Existence of Solutions for Sublinear Kirchhoff Problems with Sublinear Growth

1 School of Insurance, Southwestern University of Finance and Economics, Chengdu, 611130, China 2Collaborative Innovation Center of Financial Security, Southwestern University of Finance and Economics, Chengdu, 611130, China 3School of Economics, Southwestern University of Finance and Economics, Chengdu, 611130, China 4High Temperature Materials and Magnesite Resources Engineering, University of Science and Technology Liaoning, Anshan, Liaoning 114001, China

condition, Wang and Han considered a class of sublinear nonlinearities for problem (1) and showed the existence of infinitely many solutions when (, ) is odd in  which is the following theorem.

𝑡󳨀→0
( inf where Then Equation ( 1) possesses a sequence of weak solutions With coercive condition (), the authors obtained a new compact embedding theorem, stated as follows.
Remark 3. From Lemma 2, for any  ∈ [1, 2 * ), there exists a constant   > 0 such that A natural question is whether there exists solution for problem (1) if there is no symmetrical condition on (, ) and (, ) is not integrable in .In this paper, we try to give an existence theorem on this problem.Motivated by the above papers, we consider problem (1) with some new sublinear nonlinearities and, before we state our result, we assume that Γ is a continuous function space such that, for any () ∈ Γ, there exists a constant  0 > 0 such that (i) () > 0 for all  > 0; (ii) ∫   0 (1/()) → +∞ as  → +∞.
In order to obtain the critical points of the corresponding functional, we consider the following function space in this paper: with the inner product For any  ∈ , it follows from ( 6) that In order to show that the infimum can be achieved, consider a minimizing sequence It is easy to see that there exists ũ ∈  such that   ⇀ ũ.
where () ∈  1 (R) such that (, ) is a  1 class function.It is easy to see that (, ) satisfies (1)-(4).However, since lim →0 ((, )/ 2 ) = c2 ( + 1), we see that (, ) does not satisfy (2) Remark 6. Condition (1) is introduced by Wang and Xiao in [26] to prove the existence of periodic solutions for a class of subquadratic Hamiltonian systems.As we know, this is the first time to use such condition on the existence of solutions for sublinear Kirchhoff equations.

Proof of Theorem 1
By standard arguments, we know that the functional  :  → R, defined by is well defined and the critical points of  are the solutions for problem (1).
By above discussions, we can see that  is of  1 class and satisfies () condition.Similar to we obtain that  is bounded from below.Then  = inf  () is a critical value of  and there exists  such that () = .Finally, we show that  ̸ ≡ 0. With  > 0 being small enough, it follows from (2) and (11) that which implies that  is bounded from below on .Since we have Lemma 2, it follows from a standard argument, we obtain that ‖  −  0 ‖ → 0 as  → ∞.Hence  satisfies the () condition.Proof of Theorem 1.