Sign-Changing Solutions of Fractional 𝑝-Laplacian Problems

Theorem 1.1.

Suppose that $f_{1}$–$f_{3}$ hold. Then problem (1.1) admits a sign-changing solution. Furthermore, if f is odd in u, then problem (1.1) admits infinitely many sign-changing solutions.

Theorem 1.2.

Suppose that f∈C1⁢(Ω¯×R,R) and $f_{1}$, $f_{2}$, $f_{4}$, $f_{5}$ hold. Then problem (1.1) admits one sign-changing solution u*∈E such that Φ⁢(u*)=mα. Moreover, mα>2⁢cα.

Definition 2.1 (see [26]).

{ P 1 , P 2 } is called an admissible family of invariant sets with respect to Φ at level c provided the following deformation property holds: If Kc∖W=∅, then there exists ϵ0>0 such that, for ϵ∈(0,ϵ0), there exists a continuous map σ:X→X satisfying

1. σ ⁢ ( P 1 ¯ ) ⊂ P 1 ¯ , σ⁢(P2¯)⊂P2¯,

2. σ | Φ c - ϵ = id ,

3. σ ⁢ ( Φ c + ϵ ∖ W ) ⊂ Φ c - ϵ .

Theorem 2.2 (see [26]).

Let Φ∈C1⁢(X,R) and P1,P2 be open subsets of X. Assume that {P1,P2} is an admissible family of invariant sets with respect to Φ at level c≥c*≐infu∈Σ⁡Φ⁢(u) and there exists a continuous map φ0:Δ→X such that

1. φ 0 ⁢ ( ∂ 1 ⁡ Δ ) ⊂ P 1 , φ0⁢(∂2⁡Δ)⊂P2,

2. φ 0 ⁢ ( ∂ 0 ⁡ Δ ) ∩ M = ∅ ,

3. sup u ∈ φ 0 ⁢ ( ∂ 0 ⁡ Δ ) ⁡ Φ ⁢ ( u ) < c * ,

where

Define

where Γ≐{φ∈C⁢(Δ,X):φ⁢(∂1⁡Δ)⊂P1,φ⁢(∂2⁡Δ)⊂P2,φ|∂0⁡Δ=φ0|∂0⁡Δ}. Then c0≥c* and Kc0∖W≠∅.

Definition 2.3 (see [26]).

We say that {P1,P2} is a G-admissible family of invariant sets with respect to Φ at level c provided the following deformation property holds: There exists ϵ0>0 and a symmetric open neighborhood Nc of Kc∖W with γ⁢(Nc¯)<+∞ (Nc=∅ if Kc∖W=∅) such that, for ϵ∈(0,ϵ0), there exists a continuous map σ:X→X satisfying

1. σ ⁢ ( P 1 ¯ ) ⊂ P 1 ¯ , σ⁢(P2¯)⊂P2¯,

2. σ ∘ G = G ∘ σ ,

3. σ | Φ c - 2 ⁢ ϵ = id ,

4. σ ⁢ ( Φ c + ϵ ∖ ( N c ∪ W ) ) ⊂ Φ c - ϵ .

Theorem 2.4 (see [26]).

Assume that Φ∈C1⁢(X,R) is a G-invariant functional, P1,P2 are open subsets of X, FG≐{u∈X:G⁢u=u}. Assume that {P1,P2} is a G-admissible family of invariant sets with respect to Φ at level c≥c*≐infu∈Σ⁡Φ⁢(u). Suppose that there exists a continuous map φn:Bn→X such that

1. φ n ⁢ ( 0 ) ∈ M , φ n ⁢ ( - t ) = G ⁢ φ n ⁢ ( t ) for all t ∈ B n ,

2. φ n ⁢ ( ∂ ⁡ B n ) ∩ M = ∅ ,

3. c 0 ≐ sup u ∈ F G ∪ φ n ⁢ ( ∂ ⁡ B n ) ⁡ Φ ⁢ ( u ) < c * ≐ inf u ∈ Σ ⁡ Φ ⁢ ( u ) .

For j∈N, define

where Γj≐{B:B=φ(Bn∖Y) for some φ∈Gn, n≥j, and open Y⊂Bn such that -Y=Y and γ(Y¯)≤n-j} and Gn≐{φ:φ∈C(Bn,X),φ(-t)=Gφ(t) for t∈Bn, φ⁢(0)∈M and φ|∂⁡Bn=φn|∂⁡Bn}. Then, for j≥2, cj≥c* and Kcj∖W≠∅. Furthermore, cj→+∞ as j→+∞.

Lemma 2.5 (see [5]).

For p∈(2,+∞), there exist positive constants d1,d2 such that, for all ξ,η∈RN,

For p∈(1,2], there exist positive constants d3,d4 such that, for all ξ,η∈RN,

Lemma 3.1.

The operator A is well-defined, continuous and compact.

Proof.

Firstly, we prove that A is well-defined. For given u∈E, consider the minimization problem

where J1⁢(w)≐1p⁢∥w∥βp-∫Ωg⁢(x,u)⁢w⁢d⁢x,w∈E. Clearly, J1 is C1, coercive, bounded from below and weakly lower semicontinuous, which implies that there exists w*∈E to be a minimizer of J1. Furthermore, we can see that J1 admits a unique minimizer. In fact, if we assume that w1,w2∈E are two different minimizers of J1, then, together with Lemma 2.5 and (3.1), defining w~i⁢(x,y)=wi⁢(x)-wi⁢(y),i=1,2, we can get, for p>2,

for some C1>0. On the other hand, for p∈(1,2], using Lemma 2.5 and (3.1) again, it follows that

for some C2>0, which implies that ∥w1-w2∥β=0 and thus w1≡w2. Hence, both of these two cases imply that w1≡w2.

Secondly, we prove that A is continuous. Let {un}⊂E such that un→u∈E strongly in E. Clearly, {un} is uniformly bounded in E. Then un→u strongly in Lr⁢(Ω) for r∈[1,pα*), un⁢(x)→u⁢(x) for a.e. x∈Ω. Denote w=A⁢(u), wn=A⁢(un). By (3.1),

In view of $f_{1}$–$f_{3}$, we can see that {wn} is uniformly bounded in E, i.e., ∥wn∥β≤M1 for all n∈ℕ and some M1>0. When p>2, using Lemma 2.5, the Hölder inequality, (3.1) and defining w~n⁢(x,y)=wn⁢(x)-wn⁢(y), w~⁢(x,y)=w⁢(x)-w⁢(y), we obtain

for some C3>0, where I1≐∫Ω(g⁢(x,un)-g⁢(x,u))⁢(wn-w)⁢d⁢x. When p∈(1,2], we get from Lemma 2.5 again that

for some C4>0. Take R≥r. Define χ∈C0∞⁢(ℝ) such that χ⁢(t)=1 for |t|≤R, χ⁢(t)=0 for |t|≥R+1 and 0≤χ⁢(t)≤1 for all t∈ℝ. Set

By $f_{1}$, $f_{2}$, there exists C5>0 such that

Then, by the Hölder and Sobolev inequalities, we get

for some C6>0. Hence, in both of there two cases we can apply the dominated convergence theorem to obtain that ∥wn-w∥→0 as n→+∞.

Now we prove that A is compact. Assume that {un}⊂E is uniformly bounded. Let wn=A⁢(un). By (3.1) and similar arguments as above, it follows that ∥wn∥β≤M2 for some M2>0 independent of n. Thus, there exist subsequences of {wn} and {un}, which are still denoted by {wn} and {un} for simplicity, such that wn⇀w¯∈E, un⇀u¯∈E weakly in E. Then wn→w¯, un→u¯ strongly in Lr⁢(Ω) for r∈[1,pα*), wn⁢(x)→w¯⁢(x), un⁢(x)→u¯⁢(x) for a.e. x∈Ω. By similar arguments as above, we can get that ∥wn-w¯∥β→0 as n→+∞. This completes the proof. ∎