Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Theorem 1. Assume that F is even in u and the following conditions hold:

•  

  F₁

  $$ ()

•  

  uniformly in x ∈ Ω;

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  F₂ there exist two constants 2 < θ < 2N/(N − 4) = 2^** and k₁ > 0 such that

  $$ ()

•  

  uniformly for all x ∈ Ω;

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  F₃ there exists a constant γ > N(θ − 2)/4 such that

  $$ ()

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  uniformly for all x ∈ Ω.

Then problem (1) has infinitely many nontrivial solutions.

Theorem 2. Assume that F satisfies (F₁) and

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  F₄ there exist three positive constants L, m₁, and m₂ such that

  •  

    j₁ f(x, u)u − 2F(x, u) ≥ m₁|u|², if |u| ≥ L;

  •  

    j₂ |f(x,u)|^σ/|u|^σ ≤ m₂(f(x, u)u − 2F(x, u)), if |u| ≥ L, where σ > (N − 2)/2.

If F is even in u, problem (1) has infinitely many nontrivial solutions.

Remark 3. For Schrödinger equation, the corresponding condition (F₄) is due to Ding and Luan [11]. The condition (F₄) is weaker than the usual Ambrosetti-Rabinowitz-type condition (see [11, 12]).

Theorem 4. Assume that F satisfies (F₁), (F₂), (F₃), and

•  

  F₅

  $$ ()

•  

  uniformly in x ∈ Ω.

If 0 is an eigenvalue of Δ² + cΔ + a (with Navier boundary condition), assume also the condition that

•  

  F₆ there exists δ > 0 such that

  • (i)

    F(x, u) ≥ 0,  for all |u| ≤ δ,  x ∈ Ω; or

  • (ii)

    F(x, u) ≤ 0,  for all |u| ≤ δ,  x ∈ Ω.

Then problem (1) has at least one nontrivial solution.

Theorem 5. Suppose that F satisfies (F₁), (F₄), and (F₅). If 0 is an eigenvalue of Δ² + cΔ + a (with Navier boundary condition), assuming also (F₆), then problem (1) has at least one nontrivial solution.

Theorem A (Fountain theorem). Assume that φ ∈ C¹(X, ℝ) satisfies the Cerami condition (C), φ(−u) = φ(u). If for almost every k ∈ ℕ, there exist ρ_k > r_k > 0 such that

•  

  A₁

  $$ ()

•  

  A₂

  $$ ()

•  

  then φ has an unbounded sequence of critical values.

Theorem B (see [12], Theorem 2.2.)Suppose that φ ∈ C¹(X, ℝ) satisfies the following assumptions:

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  i₁ X ≠ {0} and φ has a local linking at 0; that is, for some r > 0,

  $$ ()

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  i₂ φ satisfies (C^*) condition;

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  i₃ φ maps bounded sets into bounded sets;

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  i₄ for every m ∈ ℕ,   φ(u)→−∞ as |u| → ∞, on $$.

Then φ has at least one nonzero critical point.

Proof of Theorem 1. At first, we claim that φ satisfies the Cerami condition (C). Consider a sequence {u_n} such that φ(u_n) is bounded and ∥φ^′(u_n)∥(1 + ∥u_n∥) → 0 as n → ∞. Then there exists a constant M₁ > 0 such that

By a standard argument, we only need to prove that {u_n} is a bounded sequence in E. Otherwise, going if necessary to a subsequence, we can assume that ∥u_n∥ → ∞ as n → ∞. From (F₃), there exist two constants k₂, k₃ > 0 such that So, by (21) and (22), we have which implies that for all n ∈ ℕ and some positive constant k₄.

Since

On the one hand, we consider the case

Putting

one has 0 < α < 1. Let where E⁰ = ker (Δ² + cΔ + a). We can obtain from Hölder′s inequality, (15), and (24) that for all n, where q = p/(p − 1) = 2^**/(α + 1).

By (F₂) and (29), one has

for all n. Since α < 1, we have

On the other hand, if γ satisfies,

then one sees 1 ≤ γ/(γ − θ + 1) ≤ 2N/(N − 4). So, we get It follows from (24) that By (F₂) and (33), we obtain for all n. Note that γ ≥ 2N(θ − 1)/(N + 4) and N > 4 imply that γ > θ − 1. So, it follows from (34) and the above expression that Hence, we conclude from (31) and (36) that Similarly for $$, we get

It follows from the equivalence of the norms on the finite dimensional space E⁰ that there exists K₁ > 0 such that

Putting τ = 2N + (5 − N)γ + 1, one has It follows from (24), (39), and Hölder′s inequality that and consequently Hence, by (37), (38), and (42), one sees as n → ∞, which is a contradiction. So, we obtain that {u_n} is bounded in E. By a standard argument, we get that φ satisfies the condition (C).

Let $$ with dim X_j < ∞ for any j ≥ 1. Set

Since dim (Y_k) < ∞, all the norms are equivalent. For u ∈ Y_k, there exists a constant K₂ > 0 such that From condition (F₁), there exists k₅ > 0 such that For u ∈ Y_k, it follows from (45) and (46) that which implies that So, (A₁) of Theorem A is satisfied for every ρ_k > 0 large enough.

Here, we obtain from (F₂) that there exists a positive constant k₆ such that

uniformly for all x ∈ Ω. Let us define For k large enough, one has Z_k ⊂ E⁺. By (49), on Z_k, we have Choosing $$, we obtain, if u ∈ Z_k and ∥u∥ = r_k, Since, by Lemma 3.8 of [14], β_k → 0 as k → ∞, (A₂) is proved. Hence, the proof is completed by using Fountain theorem.

Proof of Theorem 2. Firstly, we claim that φ satisfies the Cerami condition (C). Consider a sequence {u_n} such that φ(u_n) is bounded from above and ∥φ^′(u_n)∥(1 + ∥u_n∥) → 0 as n → ∞. By a standard argument, we only need to prove that {u_n} is a bounded sequence in E. For otherwise, we can assume that ∥u_n∥ → ∞ as n → ∞.

From assumption (F₄), there exist two positive constants m₃ and m₄, such that

So, one has for all n and some positive constant m₅.

Let v_n = u_n/∥u_n∥; then ∥v_n∥ = 1 and $$ for all r ∈ [1,2N/(N − 4)). By (54), we have

as n → ∞. So, for r ∈ (2, (2N − 4)/(N − 4)), it follows from Hölder′s inequality and the above expression that as n → ∞. It follows from (39) that Hence, we get From (F₄), (53), and (56), there exists a positive constant m₆ such that as n → ∞; therefore, 1 = o(1), which is a contradiction. Hence, {u_n} is bounded.

In a way similar to the proof of Theorem 1, we can obtain that φ satisfies (A₁) of Theorem A.

It follows from (F₁) that there is L₁ > 0 such that

Then, by (F₄) and (60), for |u| ≥ L₂ = max {L, L₁},   x ∈ Ω, one has which implies that for |u| ≥ L₂.

Therefore, there exist two positive constants m₇ and m₈ such that

where 2σ/(σ − 1) < 2N/(N − 4). As the proof of Theorem 1, we can get (A₂). Therefore, Theorem 2 holds.

Proof of Theorem 4. The proof of this theorem is divided in several steps.

Step  1. We claim that φ has a local linking at zero with respect to (X¹, X²).

By (F₅), for any ε > 0, there exists δ₁ > 0 such that

We obtain from the above expression and (49) that where $$. Hence, we can get from (15) and (65) that for all u ∈ E.

Here, we consider only the case where 0 is an eigenvalue of Δ² + cΔ + a and case (ii) of (F₆) holds. The case (i) is similar.

Let X = E,   X¹ = E⁺ ⊕ E⁰, and X² = E⁻, where E⁰ = ker (Δ² + cΔ + a). Choose a Hilbertian basis {e_n} _n≥0 for X¹ and define

Now, by (66), for each u ∈ X² = E⁻, one has Letting ε = 1/(8K²) and by θ > 2, we have for δ₂ > 0 small enough.

Let u = u⁰ + u⁺ ∈ E⁰ ⊕ E⁺ = X¹ be such that ∥u∥ ≤ δ₃ : = δ/(2K₁). Put

Then, for all ∥u∥ ≤ δ₃ and x ∈ Ω, by (39), one sees On one hand, from above expression, for any x ∈ Ω₁, we have Hence, by condition (ii) of (F₆), we get On the other hand, for any x ∈ Ω₂, one has Hence, for all x ∈ Ω₂ and u ∈ X¹ with ∥u∥ ≤ δ₃, we can obtain from (65) that which implies that Letting ε = 1/(16K²) in above expression, then for all x ∈ Ω₂ and u ∈ X¹ with ∥u∥ ≤ δ₃, we have which implies that for δ₄ > 0 small enough. Hence, φ has a local linking at zero with respect to (X¹, X²) for δ₅ = min {δ₂, δ₄} small enough.

Step  2. In a way similar to the proof of Theorem 1, we can get that φ satisfies the (C^*) condition.

Step  3. Now, we claim that for each m ∈ ℕ, one has

Since dim (E⁰) < ∞ and $$, all the norms are equivalent. For $$, there exists a constant k₇ > 0 such that From condition (F₁), there exists a constant k₈ > 0 such that For $$, it follows from (80) and (81) that which implies that Hence, all the assumptions of Theorem B are verified. Then, the proof of Theorem 4 is completed.

Proof of Theorem 5. In a way similar to the proof of Theorems 2 and 4, we can obtain that φ satisfies (i₁), (i₂), (i₃), and (i₄) of Theorem B. Therefore, Theorem 5 holds.