Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian

Lemma 3.1

For any ε ∈ (0, 2s∗-ζ), the functionalI_εsatisfies the Palais-Smale ((PS) for short) condition.

Proof

Suppose {u_n} ⊂ H0s(Ω) is a (PS) sequence for I_ε, i.e.,

We have

which implies that {u_n} is bounded sequence in H0s(Ω). Hence, there exists u₀ ∈ H0s(Ω) such that

By Lemma 2.1, we have

and

It follows that

which implies that

and the proof is completed.□

Lemma 3.2

Suppose m ≥ 1. Then there exists R = R(E_m) > 0, such that for allε ∈ (0, 2s∗ − ζ),

whereBRc:=H0s(Ω)∖BRandBR={u∈H0s(Ω):∥u∥H0s(Ω)≤R}.

Proof

Define an auxiliary functional I_* : H0s (Ω) → ℝ given by

Noting that

it is easy to check that

holds for any ε ∈ (0, 2s∗ − ζ). Since any norm in finite dimensional space is equivalent,

for any fixed m ≥ 1. Thus the result follows.□

Lemma 3.3

For anyε ∈ (0, 2s∗ − ζ), there existρ_ε > 0 andα_ε > 0 such that

whereBρε={u∈H0s(Ω):∥u∥H0s(Ω)≤ρε}.

Proof

For u ∈ H0s(Ω), by Lemma 2.1, there exists C(ε) > 0 such that

Noting that 2 < 2s∗ − ε, we conclude that there exist ρ_ε > 0 and α_ε > 0 such that

as required.□

Definition 3.1

Given g ∈ H^−s(Ω), define the operator T_s : H^−s(Ω) → H0s(Ω), by T_s(g)= tr_Ωw, where w ∈ H0,Ls(C) solves

Lemma 3.4

There exists ϑ₀ > 0 such that for any ϑ ∈ (0, ϑ₀], there holds

and

Moreover, every nontrivial solutionsu∈Pϑ+andu∈Pϑ−of (2.3) are positive and negative, respectively.

Remark 3.1

Note that there exists a constant C > 0 independent of p ∈ [2, 2^*] such that ∥u∥_p ≤ C ∥u∥_2^* for all p ∈ [2, 2^*], as in the proof of Lemma 5.2 in [16], one can show that there exists ϑ₀ > 0 such that for any ϑ ∈ (0, ϑ₀], there holds Aε(∂Pϑ±)⊂Pϑ± for all ε > 0 small enough.

Lemma 3.5

DefineKε,c1:=Kε,c∩W,Kε,c2:=Kε,c∩Q,where 𝓚_ε,c := {u∈H0s(Ω):Iε(u)=c,Iε′(u)=0}.Letϱ > 0 be such that(Kε,c1)ϱ⊂Wwhere(Kε,c1)ϱ:={u∈H0s(Ω): dist(u,Kε,c1)<ϱ}.Then there exists anδ₀ > 0 such that for any 0 < δ < δ₀, there existsη ∈ C([0, 1] × H0s(Ω),H0s(Ω))satisfying:

1. η(t, u) = u for t = 0 oru∉Iε−1([c−δ0,c+δ0])∖(Kε,c2)ϱ.

2. η(1,Iεc+δ∪W∖(Kε,c2)3ϱ)⊂Iεc−δ∪Wandη(1,Iεc+δ∪W)⊂Iεc−δ∪WifKε,c2=∅.HereIεd={u∈H0s(Ω):Iε(u)≤d}for any d ∈ ℝ.

3. η(t, ⋅) is odd and a homeomorphism ofH0s(Ω) for t ∈ [0, 1].

4. I_ε(η(⋅, u)) is non-increasing.

5. η(t, W) ⊂ W for any t ∈ [0, 1].

Proof of Theorem 1.1

Here and in the sequel, we fix λ ∈ (0, λ1s). As in [30], we divide the proof into three steps.

Step 1. For any ε ∈ (0, 2^* − ζ) small, we define the minimax value c_ε,k for the perturbed functional I_ε(u) with k = 2, 3, ⋯. Set

where R > 0 is given by Lemma 3.2. Note that id ∈ G_m, thus G_m ≠ ∅. For k ≥ 2, we define

where γ(K) is the Krasnoselskii genus of the symmetric closed se K, i.e. the smallest integer n such that there exists and odd continuous map σ : K → Sⁿ⁻¹. From [22], Γ_k possess the following properties:

1. Γ_k≠ ∅ and Γ_k+1⊂ Γ_k for all k ≥ 2.

2. If ϕ ∈ C(H0s(Ω),H0s(Ω)) is odd and ϕ = id on ∂𝓑_R ∩ E_m, then ϕ(A) ∈ Γ_k if A ∈ Γ_k for all k ≥ 2.

3. If A ∈ Γ_k, Z = −Z is open and γ(Z) ≤ s < k and k − s ≥ 2, then A \ Z ∈ Γ_k−s.

Now, for k = 2, 3, ⋯, we can define the minimax value c_{ε, k} given by

We need to show that c_{ε, k} (k ≥ 2) are well defined (that is for any A ∈ Γ_k, A ∩ Q ≠ ∅) and c_{ε, k} ≥ α_ε > 0, where α_ε is given by Lemma 3.3.

Consider the attracting domain of 0 in H0s (Ω):

Note that 𝓓 is open, since 0 is a local minimum of I_ε and by the continuous dependence of ODE on initial data. Moreover, ∂𝓓 is an invariant set and

In particular, there holds

by the similar arguments to Lemma 3.4 in [4]. Now we claim that for any A ∈ Γ_k with k ≥ 2, it holds

Set

with γ(Y) ≤ m − k and k ≥ 2. Define

Obviously, 𝓞 is a bounded open symmetric set with 0 ∈ 𝓞 and 𝓞 ⊂ 𝓑_R ∩ E_m. Thus, by the Borsuk-Ulam theorem that γ(∂𝓞) = m and by the continuity of h, h(∂𝓞) ⊂ ∂𝓓. As a consequence,

and therefore

by the “monotone, sub-additive and supervariant” property of the genus (cf. Proposition 5.4 in [29]). Since Pϑ+∩Pϑ−∩∂D=∅, one has

Thus for k ≥ 2, we conclude that

which proves (3.3). Therefore, it follows from (3.3) that A ∩ Q ≠ ∅. Moreover, we have

because ∂𝓑_ρε ⊂ 𝓓 and sup_{A ∩ Q}I_ε ≥ inf_∂𝓓I_ε ≥ inf_∂𝓑ρεI_ε ≥ α_ε > 0 by Lemma 3.3.

Hence, c_{ε, k} are well defined for all k ≥ 2 and 0 < α_ε ≤ c_{ε, 2} ≤ c_{ε, 3} ≤ ⋯.

Now, we claim

which implies that there exists a sign-changing critical point u_{ε, k} such that

and c_{ε, k} → ∞, as k → ∞. It can be done, using deformation Lemma 3.5 following the same arguments as in the proof of Step 1 in [30].

Step 2. We show that for any fixed k≥2,∥uε,k∥H0s(Ω) is uniformly bounded with respect to ε, and then u_{ε, k} converges strongly to u_k in H0s (Ω) as ε → 0.

In fact, using the same Γ_k above, we can also define the minimax value for the auxiliary functional I_* (see (3.1) by

Here, choosing R > 0 sufficiently large if necessary, we point out that Lemma 3.2 also holds for I_*. Then from a ℤ₂ version of the Mountain Pass Theorem (see Theorem 9.12 in [22]), for each k ≥ 2, β_k > 0 is well defined and

Since

holds for any ε∈(0,2s∗−ζ), by the definition of c_{ε, k} and β_k, we have

Therefore, for fixed k ≥ 2, c_{ε, k} is uniformly bounded for ε ∈ (0, 2s∗ − ζ), i.e., there exists C = C(β_k, Ω) > 0 independent on ε, such that c_{ε, k} ≤ C uniformly for ε. Since u_{ε, k} is a nodal solution of (2.3) and I_ε(u_{ε, k}) = c_{ε, k}, one concludes that

which implies that ∥uε,k∥H0s(Ω)≤C uniformly with respect to ε. Denote

Then, w_{ε, k} is a solution of (2.4) satisfying ∥wε,k∥H0,Ls(C)≤C uniformly with respect to ε. So we can apply Theorem 2.1 and obtain a subsequence {w_εn,k}_n∈ℕ, such that

for some wk∈H0,Ls(C). We set

Then,

by the trace inequality and also c_εn,k → c_k. Thus u_k is a solution of (1.1) and I(u_k) = c_k. Moreover, since u_εn,k is sign-changing, similar to Step 1, by Lemma 2.1, we can show that u_k is still sign-changing.

Step 3. We are in a position to prove that the functional has infinitely many sign-changing critical points. Recalling that c_k is non-decreasing with respect to k, we distinguish two cases:

Case I: There exist 2 ≤ k₁ < ⋯ < k_i < ⋯, satisfying c_k1 < ⋯ < c_ki < ⋯.

Case II: There is a positive integer l such that c_k = c for all k ≥ l.

Obviously, in Case I, problem (1.1) has infinitely many sign-changing solutions such that I(u_i) = c_ki, thus we are done. So we assume in the sequel that Case II holds. From now on, we suppose that there exists a δ > 0, such that I(u) has no sign-changing critical point u with

Otherwise, the result follows.

We claim that

where 𝓚_c := {u ∈ H0s (Ω) : I(u) = c, I′(u) = 0} and Kc2 = 𝓚_c ∩ Q. If it is true, I(u) has infinitely many sign-changing critical points and thus we are done. Here we borrow some ideas used in [11]. Arguing by contradiction, suppose that

(note that Kc2 ≠ ∅). Moreover, we assume Kc2 contains only finitely many critical points, otherwise the proof is completed. As a consequence, Kc2 is compact. Clearly, 0 ∉ Kc2. Thus there exists a open neighborhood N in H0s (Ω) with Kc2 ⊂ N such that γ(N) = γ( Kc2).

Define

Now we claim that for ε > 0 small, I_ε has no sign-changing critical point in U_ε. Indeed, if not, we suppose that there exist ε_n → 0 and u_n∈ U_εn satisfying Iεn′(un)=0,withun±≠0,andun∉N. Obviously,

Then, similar to (3.5), one can obtain that ∥un∥H0s(Ω)≤C uniformly with respect to n. Set

By Lemma 2.1w_n is a solution of (2.4) satisfying ∥wn∥H0,Ls(C)≤C uniformly with respect to n. Therefore, by Theorem 2.1, we obtain, up to a subsequence, that

for some w ∈ H0,Ls (𝓒). We set

Clearly, u_n → u strongly in H0s (Ω). Thus

But u is still sign-changing, a contradiction.

From the above observation, one can easily show that for any ε > 0 small, there exists a constant α_ε > 0 such that

Then, as in [11], standard techniques show that for ε > 0 small enough, there exists an odd homeomorphism η ∈ C( H0s(Ω), H0s(Ω)) such that η(u) = u for u∈Iεc−2δ and

See for example the proof of Theorem A.4 in [22] and also Lemma 5.1 in [20].

Now fix k > l. Since c_{ε, k}, c_ε,k+1 → c as ε → 0, we can find an ε > 0 small, such that

By the definition of c_ε,k+1, we can find a set A ∈ Γ_k+1, A = h(𝓑_R ∩ E_m ∖ Y), where h ∈ G_m, m ≥ k + 1, γ(Y)≤ m − (k + 1), such that

for any u ∈ A ∩ Q, which implies, A⊂Iεc+δ2∪W Then by (3.6), we have

Let Y~ = Y ∪ h⁻¹(N). Then Y~ is symmetric and open, and

Therefore one can obtain that A^:=η(h(BR∩Em∖Y~))∈Γk by (2^∘) and (3^∘) above. Consequently, by (3.7),

which contradicts to

Hence the proof is completed and the functional I has infinitely many sign-changing critical points.  □