Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Moreover, with the aid of variational methods, the multiplicity of periodic solutions for Hamiltonian systems has also been extensively investigated in many papers and books (see [15][16][17][18][19][20][21][22][23] and the references therein). In particular, in [24], using the linking theorem of Schechter [19,20], Bonanno et al. have discussed the existence of two nontrivial solutions for second-order Hamiltonian systems with subquadratic potentials at zero. For infinitely many solutions of subquadratic second-order Hamiltonian systems, Zou and Li [22] obtained two existence theorems via the minimax technique of Fei [25], and Zhang and Liu [23] obtained an existence theorem via the variant fountain theorem of Zou [26]. After that, by the symmetric mountain pass theorem of Kajikiya [27], Yi and Tang [21] obtained an existence theorem, which unifies and improves upon theorems of Zou and Li and Zhang and Liu [22,23].
Inspired by the ideas of Ye and Tang and Bonanno et al. [21,24], in this paper, we shall study the existence of two nontrivial solutions and infinitely many solutions for problem (1), where V(t, x) is subquadratic at zero. Based on the index theory of Dong [15,16], the linking theorem of Schechter [19,20], and the symmetric mountain pass theorem of Kajikiya [27], we will prove the existence of two nontrivial solutions and infinitely many solutions. Applying the results to problems (3) and (4), we obtain some new theorems. Meanwhile, some examples of problems (3) and (4) are given to illustrate the validity of our result and point out that linear terms B(t) < 0 and λ ∈ (0, +∞) are allowed, which show that our results are also new even in the cases of (3) and (4).

Remark 1.
In eorem 1, the operator Λ B is defined as and a(x, x) are defined by (13) and (15) in Section 2, respectively. e existence of δ 0 is proved in (33) of Section 3.

Remark 2.
In eorem 2, we do not need any restrictions on i P α,β (B) and ] P α,β (B), which means that α,β (B) � 0 are allowed. e paper is arranged as follows. In Section 2, we recall some useful conclusions of index theory for linear secondorder Hamiltonian systems from [15,16] and verify that problem (1) possesses a variational construction in Z. In Section 3, using the linking theorem of Schechter [19,20] and the symmetric mountain pass theorem of Kajikiya [27], we prove eorems 1 and 2. In Section 4, we investigate their applications to Sturm-Liouville equations with the mixed boundary value conditions and the Neumann boundary value conditions. Meanwhile, some examples are given to show that our results are also new even in the cases of problems (3) and (4).

Preliminaries and Variational Setting
For the reader's convenience, we first recall some useful conclusions of index theory for linear second-order Hamiltonian systems given in [15,16], respectively.
By Section 2.3 in [15], we know that Λ is self-adjoint and σ(Λ) � σ d (Λ) � λ ∈ R: λ belongs to the point spectrum of Λ is bounded from below. We define a bilinear form as follows: for all x, y ∈ Z, where (·, ·) is the usual inner product in R n , c(s) � cot s as s ∈ (0, π), c(s) � 0 as s � 0 or s � π, and Similar to the proof of Proposition 1.17 in [28], we know that Z is a Hilbert space and Z � D(|Λ| 1/2 ). By Lemma 2.3.1 of [15], we find that Z � D(|Λ| 1/2 ) can be equipped with the equivalent norm for each x ∈ Z. Clearly, the embeddings Z ⟶ L 2 � X and for all x, y ∈ Z.

Journal of Mathematics
We call ] P α,β (B) and i P α,β (B) the nullity and index of B with respect to the bilinear form ψ P,B α,β (·, ·).
Next, let us consider the functional I defined by From assumption (V 0 ), using eorem 1.4 [17], it is easy to check that I is continuously differentiable and weakly lower semicontinuous on Z, and for all x, y ∈ Z. If I ′ (x) � 0, then we have Form (25), it follows that is shows that x(t) satisfies (11)- (12), which means that the critical points of I correspond to the classical solutions of problem (1).

Proofs of the Theorems
In order to prove eorem 1, we recall some results of linking given by Schechter [19,20].
Let E be a reflexive Banach space with norm ‖ · ‖. e set Φ � Γ(t): { } is to have the following properties: one has sup 0≤t≤t 0 ,u∈A Definition 2 (see [19], Theorem 3 (see [20], eorem 4.1]). Let F, J ∈ C 1 (E, R) be bounded on bounded sets and let where I is an open interval contained in (0, +∞). Assume that G μ satisfies and for each μ ∈ I.
In order to prove eorem 2, we need to recall some propositions of genus and the symmetric mountain pass theorem given by Kajikiya [27].
Let E be a real Banach space and let Σ denote the family of sets A ⊂ E\ 0 { } such that A is closed in E and symmetric with respect to 0 (i.e., x ∈ A⇒ − x ∈ A ). For A ∈ Σ, we define a genus Υ(A) of A by the smallest integer n such that there exists an odd continuous mapping from A to R n \ 0 { }. If such a n does not exist, we define Υ(A) � ∞. Moreover, set (Υ 4 ) e n-dimensional sphere S n has a genus of n + 1 by the Borsuk-Ulam theorem.
en, Ψ possesses a sequence of critical points x k such that Ψ(x k ) ≤ 0, x k ≠ 0 and lim k⟶∞ x k � 0.

Applications to Sturm-Liouville Equations and Examples
In this section, we consider the applications of eorems 1 and 2 to Sturm-Liouville equations with the mixed boundary value conditions and the Neumann boundary value conditions. Meanwhile, some examples of problems (3) and (4) are given to illustrate the validity of our result, and the result is also new even in the cases of (3) and (4). As first special case, we consider the mixed boundary value problem (3): where λ > 0, B(t) ∈ L ∞ ([0, 1], R), and P(t) ∈ C 1 ([0, 1], R) with P(t) being positive definite for t ∈ [0, 1]. In problem (1), taking n � 1, α � 0, β � π/2, the following corollary is immediately obtained from eorem 1.

s)ds. en, for problem (3), the conclusion of eorem 1 is still valid.
An example of Corollary 1 is given below.

Remark 6.
Noticing that B(t) ≡ M ≥ 0 and λ � 1 must be assumed in the condition of eorem 1 in [11,12], we easily see that eorem 1 in [11,12] cannot be applied to Example 3. is shows that Corollary 3 is also new even in the cases of (4). Taking n � 1, α � β � π/2, from eorem 2, we have the following corollary. en, for problem (4), the conclusion of eorem 2 is still valid.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All authors typed, read, and approved the final manuscript.