INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

Consider a class of nonlocal problems −(a− b ∫ Ω |∇u|dx)∆u = f(x, u), x ∈ Ω, u = 0, x ∈ ∂Ω, where a > 0, b > 0, Ω ⊂ R is a bounded open domain, f : Ω × R −→ R is a Carathéodory function. Under suitable conditions, the equivariant link theorem without the (P.S.) condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to a/(4b), and they are neither large nor small.


Introduction
Consider the following nonlocal problem where a > 0, b > 0, Ω ⊂ R N is a bounded domain, N ≥ 3, f : Ω × R −→ R is a Carathéodory function, that is, f (x, t) is measurable in x for every t ∈ R and continuous in t for a.e. x ∈ Ω.
for every t ∈ R and a.e. x ∈ Ω. ( The main result is the following theorem. Then problem (1.1) has infinitely many solutions whose energy increasingly tends to a 2 /(4b).
is nonzero and nonnegative, then problem (1.1) has infinitely many solutions whose energy increasingly tends to a 2 /(4b).
has infinitely many solutions for every λ > 0.
and to semilinear elliptic equations if b = 0, infinitely many solutions of which were obtained by using variational methods, but they are either small negative energy solutions or large energy solutions (see [2,6] for Kirchhoff type problems and see [5,8,9,12] for semilinear elliptic equations, and their references therein). Unusually, it is interesting that the infinitely many solutions in Theorem 1.1 are neither large nor small energy solutions. Comparing Corollary 1.1 with the results in [1,3,10], under the same conditions we obtain infinitely many nonzero solutions, but they obtained at least two solutions. Corollary 1.2 shows that the appearance of the nonlocal term may cause the change of the properties of the equation. In fact, as Corollary 1.2 states, when b > 0, problem (1.2) has infinitely many solutions for every λ > 0. But when b = 0, problem (1.2) has infinitely many solutions only for λ = aλ k , k = 1, 2, · · · , where {λ k } are the eigenvalues of −∆ in H 1 0 (Ω).

Proof of the main result
In this section, we will prove the main result by using the equivariant link theorem without the (P.S.) condition(see [7]). Let H 1 0 (Ω) be the usual Hilbert space with norm and inner product We choose an orthonormal basis Define the functional φ : H 1 0 (Ω) → R as follows: s)ds for all t ∈ R and a.e. x ∈ Ω. From (f 2 ) and (f 3 ), by a standard argument, we have φ ∈ C 1 (H 1 0 (Ω), R) and φ(−u) = φ(u) for any u ∈ H 1 0 (Ω). It is well-known that the weak solutions of problem (1.1) correspond to the critical points of the functional φ, and we have , then the functional φ satisfies the (P.S.) c condition.
Proof. Suppose that {u n } is a (P.S.) c sequence of the functional φ, that is, It follows from (f 1 ) that F (x, t) ≥ 0 for every t ∈ R and a.e. x ∈ Ω. Hence by φ(u n ) → c, one has for large n, which implies that {u n } is bounded. Hence, up to subsequences, there exists u 0 ∈ H 1 0 (Ω) such that u n ⇀ u 0 in H 1 0 (Ω), u n → u 0 in L p (Ω), u n (x) → u 0 (x) for a.e. x ∈ Ω. By the boundedness of {u n }, without loss of generality, we may assume that m = lim n→∞ ∥u n ∥ 2 .
For k > n, by the orthogonality of Y n and Z k , one has ⟨u k , v⟩ = 0 for every v ∈ Y n , which implies that lim k→∞ ⟨u k , v⟩ = 0 for every v ∈ Y n . It follows from the boundedness of {u k }, the density of ∪ ∞ n=1 Y n in H 1 0 (Ω) and Theorem 5.1.3 in [11] that {u k } weakly converges to zero. In the following, we give a slight generalization of Lemma 2.13 in [7].
Then there exists w k ∈ Y k with ∥w k ∥ ≤ 2r such that α k = φ(w k ). In the case that w k = 0, we have In the case that w k ̸ = 0, by (f 1 ), one has Hence, we have for any k ≥ 2.
then we have It follows from Lemma 2.2 that {u k } converges to zero weakly in H 1 0 (Ω) as k → ∞. By (f 2 ) and Lemma 2.3, one has ∫ Ω F (x, u k )dx → 0, which implies that lim k→∞ β k = 0. Hence as n → ∞ for k large enough. It is easy to see that c k is a critical value of φ by Lemma 2.1. By (2.3), (2.4) and the fact that b k ≤ c k , φ has infinitely many distinct critical values c k such that c k → a 2 4b as n → ∞, which implies that problem (1.1) has infinitely many distinct solutions. The proof of Theorem 1.1 is completed.