A Note on the Sign-Changing Solutions for a Double Critical Hardy–Sobolev–Maz’ya Problem

Assume that a⁢(x)>0, and Ω is a bounded domain such that (x-(0,z*))⋅ν≤0 in a neighborhood of (0,z*)∈∂⁡Ω, where ν is the outward normal of ∂⁡Ω. If N>6+t when μ>0, and N>6+s when μ=0, then (1.1) has infinitely many sign-changing solutions.

Our result includes the special case where k=N and t=0 in (1.1), which was studied by Bhakta in [7]. She assumed that ∂⁡Ω∈C3 and if 0∈∂⁡Ω, then the principle curvature of ∂⁡Ω at 0 is negative. However, in this case, our condition is much weaker than hers. For the details, one can refer to [45].

In fact, if we assume the weaker condition that at any point (0,z*))∈{y=0}∩∂Ω the principal curvature of ∂⁡Ω is nonpositive and the mean curvature of ∂⁡Ω is nontrivial, then the result of Theorem 1.1 can also hold. For the compactness result, one can refer to [29, Theorem 1.1].

Suppose that N≥3, a⁢(x)∈C1⁢(Ω¯) with a⁢(x)+12⁢x⋅∇⁡a≤0 for every x∈Ω and Ω is star shaped with respect to the origin. Then, (1.1) does not have any nontrivial solution.

Suppose that all the assumptions in Theorem 1.1 hold and λ1≤1. Then, (1.1) has infinitely many sign-changing solutions.

Proof.

Multiplying φ1 to equation (1.1) and integrating by parts, we find

( λ 1 - 1 ) ⁢ ∫ Ω a ⁢ ( x ) ⁢ u ⁢ φ 1 = μ ⁢ ∫ Ω | u | 2 * ⁢ ( t ) - 2 ⁢ u ⁢ φ 1 | y | t + ∫ Ω | u | 2 * ⁢ ( s ) - 2 ⁢ u ⁢ φ 1 | y | s .

So it is easy to see that if λ1≤1, then any nontrivial solution of (1.1) has to change sign. Since it follows from [46, Theorem 1.1] that (1.1) has infinitely many solutions, we have proved our result. ∎

Let λ1>1. Then, for any small enough ρ0>0, we have 𝒜n⁢(±D⁢(ρ0))⊂±D⁢(ρ)⊂±D⁢(ρ0) for some ρ∈(0,ρ0). Moreover, ±D⁢(ρ0)∩Kn⊂𝒫.

Proof.

First, we observe that 𝒜n can be decomposed as 𝒜n⁢(u)=B⁢(u)+Fn⁢(u), where B⁢(u),Fn⁢(u)∈H01⁢(Ω) are, respectively, the unique solutions of the following two equations:

- Δ ⁢ ( B ⁢ ( u ) ) = a ⁢ ( x ) ⁢ u , - Δ ⁢ ( F n ⁢ ( u ) ) = μ ⁢ | u | 2 * ⁢ ( t ) - 2 - ϵ n ⁢ u | y | t + | u | 2 * ⁢ ( s ) - 2 - ϵ n ⁢ u | y | s .

In other words, B⁢(u) and Fn⁢(u) can be uniquely determined by

and

Now we claim that if u∈𝒫, then B⁢(u),Fn⁢(u)∈𝒫. To see this, let u∈𝒫. Then,

- ∫ Ω | ∇ B ( u ) - | 2 = 〈 B ( u ) , B ( u ) - 〉 H 0 1 ⁢ ( Ω ) = ∫ Ω a ( x ) u B ( u ) - ≥ 0 ,

which yields B⁢(u)∈𝒫, and similarly Fn⁢(u)∈𝒫.

Note that

∥ B ⁢ ( u ) ∥ 2 = 〈 B ⁢ ( u ) , B ⁢ ( u ) 〉 H 0 1 ⁢ ( Ω ) = ∫ Ω a ⁢ ( x ) ⁢ u ⁢ B ⁢ ( u ) ≤ ( ∫ Ω a ⁢ ( x ) ⁢ u 2 ) 1 2 ⁢ ( ∫ Ω a ⁢ ( x ) ⁢ B ⁢ ( u ) 2 ) 1 2 .

So from (2.2), it follows that

∥ B ⁢ ( u ) ∥ 2 ≤ 1 λ 1 ⁢ ∥ u ∥ ⁢ ∥ B ⁢ ( u ) ∥ ,

and then

∥ B ⁢ ( u ) ∥ ≤ 1 λ 1 ⁢ ∥ u ∥ .

For any u∈H01⁢(Ω), we consider v∈𝒫 such that dist⁡(u,𝒫)=∥u-v∥. Then,

On the other hand, using inequality (1.2), we find

which implies that for any u∈H01⁢(Ω),

since

| u - | 2 * ⁢ ( t ) - ϵ n , t , Ω = min v ∈ 𝒫 | u - v | 2 * ⁢ ( t ) - ϵ n , t , Ω ≤ C min v ∈ 𝒫 ∥ u - v ∥ = C dist ( u , 𝒫 ) .

Since λ1>1, we can choose γ∈(1λ1,1). Then, from (2.7), there exists small enough ρ0>0 such that for any u∈D⁢(ρ0),

Combining (2.6) and (2.8), one has

dist ⁡ ( 𝒜 n ⁢ ( u ) , 𝒫 ) ≤ dist ⁡ ( B ⁢ ( u ) , 𝒫 ) + dist ⁡ ( F n ⁢ ( u ) , 𝒫 ) ≤ γ ⁢ dist ⁡ ( u , 𝒫 )   for all ⁢ u ∈ D ⁢ ( ρ 0 ) .

Hence, we can get 𝒜n⁢(D⁢(ρ0))⊂D⁢(γ⁢ρ0)⊂D⁢(ρ0). Also if dist⁡(u,𝒫)<ρ0 and Iϵn′⁢(u)=0, that is, u=𝒜n⁢(u), then dist⁡(u,𝒫)=dist⁡(𝒜n⁢(u),𝒫)≤γ⁢dist⁡(u,𝒫), which yields u∈𝒫. Similarly, we can prove that 𝒜n⁢(-D⁢(ρ0))⊂-D⁢(ρ)⊂-D⁢(ρ0) for some ρ∈(0,ρ0) and -D⁢(ρ0)∩Kn⊂𝒫. This completes our proof. ∎

Let λ1>1. Then, for any ℓ, lim∥u∥→∞,u∈Eℓ⁡Iϵn⁢(u)=-∞.

Proof.

For each n, ∥⋅∥n,* defines a norm on H01⁢(Ω). Now since Eℓ is finite dimensional, there exists a constant C>0 such that ∥u∥≤C⁢∥u∥*,n for all u∈Eℓ. So,

As a result,

lim ∥ u ∥ → ∞ , u ∈ E ℓ ⁡ I ϵ n ⁢ ( u ) = - ∞ ,

since 2*⁢(s)-ϵn>2. ∎

For any α1,α2>0, there exists α3 depending on α1 and α2 such that ∥u∥≤α3 for all u∈Iϵnα1∩{u∈H01⁢(Ω):∥u∥*,n≤α2}, where Iϵnα1={u∈H01⁢(Ω):Iϵn⁢(u)≤α1}.

Proof.

Noting that ∥u∥*,n≤α2, we have

μ 2 * ⁢ ( t ) - ϵ n ⁢ ∫ Ω | u | 2 * ⁢ ( t ) - ϵ n | y | t + ( ∫ Ω | u | 2 * ⁢ ( s ) - ϵ n | y | s ) 2 * ⁢ ( t ) - ϵ n 2 * ⁢ ( s ) - ϵ n ≤ α 2 2 * ⁢ ( t ) - ϵ n ,

and then

μ ⁢ ∫ Ω | u | 2 * ⁢ ( t ) - ϵ n | y | t + ∫ Ω | u | 2 * ⁢ ( s ) - ϵ n | y | s ≤ C ⁢ α 2 2 * ⁢ ( t ) - ϵ n .

So from the fact that Iϵn⁢(u)≤α1, we find

Using (2.9) and (2.2), we have

1 2 ( 1 - 1 λ 1 ) ∫ Ω | ∇ u | 2 + μ ( 1 2 * ⁢ ( s ) - ϵ n - 1 2 * ⁢ ( t ) - ϵ n ) ∫ Ω | u | 2 * ⁢ ( t ) - ϵ n | y | t ≤ C ,

which implies the desired result. ∎

Let λ1>1. Then, for each n, (1.4) has infinitely many sign-changing solutions {un,l}l=1∞ such that for each l, the sequence {un,l} is bounded in H01⁢(Ω) and the augmented Morse index of {un,l} is greater than or equal to l.

Proof.

First, from Propositions 2.2–2.4, it follows that Iϵn satisfies all the conditions of [42, Theorem 2]. Thus, Iϵn has a sign-changing critical point un,l∈H01⁢(Ω) at the level cn,l with cn,l≤supEl+1⁡Iϵn and m*⁢(un,l)≥l. Now it remains to show that the sequence {un,l} is bounded for each l.

We claim that there exists T1>0, independent of n and l, such that

c n , l ≤ T 1 ⁢ λ l + 1 2 * ⁢ ( s ) - ϵ 0 2 ⁢ ( 2 * ⁢ ( s ) - ϵ 0 - 2 ) .

To see this, since 2*⁢(s)-ϵ0>2, we have

where C>0 is a constant independent of n and l. Moreover, from the fact that 2*⁢(s)-ϵ0<2*⁢(s)-ϵn, we get that there exist C1,C2>0, independent of n and l, such that

| u | 2 * ⁢ ( s ) - ϵ 0 , s , Ω ≤ C 1 ⁢ | u | 2 * ⁢ ( s ) - ϵ n , s , Ω + C 2 .

Therefore,

where C3,C4>0 are independent of n and l.

So from (2.10) and (2.11), for any u∈El+1 we obtain

where Ci>0 (i=1,…,6) and T1>0 are independent of n and l.

Moreover, we have

Then, from (2.12) and (2.13), we conclude that

I ϵ n ⁢ ( u n , l ) ∈ ( 0 , T 1 ⁢ λ l + 1 2 * ⁢ ( s ) - ϵ 0 2 ⁢ ( 2 * ⁢ ( s ) - ϵ 0 - 2 ) ) .

So,