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

Definition 1.1

A continuous functional J is said to be E-differentiable if

(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_EJ(u), φ:

(2) The map (u, φ) → D_EJ(u), φ satisfies:

1. D_EJ(u), φ is linear in φ ∈ E,

2. D_EJ(u), φ is continuous in u, that is D_EJ(u_n), φ → D_EJ(u), φ as u_n → u in X.

Definition 1.2

The slope of an E-differentiable functional J at u denoted by |D_EJ(u)| is an extended number in [0,∞]:

A point u ∈ X is said to be a critical point of J at the level c if |D_EJ(u)| = 0 and J(u) = c.

Definition 1.3

I_ε(u) is said to satisfy (P-S) conditions uniformly on ε if a sequence (ε_n , u_n) ∈ [0, 1]×X satisfies that

then (ε_n , u_n) has a convergent subsequence.

We now introduce the variant of Clark’s theorem.

Theorem 1.1

Assume that Condition (I) holds. Denote

Let k ∈ N \ {0} satisfying d_k < d_k+1. Then there exist two constants ε_k+1, c_k+1 such that 0 < ε_k+1 ≤ 1, d_k+1 ≤ c_k+1 < −ψ(ε) for ε ∈ [0, ε_k+1] and for any ε ∈ [0, ε_k+1], I_ε has a critical value in the interval [d_k+1 − ψ(ε), c_k+1 + ψ(ε)].

From the above abstract theorem, we have the following corollary.

Corollary 1.1

Assume the condition (I) holds. Then, for any k ∈ N, there exists an ε_k > 0 such that when 0 ≤ ε ≤ ε_k, functional I_ε has at least k distinct critical points with negative critical values.

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 C¹ 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

where N≥3,Dsaij(x,u)=∂∂uaij(x,u),Diu=∂∂xiu(x)andDju=∂∂xju(x). Denoted F(x,u)by∫0uf(x,s)ds. For given a > 0, let f (x, u) ∈ C(ℝ^N × [−a, a]) and satisfy (f₁)− (f₄) below

(f₁): f (x, −u) = −f (x, u) for x ∈ ℝ^N and |u| ≤ a;

(f₂): uf (x, u) − 2F(x, u) < 0when 0 < |u| < a and x ∈ ℝ^N;

(f₃): |f (x, u)| ≤ C|u|^r−1 for x ∈ ℝ^N and |u| ≤ a, where r ∈ [1, 2);

(f4):limu→0(minx∈RNu−2F(x,u))=∞.

Besides, a_ij(x, u) satisfies

(a): a_ij ∈ C¹(ℝ^N × [−a, a], ℝ), a_ij is even, a_ij = a_ji, for all x ∈ ℝ^N and |s| ≤ a, D_sa_ij(x, s)s ≥ 0 and there exist C₀, C₁ such that C₀|ξ|² ≤ a_ijξ_iξ_j ≤ C₁|ξ|².

And V, K ∈ C(ℝ^N , ℝ) satisfy:

(V): V(x)⁻¹ ∈ L^r/(2−r)(ℝ^N) and there exists a constant V₀ such that

(K): K(x) ∈ L¹(ℝ^N) ∩ L^∞(ℝ^N).

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) = (1 + 2t²)δ_ij. The problem (1.2) arose in several models of physical phenomena, such as superfluid films 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 = 0, limited work has been done in the general form of the quasi-linear problem (1.1). In [13], a least energy sign-changing solution of (1.1) with ε = 0 is obtained via the Nehari manifold method. Multiple solutions for (1.1) with ε = 0 was first proved in [16], where a 4-Laplacian perturbation term is added to (1.1) so that the associated functionals are well-defined on W01,4(Ω). Then in [15], they obtained multiplicity of sign-changing solutions for general form of the quasi-linear problem (1.1) with ε = 0. 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−2u (4 < s ≤ 2 · 2^*). The case 1 < s < 2 is investigated in [18], by using the variants of Clark’s theorem, the quasi-linear problem (1.1) with ε = 0 has a sequence of solutions with L^∞-norms tending to zero. For 2 < s < 4, 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 = 0 has at least six solutions.

When ε ≠0, 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 infinitely many solutions.

Next, we give our second main result.

Theorem 1.2

Let g(u) be continuous on [−a, a]. Assume that conditions (a), (V), (K) and (f₁)-(f₄) hold. Then, for any k ∈ N and any b > 0, there exists an ε₀(k, b) > 0such that when |ε| ≤ ε₀(k, b) the problem (1.1) possesses at least k distinct solutions whose L^∞-norms are less than b.

Remark 1.1

We do not need any growth condition on the nonlinear term f at infinity. Thus there exist some supercritical growth examples satisfying our conditions (f₁)− (f₄):

1. f (x, u) = |u|^r−1sgn u with r ∈ (1, 2);

2. f (x, u) = |u|^r−1sgn u + |u|ⁱ⁻¹sgn u, where 1 < r < 2 , i ≥ 2 ∗ := 2 N N − 2 .

Remark 1.2

Since our theorem does not need any growth or odd condition on the perturbation term g. The Euler-Lagrange functional I_ε(u) may not satisfy symmetrical condition. By applying our abstract results, for |ε| small enough, I_ε(u) still possesses infinite many critical points.

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.

Definition 2.1

Let M be a compact subset of X such that 0 ∉M. Denote

1. ℝM := {ru : r ∈ ℝ, u ∈ M};

2. O := {u ∈ X : I₀(u) < 0}.

Lemma 2.1

[Topological Lemma] (1) There exists v ∈ X such that

(2) Let v be mentioned in Lemma 2.1-(1), there exists a δ₀ > 0 such that

Proof. Notice that condition (I₅) means that

which plays a key role in the proof of this lemma. The details in the proof see also [10].

Recall the definitions in Section 1, as follows:

and

Using these definitions and the condition (I₅), 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 fixed positive integer. For any α ∈ A_k, by the condition (I₅), there exists t(α) such that

Then we can take αk=t(α)2α, which satisfies

For any k ∈ N, it follows from Lemma 2.2 and the definition of d _k that

Next we suppose that there exists a positive integer k such that d _k < d_k+1 < 0. Define

(ℌ₁) h(−x) = −h(x) for x ∈ S^k.

(ℌ₂) I₀(h(x)) < d_k + r for x ∈ S^k, where the constant r > 0 small enough such that d_k + r < d_k+1.

Using the above definitions 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+1 ∈ A_k+1 ∩ℌ_k+1 such that

Proof. Let d _k and r > 0 be defined in (ℌ₂). From the definition 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 0 ∉M. From Lemma 2.1-(2), there exist v ∈ X and δ₀ > 0 such that

This means

Next, we denote

Then for x∈S+k+1, we take

We only need extend the continuous function f_k+1(x) onto S^k+1 as an odd mapping f˜k+1(x).Thenf˜k+1∈Ak+1∩ ℌ_k+1 and (2.3) imply that

Lemma 2.4

Each d _k is a critical value of I₀(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-differentiable functional defined on X and satisfies the (P-S) conditions. For some c ∈ ℝ, let N be a neighborhood of K_c = {u ∈ X : J(u) = c, |D_EJ(u)| = 0}. Then there exists a deformation map σ : [0, 1] × X → X and δ > δ > 0 such that

1. If J(u) ≤ c − δ, then σ(t, u) = u.

2. If J(u) ≤ c + δ and u ∉N, then J(σ(1, u)) ≤ c − δ.

3. If J(·) is even functional, then σ(t, ·) is odd mapping.

4. If c is a regular value of J with J(u) ≤ c + δ, then J(σ(1, u)) ≤ c − δ.

Definition 2.2

Denote Γ = {A ⊂ X \ {0} : A is closed, −A = A}. Let A ∈ Γ, define a genus γ(A) of A by

If such a minimum does not exist, then we define γ(A) = +∞. Moreover, set γ(∅) = 0. For all k ∈ N, let Γ_k = {A ∈ Γ : γ(A) ≥ k}. We define

Remark 2.1

Due to the Borsuk-Ulam theorem, we have γ(S^k) = k +1. Then from the property of genus, for any α ∈ A_k we have γ(α(S^k))≥ k + 1. Besides, since d_k < 0, we can suppose 0 ∉ α(S^k). Thus, we deduce A_k ⊂ Γ_k and then e_k ≤ d_k < 0.

With the help of lemma 2.5, and using the standardmethod 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 I₀(u) and e_k → 0 as k →∞.

Proof. Under the condition (I₁), we know that

Remark 2.1 means e_k ≤ d_k < 0. Since Γ_k+1 ⊂ Γ_k, it follows that e_k ≤ e_k+1. Then from the above facts, we have

In contradiction to Lemma 2.6, we suppose that

Next we use the following notation:

By (P-S) conditions on I₀, we have K is compact. It is clear that K is symmetric. Due to (2.4), we have 0 ∉ K.

Then by the properties of genus, there exists a δ^′ > 0 small enough such that

where N_δ′ (K) = {u ∈ X : dist(u, K) < δ^′}.

By Lemma 2.5 with c = e_∞, there exists δ > 0 such that

Now we fix an integer j ∈ N such that

By the definition 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

Since γ(σ(1,Q))≥γ(Q)≥γ(P)−γ(Nδ′(K)‾)≥j, 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 definition of d _k, we know that for any ϵ > 0, there exists an α_k ∈ A_k such that

Here we fix ϵ = δ which is mentioned in Lemma 2.5. Assume to the contrary that the conclusions are false. d _k is a regular value. Using Lemma 2.5 with c = d_k, there exists σ : [0, 1] × X → X satisfying

On the other hand, it is straightforward to show that σ(1, α_k) ∈ A_k. Then by the definition of d _k, it implies

which contradicts (2.9). Hence d _k is a critical value of I₀.

Next we shall prove that d _k → 0 as k → ∞. Due to e_k ≤ d _k < 0, 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

From Lemma 2.3 and I₀(u) is even on u, we have

Choose ε_k+1 ∈ (0, 1] so small that

For all ε ∈ [0, ε_k+1], define

On one hand, from condition (I₁), it implies

On the other hand, for any h ∈ ℌ_k+1 fixed, denote the odd extension of honSk+1byh‾, then h‾∈Ak+1.

Since I₀(u) is even, it holds that

Then

Taking the infimum on h ∈ ℌ_k+1 in the above inequality, we have

Then, from (2.10) and (2.11), it implies

Next we shall prove b_k+1(ε) is critical value of I_ε. Assume to the contrary that the conclusions are false. b_k+1(ε) is a regular value. Then by Lemma 2.5 with c=bk+1(ε)andc−δ‾=dk+r+ψ(ε), we have an δ∈(0,δ‾) and σ : [0, 1] × X → X satisfying the conditions below:

1. If I_ε(u) ≤ b_k+1(ε) + δ, then I_ε(σ(1, u)) ≤ b_k+1(ε) − δ.

2. If I_ε(u) ≤ d_k + r + ψ(ε), then σ(1, u) = u.

By the definition of b_k+1(ε), there exists an h₀ ∈ ℌ_k+1 such that

By the deformation property (i), we have

Since h₀ ∈ ℌ_k+1, we get

From this, we have

Thus σ(1, h₀(x)) satisfies (ℌ₁) and (ℌ₂) and then

Then, by the definition of b _k+1(ε), we obtain

which contradicts (2.12).

The proof of Corollary 1.1

From Theorem 1.1, there exist sequences {ε_k+1}, {d_k+1} and {b_k+1(ε)} such that b_k+1(ε) is a critical value of I_ε for ε ∈ [0, ε_k+1] and

For any δ > 0 and k ∈ N fixed. Let n(i) (i ∈ {1, 2, · · · , k}) be a increasing positive integers sequence, such that

Then for all i ∈ {1, 2, · · · , k}, there exists ε_k > 0 small enough such that

Combining (2.13) and (2.14), for all ε ∈ [0, ε_k], we have

which means I_ε has at least k distinct critical values.

Definition 3.1

Areal number c is said to be an essential value of I, if for every ε > 0there exist a, b ∈ (c−ε, c+ε) with a < b such that the pair (I^b, I^a) is not trivial.

Definition 3.2

Let a, b ∈ ℝ∪{−∞, +∞} with a ≤ b. The pair (I^b, I^a) is said to be trivial, if for any neighborhood [α^′, α^′′] of a and [β^′, β^′′] of b, there exist two closed subsets A and B such that I^α′ ⊂ A ⊂ I^α′′, I^β′ ⊂ B ⊂ I^β′′ 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.

Lemma 3.1

[see Theorem 2.6 in [6]] Let c be an essential value of I. Then for every ε > 0 there exists δ > 0 such that every J ∈ C(X, ℝ) with sup |J(u) − I(u)| < δ admits an essential value in (c − ε, c + ε). _u∈_X

In what follows, let I be a E-differentiable functional defined 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-differentiable functional defined 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 : [0, 1] × X → X such that

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 C¹ 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 = 1} and O = {tu : 0 < t < t(u), u ∈ S^∞}, where for u ∈ S^∞, t(u) is mentioned in condition (I₅). From the above definitions, we first show O is contractible, which will be used in the sequel. Define G = {t(u)u/2 : u ∈ S^∞}. Consider g : S^∞ → G as g(u) = t(u)u/2. Then the inverse of g is given by g⁻¹ : G → S^∞, g⁻¹(u) = u/‖u and both g and g⁻¹ 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−1 be the unit sphere in ℝ^k and define

and

It is easy to see p₁ ≤ p₂ ≤ · · · ≤ p_k. For any k ∈ N, by Lemma 3.1 and 3.2, there exists ε_k > 0 such that if 0 ≤ ε ≤ ε_k then

Recall the definition of e_k in Definition 2.2. By {h(S^k−1) : h ∈ H _k}⊂ γ_k, it implies that p_k ≥ e_k. From Lemma 2.6, we see that e_k → 0as k →∞. So p_k → 0as k →∞.

Finally, we set E = {c < 0 : c is an essential value of I₀}. It suffices to prove E ≠ ∅ and sup E = 0. By contradiction, there exists k ∈ N such that p _k < p_k+1 and [p_k, 0) ∩ E = ∅. Then there exist constants α^′, a, α^′′ such that

By the definition of p _k, we can choose h ∈ H _k satisfying

Let S + k = { x : x = ( x ′ , x k + 1 ) , x ′ ∈ R k , x k + 1 ≥ 0 , | x | = 1 } . From O is contractible, it implies that h can extend to h′∈C(S+k,O). Take β=max{I0(h′(x)):x∈S+k}. Since h′∈C(S+k,O), we know β < 0. Then there exist constants b and β^′ such that

Due to [p_k, 0) ∩ E = ∅, the pair (I^b₀ , I^a₀) is trivial. This means that we can choose two closed subsets A and B of X satisfying

and there exists a deformation map σ : [0, 1] × B → B such that

Then we consider h^′′(x) = σ(1, h^′(x)) which satisfies

Denote the odd extension of h^′′ on S^k by h^*, which satisfies I0(h∗(Sk))⊂I0α′′. However this contradicts the definition of p _k+1, i.e.

Consequently, there exists {c_k}⊂ E such that

For any k ∈ N, by Lemma 3.1 and 3.2, there exists ε_k > 0 such that if |ε| ≤ ε_k then I_ε has at least k distinct critical points with negative critical values.

Lemma 4.1

f˜(x,u)andg˜(u) are continuous functions defined on ℝ^N × R and satisfy the conditions below

(f1′):f˜(x,−u)=−f˜(x,u) for (x, u) ∈ ℝ^N × ℝ;

(f2′):uf˜(x,u)−2F˜(x,u)<0 when 0 < |u| < a and x ∈ ℝ^N;

(f3′):f˜(x,u)=g˜(u)=F˜(x,u)=0 when |u| ≥ a and x ∈ ℝ^N;

(f4′):|f˜(x,u)|≤C|u|r−1for(x,u)∈RN×R;

(f5′):limu→0(minx∈RNu−2F˜(x,u))=∞.

Proof. From the definition off˜(x,u)andg˜(u), it is easy to show that (f1′),(f3′),(f4′)and(f5′) hold. To obtain (f2′) a little manipulation is needed. For 0 < u < a and x ∈ ℝ_N, from the definition of η, we consider

On the other hand, ∂∂u(u−2F˜(x,u)) can also be denoted by

Combing the above two equations and the fact that uf˜(x,u)−2F˜(x,u) is evenon u,we see thatf˜(x,u) satisfies condition (f2′).

Lemma 4.2

a˜ij(x,u) satisfies

(a0′):a˜ij∈C1(RN×R,R),a˜ij is even, a˜ij=a˜ji and there exist two constants C₀ and C₁ such that

(a^′₁): there exists some C > 0, such that

Proof. It is clear that a˜ij satisfies (a_′0) and (a^′₁). We now verify (a^′₂). Observe the relation,

Using condition (a) and the definition of η, we get

We set X := {W^1,2(ℝ^N) : ℝ_N V(x)u²dx < ∞} in which the norm is given by

and E := X ∩ L^∞(ℝ^N). For u ∈ X, we consider the following modified functional corresponding to (1.1),

Now by Lemma 4.1, 4.2, I_ε is well defined in X. And it is easy to say functionals I_ε are E-differentiable in the directions φ ∈ E.

Next, we verify the (P-S) conditions for functional I_ε.

Lemma 4.3

For ε ∈ [0, 1], I_ε satisfies (P-S) conditions uniformly on ε.

Proof. For any fixed u ∈ X and ε ∈ [0, 1], we have

and

Thus,

and then I_ε(u) is coercive and bounded below. Let (ε_n , u_n) ∈ [0, 1] × X be any sequence such that

Then {u_n} and {ε_n} are bounded. Therefore, there exists a subsequence of {ε_n} converges to ε and a subsequence of {u_n} 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 I_ε.

Take T > a, and define

Choose φ ∈ E, φ ≥ 0 and ψn=φexp(−HunT) where H > 0 large enough such that −Ha˜ij+12Dsa˜ij is negatively definite.

Obviously, ψ_n can be seen as a test function in D_EI_ε(u_n), ψ_n → 0, that is