Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential

Proposition 2.1

[21]. The space (L^p(.)(Ω), | ⋅|_p(.)) is separable, uniformly convex, reflexive and its conjugate dual space is L^p′(⋅)(Ω) where p′(⋅) is the conjugate function of p(⋅), i.e.,

For u ∈ L^p(⋅)(Ω) and v ∈ L^p′(⋅)(Ω) we have the "equivalent" Hölder’s inequality

Proposition 2.2

([8, Theorem 1.3]). Let u_n, u ∈ L^p(⋅), we have

1. |u|_p(⋅) = a ⇔ ρ_p(x) (ua) = 1 for u ≠ 0 and a > 0.

2. |u|_p(⋅) < (=; > 1) ⇔ ρ_p(x)(u) < (=; > 1).

3. |u_n| → 0 (resp → +∞) ⇔ ρ_p(x)(u_n) → 0, (resp → +∞).

4. the following statements are equivalent.

   1. lim_n→+∞ |u_n − u|_p(⋅) = 0,

   2. lim_n→+∞ ρ_p(x)(u_n − u) = 0,

   3. u_n → u in measure in Ω and lim_n→+∞ ρ_p(x)(u_n) = ρ_p(x)(u).

Remark 2.3

According to [21] the norm ∥.∥_2,p(⋅) is equivalent to the norm ∥ Δ.∥_p(⋅) in the space W02,p(⋅) (Ω), for the Lipschitz boundary and the exponent p(⋅) is in a class functions for which 1p(⋅) is globally log-Hölder continuous. Consequently, the norms ∥.∥_2,p(⋅), ∥.∥ and ∥ Δ .∥_p(⋅) are equivalent. See also [2].

Remark 2.4

In view of Proposition 2.2, for every u ∈ L^p(.)(Ω), then

Theorem 2.5

([1, Theorem 3.2]). Let p, q ∈ C1+ (Ω). Assume p(x) < N2 and q(x) < p2∗ (x). Then there is a continuous and compact embedding of W02,p(x) (Ω) into L^q(x)(Ω).

Definition 2.6

A function u ∈ W02,p(⋅) (Ω) is said to be a weak solution of (1.1) if

We point out that in the case u is nontrivial we say that λ is an eigenvalue of (1.1) corresponding to the eigenfunction u.

Lemma 3.1

Assume that H(p, q) holds. Then there exists a positive constant C such that the q(.)-Hardy-Rellich inequality

holds for all u ∈ W02,p(⋅) (Ω) in one of the following cases:

1. |u| ≤ δ(x)² and |△ u| ≥ 1;

2. |u| ≥ δ(x)² and |△u| ≤ 1.

Proof

Let u ∈ W02,p(⋅) (Ω) such that 1 < q⁻ ≤ q⁺ < p⁻ ≤ p⁺ < N2 .

From (a) we have |△ u| ≥ 1. Thus

In view of (3.2), we deduce that

where C−=(N(q−−1)(N−2q−)(q−)2)q−. This and (3.4) permit us to write

Since |u| ≤ δ(x)², we obtain

Therefore

Therefore

That is

Hence we conclude that

where C1=q−p+C−.

From (b) w have |△ u| ≤ 1. Thus in this case

Using again (3.2), we have

where C+=(N(p+−1)(N−2p+)(p+)2)p+. This and (3.5) give

Since |u| ≥ δ(x)², we get

Thus

Therefore

Then

Hence we deduce that

where C2=q−p+C+. The result follows. □

Remark 3.2

In view of (3.3), λ₁ > 0 if and only if the q(.)-Hardy-Rellich inequality holds in Ω. Moreover, λ₁ is exactly the best constant in the inequality (3.3). This makes our problem (1.1) naturally well defined.

Definition 3.3

Let X be a real reflexive Banach space and let X^* stand for its dual with respect to the pairing 〈⋅, ⋅〉. We shall deal with mappings T acting from X into X^*. The strong convergence in X (and in X^*) is denoted by → and the weak convergence by ⇀. T is said to belong to the class (S⁺) if for any sequence u_n in X converging weakly to u ∈ X and limsup_n→+∞ 〈Tu_n, u_n − u〉 ≤ 0, u_n converges strongly to u in X. We write T ∈ (S⁺).

Lemma 3.4

We have the following statements:

1. Φ and φ are even, and of class C¹ on W02,p(⋅) (Ω).

2. 𝓗 is a closed C¹-manifold.

Proof

It is clear that φ and Φ are even and of class C¹ on W02,p(⋅) (Ω) and 𝓗 = φ⁻¹ {1}. In view of (3.3) 𝓗 is closed. The derivative operator φ′ satisfies φ′(u) ≠ 0 ∀ u ∈ 𝓗 (i.e., φ′(u) is onto for all u ∈ 𝓗). Hence φ is a submersion, which proves that 𝓗 is a C¹-manifold. □

Lemma 3.5

The following statements hold:

1. T is continuous, bounded and strictly monotone.

2. T is of (S₊) type.

3. T is a homeomorphism.

Proof

1. Since T is the Fréchet derivative of Φ, it follows that T is continuous and bounded.

   Using the following elementary inequalities, which hold for any three real x, y and γ.

   |x−y|γ≤2γ(|x|γ−2x−|y|γ−2y)(x−y)if γ≥2,|x−y|2≤1(γ−1)(|x|+|y|)2−γ(|x|γ−2x−|y|γ−2y)(x−y)if 1<γ<2,

   we obtain for all u, v ∈ W02,p(⋅) (Ω) such that u ≠ v,

   〈T(u)−T(v),u−v〉>0,

   which means that T is strictly monotone.

2. Let (u_n)_n be a sequence of W02,p(⋅) (Ω) such that

   un⇀u weakly in W02,p(⋅)(Ω) and lim supn→+∞〈T(un),un−u〉≤0.

   In view of the monotonicity of T, we have

   〈T(un)−T(u),un−u〉≥0, (3.6)

   and since u_n ⇀ u weakly in W02,p(⋅) (Ω), it follows that

   lim supn→+∞〈T(un)−T(u),un−u〉=0. (3.7)

   Thanks to the above inequalities,

   ∫{x∈Ω:p(x)≥2}|Δun−Δu|p(x)dx≤2(p−−2)∫ΩA(un,u)dx,∫{x∈Ω:1<p(x)<2}|Δun−Δu|p(x)dx≤(p+−1)∫Ω(A(un,u))p(x)2(B(un,u))(2−p(x))p(x)2dx,

   where

   A(un,u)=(|Δun|p(x)−2Δun−|Δu|p(x)−2Δu)(Δun−Δu),B(un,u)=(|Δun|+|Δu|)2−p(x).∫ΩA(un,u)dx=〈T(un)−T(u),un−u〉,

   we can consider

   0≤∫ΩA(un,u)dx<1

   and we distinguish two cases.

   First, if ∫_Ω A(u_n, u) dx = 0, then A(u_n, u) = 0, since A(u_n, u) ≥ 0 a.e. in Ω.

   Second, if 0 < ∫_Ω A(u_n, u) dx < 1, then

   tp(x):=(∫{x∈Ω:1<p(x)<2}A(un,u)dx)−1

   is positive and by applying Young’s inequality we deduce that

   ∫{x∈Ω:1<p(x)<2}[t(A(un,u))p(x)2](B(un,u))(2−p(x))p(x)2dx≤∫{x∈Ω:1<p(x)<2}A(un,u)(t)2p(x)+(B(un,u))p(x)dx.

   Now, by the fact that 2p(x)<2, we have

   ∫{x∈Ω:1<p(x)<2}A(un,u)(t)2p(x)+(B(un,u))p(x)dx≤∫{x∈Ω:1<p(x)<2}A(un,u)t2+(B(un,u))p(x)dx≤1+∫{x∈Ω:1<p(x)<2}(B(un,u))p(x)dx.

   Hence

   ∫{x∈Ω:1<p(x)<2}|Δun−Δu|p(x)dx≤(∫{x∈Ω:1<p(x)<2}A(un,u)dx)12(1+∫Ω(B(un,u))p(x)dx).

   Since ∫_Ω(B(u_n, u))^p(x) dx is bounded, we have

   ∫{x∈Ω:1<p(x)<2}|Δun−Δu|p(x)dx→0 as n→∞.

3. Note that the strict monotonicity of T implies that T is into an operator.

Moreover, T is a coercive operator. Indeed, from (2.1) and since p⁻ − 1 > 0, for each u ∈ W02,p(⋅) (Ω) such that ∥ u ∥ ≥ 1, we have

Finally, thanks to the Minty-Browder Theorem [22], the operator T is surjective and admits an inverse mapping.

To complete the proof of (3), it suffices then to show the continuity of T⁻¹. Indeed, let (f_n)_n be a sequence of W^{−2,p′(⋅)}(Ω) such that f_n → f in W^{−2,p′(⋅)}(Ω). Let u_n and u in W02,p(⋅) (Ω) such that

By the coercivity of T, we deduce that the sequence (u_n)_n is bounded in the reflexive space W02,p(⋅) (Ω). For a subsequence, if necessary, we have u_n ⇀ û in W02,p(⋅) (Ω) for a some û. Then

It follows by the assertion (2) and the continuity of T that

Further, since T is an into operator, we conclude that u ≡ û. □

Lemma 3.6

We have the following statements:

1. φ′ is completely continuous.

2. The functional Φ satisfies the Palais-Smale condition on 𝓗, i.e., for {u_n} ⊂ 𝓗, if {Φ(u_n)}_n is bounded and

   αn=Φ′(un)−βnφ′(un)→0asn→+∞, (3.8)

   where

   βn=〈Φ′(un),un〉〈φ′(un),un〉,

   then {u_n}_n≥1 has a convergent subsequence in W02,p(⋅) (Ω).

Proof

1. First let us prove that φ′ is well defined. Let u, v ∈ W02,p(⋅) (Ω). We have

   〈φ′(u),v〉=∫Ω|u|q(x)−2uδ(x)2q(x)vdx.

   Thus

   |〈φ′(u),v〉|≤∫x∈Ω:δ(x)>1|u|q(x)−1δ(x)2q(x)vdx+∫x∈Ω:δ(x)≤1|u|q(x)−1δ(x)2q(x)vdx.

   Therefore

   |〈φ′(u),v〉|≤∫x∈Ω:δ(x)>1|u|q(x)−1vdx+∫x∈Ω:δ(x)≤11δ(x)2|u|q(x)−1δ(x)2(q(x)−1)vdx.

   By applying Hölder’s inequality, we obtain

   |〈φ′(u),v〉|≤2(|u|r(x)q(x)−1|v|q(x)+uδ(x)2r(x)q(x)−1vδ(x)2q(x)).

   where r(x)=q(x)q(x)−1.

   This and (3.3) yield

   |〈φ′(u),v〉|≤2(|u|r(x)q(x)−1|v|q(x)+1C2|Δu|r(x)q(x)−1|Δv|q(x)).

   Then

   |〈φ′(u),v〉|≤2(k1∥u∥q(x)−1∥v∥+k2C2∥u∥q(x)−1∥v∥),

   where k₁ is a constant given by the embedding of W02,p(⋅) (Ω) in L^q(⋅)(Ω) and k₂ is given by the equivalence of the norm |Δ.|_p(⋅) and ∥.∥. Hence

   ∥φ′(u)∥∗≤2k1+k2C2∥u∥q(x)−1,

   where ∥⋅∥_∗ is the dual norm associated with ∥.∥.

   For the complete continuity of φ′, we argue as follow. Let (u_n)_n ⊂ W02,p(⋅) (Ω) be a bounded sequence and u_n ⇀ u (weakly) in W02,p(⋅) (Ω). Due to the q(⋅)-Hardy inequality (3.3) u_n ⇀ u in Lq(⋅)(Ω;1q(⋅)) and due to the fact that the embedding W02,p(⋅) (Ω) ↪ L^q(⋅)(Ω) is compact, u_n converges strongly to u in L^q(⋅)(Ω). Consequently, there exists a positive function g ∈ L^q(⋅)(Ω) such that

   ∣u∣≤g a.e. in Ω.

   Since g ∈ L^q(⋅)–1(Ω), it follows from the Dominated Convergence Theorem that

   ∣un∣q(x)−2un→∣u∣q(x)−2u in Lq′(⋅)(Ω).

   That is,

   φ′(un)→φ′(u) in Lq′(⋅)(Ω).

   Recall that the embedding

   Lq′(⋅)(Ω)↪W−2,p′(⋅)(Ω)

   is compact. Thus

   φ′(un)→φ′(u) in W−2,p′(⋅)(Ω).

   This proves the assertion (i).

2. By the definition of Φ we have that ρ_p(x)(Δu_n) is bounded in ℝ. Thus, without loss of generality, we can assume that u_n converges weakly in W02,p(⋅) (Ω) for some functions u ∈ W02,p(⋅) (Ω) and ρ_p(x)(Δu_n) → ℓ. For the rest we distinguish two cases. If ℓ = 0, then u_n converges strongly to 0 in W02,p(⋅) (Ω). Otherwise, let us prove that

   lim supn→∞〈Δp(⋅)2un,un−u〉≤0.

   Indeed, notice that

   〈Δp(⋅)2un,un−u〉=ρp(x)(Δun)−〈Δp(⋅)2un,u〉.

   Applying α_n of (3.8) to u, we deduce that

   θn=〈Δp(⋅)2un,u〉−βn〈φ′(un),u〉→0 as n→∞.

   Therefore

   〈Δp(⋅)2un,un−u〉=ρp(x)(Δun)−θn−(〈Φ′(un),un〉〈φ′(un),un〉)〈φ′(un),u〉.

   That is,

   〈Δp(⋅)2un,un−u〉=ρp(x)(Δun)〈φ′(un),un〉(〈φ′(un),un〉−〈φ′(un),u〉)−θn.

   On the other hand, from Lemma 3.6, φ′ is completely continuous. Thus

   φ′(un)→φ′(u)and〈φ′(un),un〉→〈φ′(u),u〉.

   Then

   |〈φ′(un),un〉−〈φ′(un),u〉|≤|〈φ′(un),un〉−〈φ′(u),u〉|+|〈φ′(un),u〉−〈φ′(u),u〉|.

   It follows that

   |〈φ′(un),un〉−〈φ′(un),u〉|≤|〈φ′(un),un〉−〈φ′(u),u〉|+∥φ′(un)−φ′(u)∥∗∥u∥.

   This implies that

   〈φ′(un),un〉−〈φ′(un),u〉→0 as n→∞. (3.9)

   Combining with the above equalities, we obtain

   lim supn→+∞〈Δp(⋅)2un,un−u〉≤ℓ〈φ′(u),u〉lim supn→∞(〈φ′(un),un〉−〈φ′(un),u〉).

   We deduce

   lim supn→∞〈Δp(⋅)2un,un−u〉≤0. (3.10)

   On the other hand,

   〈Φ′(un),un−u〉=〈Δp(⋅)2un,un−u〉.

   According to (3.10), we conclude that

   lim supn→∞〈Φ′(un),un−u〉≤0. (3.11)

   In view of Lemma 3.5, u_n → u strongly in W02,p(⋅) (Ω). This achieves the proof of Lemma 3.6. □

Lemma 4.1

Let X be a real Banach space and A, B be symmetric subsets of X ∖ {0} which are closed in X. Then

1. If there exists an odd continuous mapping f : A → B, then γ(A) ≤ γ(B).

2. If A ⊂ B, then γ(A) ≤ γ(B).

3. γ(A ∪ B) ≤ γ(A) + γ(B).

4. If γ(B) < +∞, then γ(A–B) ≥ γ(A) – γ(B).

5. If A is compact, then γ(A) < +∞ and there exists a neighborhood N of A, N is a symmetric subset of X ∖ {0}, closed in X such that γ(N) = γ(A).

6. If N is a symmetric and bounded neighborhood of the origin in ℝ^k and if A is homeomorphic to the boundary of N by an odd homeomorphism, then γ(A) = k.

7. If X₀ is a subspace of X of codimension k and if γ(A) > k then A ∩ X₀ ≠ ϕ.