′ ), for u ∈ X, we have Let u ∈ X be a critical point of Iε. Using the equation, for any ψ ∈ X ∩ L ∞ (R N ), we obtain Ds a ij (x, u)D i uD j uψdx Choosing ψ = |u T | η u T in (4.4), where η > (will be xed by some constants) and u T is de ned in Lemma 4.3, we obtain Ds a ij (x, u)D i uD j u|u T | η u T dx Combining the second, third term in the left side of the above equation is nonnegative and (4.3), we obtain Without loss of generality, for xed u ∈ X, we have the following estimate (4.6) Since we can assume R N |u| η+r ≥ R N |u| η+ , the case R N |u| η+r < R N |u| η+ is similarly treated by the following arguments.
On the other hand, using the Sobolev inequality, we deduce where we used that S = inf{ R N |Dv| dx : It is easy to see that η k → +∞ as k → +∞. We may assume C > , then for i < j, it follows that ( C(η i + )) (η j +r)/(η j + ) ≤ C(η i + ).
By Moser's iteration method we have u η k+ +r ≤ exp  From the above two equations, it follows that Ds a ij (x, u )u D i u D j u dx Since u ∈ X, conditions (a ′ ) and (f ′ ) imply that u ≡ . On the other hand, from un X ≥ b > and un → u in X, we have u X ≥ b > , which contradicts the fact that u ≡ . The proof is complete.
From the above two lemmas, it is straightforward to show the following corollary. Now, we are ready to prove the second main result. The proof of Theorem 1.2. Without loss of generality, we assume ε > . Because the case ε < is same studied by replacing g(u) by − g(u).
Finally, due to the arbitrariness of δ, take < δ < δ( a C * ), by Corollary 4.1, the original problem (1.1) has at least k solutions whose L ∞ -norms are less than a .