Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent

Theorem 1

Assume that hypotheses (H) hold and ∫R3k(x)|x|e14dx<0. Then there exists δ > 0 such that problem (1.5) has at least two solutions whenever λ₁ < λ < λ₁ + δ.

Remark 1

This paper considers the case with critical Hardy-Sobolev exponent and indefinite nonlinearity. This is a case excluded in [8, 22], which can be obtained in a similar argument. Compared with [8, 22], we concern the case p = 4, it occurs as a “critical” exponent for Hardy-Sobolev inequality ((2.1) below), this case does not belong to an extension of the case p > 4, since it requires substantial changes in the proofs.

Remark 2

The condition ∫R3k(x)|x|e14dx<0 has been shown to be a necessary condition for the existence of positive solutions for semi-linear elliptic equation with indefinite nonlinearity, see[1].

Lemma 3.1

Suppose that ∫R3k(x)|x||e1|4dx<0. Then there exists σ > 0 such that Lλ−¯⋂Bλ+¯=∅ whenever λ₁ < λ < λ₁ + σ.

Proof

We prove the conclusion by contradiction. Assume that Lλ−¯⋂Bλ+¯≠∅, , λ₁ < λ < λ₁ + 1n for any n ∈ ℕ. Then there exists sequences {λ_n} and {u_n} such that ∥u_n∥ = 1, λ_n → λ1+ and

Since u_n is bounded, we may assume that u_n ⇀ u in D^1,2(ℝ³). We claim that u_n → u in D^1,2(ℝ³). Otherwise, we have ∥u∥ < lim inf_n→∞∥u_n∥. This implies

which is a contradiction to λ₁. Hence u_n → ± e₁ in D^1,2(ℝ³). We may assume u = e₁ (the case when u = −e₁ is completely similar and will be omitted here entirely). Since u_n → e₁ in D^1,2(ℝ³), we have

In view of inequality (2.1), the Sobolev space D^1,2(ℝ³) is continuously embedded in weighted Lebesgue Space L4(R3,|x|−14⋅4dx). Note that L4(R3,|x|−14⋅4dx) consists of those functions u such that u|x|14 ∈ L⁴(ℝ³, dx), thus we have

Hence

which is a contradiction to the assumption. This complete the proof. □

Lemma 3.2

Suppose that Lλ−¯∩Bλ+¯=∅. Then

1. Sλ0 = {0}.

2. 0∉Sλ−¯  and  Sλ− is closed.

3. Sλ−  and  Sλ+ are separated. That is Sλ−¯∩Sλ+¯=∅.

4. Sλ+ is bounded.

Proof

We prove (i), (ii) and (iv) by contradiction. (i). Suppose that there is u ∈ Sλ0 such that u ≠ 0. Then u∥u∥∈Lλ0∩Bλ0⊂Lλ−¯∩Bλ+¯=∅ which is impossible.

(ii). Suppose that there exists {u_n} ∈ Sλ− such that u_n → 0 in D^1,2(ℝ³). Then by Lemma 2.2 (i), we have

Let vn=un∥un∥. We observe that

We may assume that v_n ⇀ v₀ in D^1,2(ℝ³). We claim that v₀ ≠ 0. Indeed, By Hardy-Sobolev inequality (2.1), there exists C > 0 such that

This implies

Hence, by (3.2) and (3.3), it follows that

that is

This implies v₀ ≠ 0.

To obtain a contradiction, we are ready to prove that v0∥v0∥∈Lλ−¯∩Bλ+¯. By (3.4), we get

This implies v0∥v0∥∈Lλ−¯.

On another hand, according to (3.3), it follows that

We claim that

We claim that

Let Ψ_R ∈ C¹(ℝ³) be such that Ψ_R(x) = 1 for |x| > R + 1, Ψ_R(x) = 0 for |x| < R and 0 ≤ Ψ_R(x) ≤ 1 on ℝ³. For every R > 1, we have

When R → ∞, we obtain, by Lebesgue theorem

So our claim holds.

By (3.6) and (3.7), we get

This implies that v0∥v0∥∈Bλ+¯. Therefore we have proved that v0∥v0∥∈Bλ+¯∩Lλ−¯, which is impossible. Hence 0∉Sλ−¯. Finally, since Sλ−¯⊆Sλ−∪Sλ0=Sλ−∪{0}  and  0∉Sλ−¯, we conclude that S⁻ is closed.

(iii). By (i) and (ii), it follows that

(iv). Suppose that Sλ+ is unbounded. Then there exists a sequence {u_n} ⊂ Sλ+ such that ∥u_n∥ → ∞ as n → ∞. It follows from Lemma 2.2 (ii) that

Let vn=un∥un∥. We may assume that v_n ⇀ v₀ in D^1,2(ℝ³). By the weak lower semi-continuity of norm, we have

Since u_n ∈ S_λ, we get

We assert that v₀ ≠ 0. Suppose v₀ = 0, it is easy to see that v_n → 0 in D^1,2(ℝ³), which a contradiction to ∥v_n∥ = 1. Indeed, using (3.8), if v_n ↛ 0, we have

a contradiction.

By (3.8) and v₀ ≠ 0, we have v0∥v0∥∈Lλ−¯.

On the other hand, according to (3.9), it follows that

According to the definitions of α_∞ and k_∞, then we have

This means v0∥v0∥∈Bλ+¯. Hence v0∥v0∥∈Lλ−¯∩Bλ+¯, it is a contradiction to Lλ−¯∩Bλ+¯=∅. Therefore Sλ+ is bounded. □

Lemma 3.3

Suppose that Lλ−¯∩Bλ+¯=∅. Then

1. Every minimizing sequence of J_λ(u) on Sλ− is bounded.

2. infu∈Sλ−Jλ(u)>0.

3. There exists a minimizer of J_λ(u) on Sλ− .

Proof

1. Let {u_n} ∈ Sλ− be a minimizing sequence for the functional J_λ(u), i.e.,

   J(un)=14∫R3|∇un|2−λh(x)un2dx=14∫R3k(x)|un|4|x|dx−b∫R3|∇un|2dx2→infu∈S−J(u).

   Suppose {u_n} is unbounded in D^1,2(ℝ³), i.e., ∥u_n∥ → ∞ as n → ∞. Let vn=un∥un∥. We may assume that v_n ⇀ v₀ in D^1,2(ℝ³). It is clear that

   ∥un∥2∫R3|∇vn|2−λh(x)vn2dx→4infu∈S−J(u), as n→∞. (3.10)

   Together this with ∥u_n∥ → ∞ give us

   limn→∞∫R3|∇vn|2−λh(x)vn2dx=0. (3.11)

   Suppose v₀ = 0, it is easy to see that v_n → 0 in D^1,2(ℝ³). This, by contrary, together with (3.11) implies that

   0=∫R3|∇v0|2−λh(x)v02dx<lim infn→∞∫R3|∇vn|2−λh(x)vn2dx=0,

   it is impossible. However, v_n → 0 in D^1,2(ℝ³) with contradicts to ∥v_n∥ = 1.

   Hence, v₀ ≠ 0, and v0∥v0∥∈Lλ−¯. Similar to the one used in the proof of Lemma 3.2(iv), it is easy to prove that v0∥v0∥∈Bλ+¯, Hence v0∥v0∥∈Lλ−¯∩Bλ+¯, it is a contradiction to Lλ−¯∩Bλ+¯=∅.

2. It is clear that J_λ(u) ≥ 0 on Sλ− . Suppose infw∈Sλ−Jλ(w)=0. {u_n} be a minimizing sequence. According to Lemma 3.3(i), it follows that {u_n} is bounded and we may assume u_n ⇀ u₀. Clearly, we have

   limn→∞∫R3|∇un|2−λh(x)un2dx=0 (3.12)

   and

   limn→∞∫R3k(x)|un|4|x|dx−b∫R3|∇un|2dx2=0. (3.13)

   Then it is easy to prove that u0∥u0∥∈Bλ+¯∩Lλ−¯. Indeed, using the same argument in proof of Lemma 3.2(ii), by(3.12), it is easy to show that u0∥u0∥∈Lλ−¯ and by (3.13), u0∥u0∥∈Bλ+¯ follows. This complete the proof

3. Let {u_n} be a minimizing sequence. According to Lemma 3.3(i), it follows that {u_n} is bounded and we may assume u_n ⇀ u₀ in D^1,2(ℝ³). Suppose u_n ↛ u₀ in D^1,2(ℝ³), we get

   ∫R3|∇u0|2−λh(x)u02dx  <  limn→∞∫R3|∇un|2−λh(x)un2dx=limn→∞∫R3k(x)|un|4|x|dx−b∫R3|∇un|2dx2≤lim supn→∞∫R3k(x)|un|4|x|dx−lim infn→∞b∫R3|∇un|2dx2≤∫R3k(x)|u0|4|x|dx−b∫R3|∇u0|2dx2+k∞α∞+k(0)ν0≤∫R3k(x)|u0|4|x|dx−b∫R3|∇u0|2dx2

   So there exists a 0 < t < 1 such that tu₀ ∈ Sλ− . On other hand, since

   0<∫R3|∇u0|2−λh(x)|u0|2dx≤lim infn→∞∫R3|∇un|2−λh(x)|un|2dx=4infw∈S−J(w)≤∫R3|∇tu0|2−λh(x)|tu0|2dx,

   we have t ≥ 1, a contradiction. Consequently, we have u_n → u ≠ 0 in D^1,2(ℝ³). Since Sλ− is closed, then we have u₀ ∈ Sλ− and J_λ(u₀) = infw∈Sλ−Jλ(w).

   We are going to the investigation on Sλ+ .