Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.


Introduction and Main Results
In their excellent paper [1] in 1991, Brezis-Nirenberg proved the existence of two nonzero critical points under suitable assumptions by a negative gradient flow and linking. The obtained result was applied to second-order Hamiltonian systems and yielded the existence of three periodic solutions for these systems. After that, in [2], Tang obtained one existence of at least three periodic solutions for second-order Hamiltonian systems by using two nonzero critical points theorem in [1]. This result generalizes the corresponding result in [1]. In addition, in [3], one new existence of at least three periodic solutions for second-order Hamiltonian systems was obtained by using the critical point reduction method and two nonzero critical points theorem in [1,4].
Moreover, after the three critical points theorem (see [5,6]) of Ricceri appearing, it has proved to be one of the most often widely used to solve differential equations, such as elliptic partial differential equations satisfying Dirichlet boundary value conditions (see [7][8][9][10][11][12]) and second-order Hamiltonian systems satisfying periodic boundary value conditions (see [3,13]). Recently, a general three critical points theorem is given by Averna-Bonanno [14]. After that, in [15], the existence of at least three periodic solutions was established for a class of p-Hamiltonian systems by two general three critical points theorems in [6,14].
It is well known that the above-mentioned problems can be displayed as a self-adjoint operator equation. For the multiplicity of self-adjoint operator equation, Chang proved the existence of three distinct solutions under suitable assumptions via Morse theory in his excellent paper [16] in 1981. The obtained results were applied to second-order Hamiltonian systems and wave equations and yielded the existence of three periodic solutions for these systems. Similar results can be found in [17][18][19][20][21].
Inspired by the ideas of [16,18,19], in this paper, we reconsider in the framework of the operator equations some theorems proved in [1-3, 9, 10, 15, 22] and generalize these theorems to nonlinear self-adjoint operator equations. Our technical approach is based on some three critical points theorems obtained by Brezis-Nirenberg [1], Averna-Bonanno [14], and Ricceri [6] or the critical point reduction method in [4]. As applications, we consider the existence of three solutions for second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions. Meanwhile, we give some examples (see Examples 1-4 in Section 4.1) to illustrate the effectiveness of our results and point out that our results improve and generalize the corresponding results in [1-3, 9, 10, 15, 22] via several remarks (see Remarks 18,20,22,24,29, and 33 in Section 4).
Before expressing our main results, for the convenience of the readers, we give a brief introduction to the index theory and the setting of the space; for some details, we refer to [23,24] (or see Section 2).
Noticing that i A ðB 0 Þ = 0, ν A ðB 0 Þ ≠ 0, we have min σðA − B 0 Þ = 0. Then, we can define another inner product with the corresponding norm Clearly, ∥·∥ is equivalent to ∥·∥ Z . Let Since Observe that the embedding Z↪X is compact. Further, assume that there exists a Banach space X 0 satisfying Z ⊂ X 0 ⊂ X and the embeddings Z↪X 0 and X 0 ↪X are compact and continuous, respectively. Thus, there exists k 1 > 0 such that ∥x∥ X 0 ≤ k 1 ∥x∥ for all x ∈ Z. Let From (8), we have Let X 1 be a nontrivial subspace of X. For B 1 , B 2 ∈ L s ðXÞ, we write B 1 ≤ B 2 with respect to X 1 if and only if ðB 1 x, xÞ X ≤ ðB 2 x, xÞ X for all x ∈ X 1 ; we write B 1 < B 2 w.r.t. X 1 if and only if ðB 1 x, xÞ X < ðB 2 x, xÞ X for all x ∈ X 1 \ fθg. If X 1 = X, then we just write B 1 ≤ B 2 or B 1 < B 2 .
Next, we use the index ði A ðBÞ, ν A ðBÞÞ ∈ N × N defined in [23,24] (see Section 2) for all B ∈ L s ðXÞ to reach our main results.
for all x ∈ X ðΦ 2 Þ there exists a positive definite B 3 ∈ L s ðXÞ such that ΦðxÞ − ð1/2ÞðB 3 x, xÞ X is convex Then, problem (3) has at least one solution with saddle character in Z (i.e., the solution is a saddle point).
Assume in addition that for all ∥x∥≤δ. Then, problem (3) has at least three distinct solutions in Z.
for all x, y ∈ X ðΦ 5 Þ there exists c 3 > ðc 2 for all x ∈ kerðA − B 0 Þ, where the constants k 1 and ε 2 are given by (9) and (49), respectively. Then, problem (3) has at least one solution in Z. Further, assume that Φ satisfies ðΦ 3 Þ. Then, problem (3) has at least three distinct solutions in Z.
where E = fx ∈ Zjkxk X 0 ≤cg. Then, there exist an open interval Λ ⊆ ½0,+∞Þ and a positive number r 0 with the following property: for each η ∈ Λ and for every Ψ : X ⟶ R satisfies ðΨ 0 Þ, there exists ρ > 0 such that, for each μ ∈ ½0, ρ, problem (4) has at least three distinct solutions whose norms in Z are less than r 0 .

Remark 5.
For some examples of the scalar equations corresponding to Theorems 1-4, we will give in Section 4, respectively.
The paper will be organized as follows. In Section 2, we recall first a saddle point reduction theorem under rather general assumptions in [4], a general two nonzero critical points theorem in [1], and two general three critical points theorems in [6,14] as well as one proposition in [25]. Then, we recall some useful conclusions of index theory for linear self-adjoint operator equations from [23,24]. Finally, we quote a lemma in [26], which verifies that (3) and (4) possess a variational construction in Z. In Section 3, we prove Theorems 1-4. In Section 4, we will investigate its applications to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and second-order elliptic partial differential equations satisfying Dirichlet boundary value conditions. Meanwhile, we give some examples and some remarks to illustrate that the corresponding results of [1-3, 9, 10, 15, 22] are special cases of these results in a sense.

Preliminaries
In order to prove our main results, we recall first four lemmas and one proposition due to [1,4,6,14,25], respectively.
Lemma 8 (see [6], Theorem 1). Let Z be a reflexive real Banach space,Ĩ ⊆ R an interval, J : Z ⟶ R a sequentially weakly lower semicontinuous C 1 functional whose derivative admits a continuous inverse on Z * , and J bounded on each bounded subset of Z, Φ : Z ⟶ R a C 1 functional with 3 Journal of Function Spaces compact derivative. Assume that for all η ∈Ĩ and that there exists β ∈ R such that Then, there exist a nonempty open set Λ ∈Ĩ and a positive number r 0 with the following property: for every η ∈ Λ and every C 1 functional Ψ : Z ⟶ R with compact derivative, there exists ρ > 0 such that, for each μ ∈ ½0, ρ, the equation has at least three solutions whose norms in Z are less than r 0 .
Next, we recall some definitions and propositions about index theory in [23,24].
Definition 11 (see [24], page 108). For any B ∈ L s ðXÞ, we define For any x, y ∈ Z if ψ a,B ðx, yÞ = 0, we say that x and y are ψ a,B -orthogonal. For any two subspaces Z 1 and Z 2 of Z if ψ a,B ðx, yÞ = 0 for any x ∈ Z 1 , y ∈ Z 2 , we say that Z 1 and Z 2 are ψ a,B -orthogonal.
The following important lemma is an immediate conclusion of Lemma 6 in [26].  (4)).

Proofs of the Theorems
In this section, we give the proofs of Theorems 1-4, respectively. First, with the aid of the critical point reduction method in [4] and the three critical points theorem in [1], we prove Theorem 1. For Theorem 2, we use Theorem 1 in [28] and the three critical points theorem in [1] to give its proof. In the Proof of Theorem 3, our technical approach is based on the three critical points theorem of Ricceri [6]. Finally, we use Proposition 3.1 in [25] and Theorem B in [14] to prove Theorem 4.
If inf fφðx 2 Þ: x 2 ∈ Z 2 g = 0, then all x 2,1 ∈ Z 2,1 with kx 2,1 k ≤ r are minima of φ, which shows that φ has infinite critical points. If inf fφðx 2 Þ: x 2 ∈ z 2 g < 0, then φ has at least two nonzero critical points via Lemma 7. Consequently, problem (3) has at least two nontrivial solutions in Z. Further, since ΦðθÞ = 0, we know that problem (3) has trivial solution θ. Hence, problem (3) has at least three distinct solutions in Z. The proof is complete.

Proof of Theorem 3. Let
for each x ∈ Z. From (33) and (35), we can see that a solution of J ′ðxÞ − ηΦ′ðxÞ − μΨ′ðxÞ = 0 must be a critical point of I 1 ðxÞ, which means that problem (4) also has a solution via Lemma 15. Obviously, J is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. From the conditions (Φ 0 ) and (Ψ 0 ), we know that Φ and Ψ are continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Moreover, J is bounded on each bounded subset of Z via (8).
Next, we prove that J admits a continuous inverse on Z. In fact, from (8) and (35), we have for all x, y ∈ Z, which shows that J ′ is uniformly monotone on Z. Moreover, using standard arguments ensure that J ′ also turns out to be coercive and hemicontinuous in Z. Applying Theorem 26.A of [29], we know that J admits a continuous inverse on Z.
Finally, setĨ = ½0,+∞Þ, we can see that all the assumptions of Lemma 8 are satisfied. Hence, our conclusion follows from Lemma 8.
Proof of Theorem 4. In order to use Lemma 10 to prove Theorem 4, we consider the functional I 1 defined by (33) as μ = 0. From the Proof of Theorem 3, it is not difficult to see that we only need to verify the validity of (i) in Lemma 10.

Applications to Second-Order Hamiltonian Systems and Elliptic Partial Differential Equations
In this section, as three examples of applications of the main results, we investigate second-order Hamiltonian systems satisfying generalized periodic boundary value conditions, second-order Hamiltonian systems satisfying Sturm-Liouville boundary value conditions, and elliptic partial differential equations satisfying Dirichlet boundary value conditions. For systematic researches of Hamiltonian systems, we refer to the excellent books [28,30,31]. Meanwhile, for researches of elliptic partial differential equations, we refer to [32][33][34].
Assume in addition that for a.e. t ∈ ½0, 1. Then, problem (73) has at least three distinct solutions in Z 1 .
Then, problem (73) has at least one solution in Z 1 . Further, assume that Vðt, xÞ satisfies ðH 3 Þ. Then, problem (73) has at least three distinct solutions in Z 1 .

Example 2. Let
for all x ∈ R n and t ∈ ½0, 1. It is easy to see that for all x ∈ R n and a.e. t ∈ ½0, 1. Note that for ε > 0, we have where E 1 = fx ∈ R n | jxj ≤cg.
Then, there exist an open interval Λ ⊆ ½0,+∞Þ and a positive number r 0 with the following property: for each η ∈ Λ and for every Fðt, xÞ: ½0, 1 × R n ⟶ R satisfies ðH 0 Þ, there exists ρ > 0 such that, for each μ ∈ ½0, ρ, problem (74) has at least three distinct solutions whose norms in Z 1 are less than r 0 .
Remark 22. In particular, set B 0 ðtÞ ≡ 0 and M = I n . Then, the following problem x ð Þ, a:e:t ∈ 0, 1 ½ , a similar conclusion of Theorem 21, wherek = ffiffi ffi 2 p . In addition, our result slightly generalizes Theorem 1 in [15] as T = 1, p = 2 in two aspects. At first, Theorem 21 requires Vðt, xÞ being measurable in t for every x ∈ R n instead of continuous in t for every x ∈ R n ; secondly Theorem 21 requires B 1 ðtÞ ∈ L ∞ ð½0, 1, L s ðR n ÞÞ instead of B 1 ðtÞ ∈ Cð½0, 1, L s ðR n ÞÞ: An example of problem (106) corresponding to Theorem 21 is given below.
Proof. Similar to the proof of Theorem 21, we need only to show that ðΦ 9 Þ follows from (H 9 ). Noticing that E ⊂ E 1 , we have which means that ðΦ 9 Þ holds and Therefore, we can apply Theorem 4 to obtain Theorem 23.
Remark 24. In particular, set B 0 ðtÞ ≡ 0 and M = I n . Then, the following problem (106) has a similar conclusion of Theorem