Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

which was first presented by Kirchhoff [1] as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings, where u = uðx, tÞ denotes the lateral displacement, ρ the mass density, P0 the initial tension, h the crosssection area, E the Young modulus of the material, and L the length of the string. Under different assumptions on f ðx, uÞ, many interesting results on the existence of solutions to (2) were obtained. We refer the interested readers to [2–14] and the references therein. However, we now face a new nonlocal term a − b Ð


Introduction and Main Results
This paper is concerned with the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions: where a, b > 0, 1 < p < 2, λ > 0, f ∈ L ∞ ðΩÞ is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3.
In the past two decades, the following Kirchhoff type problems with Dirichlet boundary value conditions have attracted great attention of many researchers. Such problems are often viewed as nonlocal because of the appear-ance of the term a + b Ð Ω j∇uj 2 dx, which implies that (2) is no longer a pointwise identity. It is worthwhile pointing out that the equation in (2) arises in various models of physical and biological systems. Indeed, problem (2) is related to the stationary analogue of the following equation: which was first presented by Kirchhoff [1] as an extension of the classical d'Alembert wave equation for free vibrations of elastic strings, where u = uðx, tÞ denotes the lateral displacement, ρ the mass density, P 0 the initial tension, h the crosssection area, E the Young modulus of the material, and L the length of the string. Under different assumptions on f ðx, uÞ, many interesting results on the existence of solutions to (2) were obtained. We refer the interested readers to [2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. However, we now face a new nonlocal term a − b Ð Ω j∇uj 2 dx, which is different from the well-known Kirchhoff type nonlocal term a + b Ð Ω j∇uj 2 dx. Now, there has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem (see [15][16][17][18][19][20][21][22]).
In particular, Yin and Liu [15] firstly studied this kind of problem: where 2 < p < 2 * , and obtained the existence and multiplicity of solutions for the problem.
In [16], Lei et al. considered where 1 < p < 2, and proved under certain condition on f λ ðxÞ , that there are at least two positive solutions. After this, the authors also studied the problem with singularity [17].
Wang et al. [20] investigated the nonlocal problem with critical exponent When μ is a nonnegative parameter and gðxÞ ∈ L 4/3 ðℝ 4 Þ, they showed the existence of multiple positive solutions.
Recently, Zhang and Zhang [22] studied the nonlocal problem where the parameter 0 < λ < aλ 1 , λ 1 is the first eigenvalue of operator −Δ, and g is a superlinear function with subcritical growth. By using the Mountain Pass Theorem, the authors obtained the existence of a nontrivial solution.
As far as we know, there is no work on the existence of solution to (1), which is just our purpose here. Moreover, we extend λ ≥ aλ 1 and without assuming nonlinearity is superlinear.
Our main result can be stated as follows.
is a positive function, then problem (1) has at least a pair of nontrivial solutions if j f j ∞ is small enough.
The paper is organized as follows. In Section 2, we give some notations and preliminaries. In Section 3, we prove Theorem 1 for the case of 0 < λ < aλ 1 . Section 4 is devoted to the proof of Theorem 1 for the case of λ ≥ aλ 1 .

Notations and Preliminaries
Throughout this paper, we make use of the following notations. H 1 0 ðΩÞ and L q ðΩÞ are standard Sobolev spaces with the usual norm ∥u∥ 2 = Ð Ω j∇uj 2 dx, juj q q = Ð Ω juj q dx. B r ðxÞ (∂B r ðxÞ) denotes an open ball (the sphere) centered at x with radius r > 0. ⟶ and ⇀ denote strong and weak convergence, respectively. For each 1 ≤ r ≤ 2 * , we denote by S q the best Sobolev constant for the embedding of H 1 0 ðΩÞ into L q ðΩÞ. Let 0 < λ 1 < λ 2 ≤ ⋯≤λ k <⋯ be the sequence of eigenvalues of −Δ on H 1 0 ðΩÞ satisfying lim k⟶∞ λ k = +∞, and let e 1 , e 2 , ⋯, be the corresponding orthonormal eigenfunctions in L 2 ðΩÞ.
By the Sobolev Theorem and f ∈ L ∞ ðΩÞ, the functional is well defined on H 1 0 ðΩÞ. Furthermore, it belongs to C 1 ðH 1 0 ðΩÞ, ℝÞ, and its critical points are precisely the weak solutions of (1). Here, we say u ∈ H 1 0 ðΩÞ is a weak solution to Following [15], we first prove that the functional I satisfies the ðPSÞ c condition for any c < a 2 /4b.

Lemma 2.
Under the assumptions of Theorem 1, I satisfies the ðPSÞ c condition with c < a 2 /4b.
Proof. Let fu n g ⊂ H 1 0 ðΩÞ be a ðPSÞ c sequence for I with c < a 2 /4b; that is, By the Sobolev Theorem and (10), Since p < 2, we conclude that fu n g is bounded in H 1 0 ðΩÞ. Up to a subsequence (still denoted by fu n g), we may assume that 2 Advances in Mathematical Physics u n ⇀ u, u n ⟶ u, a:e:in Ω: By using Hölder's inequality, it follows from (12) that as n ⟶ ∞. Similarly, we also have From the two above convergences, we get as n ⟶ ∞. We claim that a − b∥u n ∥ 2 ⟶ 0 is false. If, to the contrary, namely, ∥u n ∥ 2 ⟶ ab, define a functional by Then, By using Hölder's inequality again, we obtain Hence, by using (12), we obtain as n ⟶ ∞. This shows ϕ ′ ðu n Þ ⟶ ϕ ′ ðuÞ.
On the other hand, from and ∥u n ∥ 2 ⟶ a/b, we have ϕ ′ ðu n Þ ⟶ 0. Thus, we can deduce that By the variational method fundamental lemma [23], we further obtain Since f ðxÞ > 0, it then follows that u = 0. By (12) and f ∈ L ∞ ðΩÞ, we can use the Vitali Convergence Theorem to obtain and consequently, This and ∥u n ∥ 2 ⟶ a/b provide which contradicts Iðu n Þ ⟶ c < a 2 /4b. Thus, the claim follows. In turn, we have from (15) that Ð Ω ∇u n ∇ðu n − uÞdx ⟶ 0, and hence, ∥u n ∥⟶∥u∥. Combining this with the weak convergence of fu n g in H 1 0 ðΩÞ, we deduce that u n ⟶ u in H 1 0 ðΩÞ. ☐ 3. Proof of Theorem 1 for 0 < λ < aλ 1 In this section, we will use the Mountain Pass Theorem to prove the existence of a pair of nontrivial solutions of the considered problem for 0 < λ < aλ 1 .
By IðuÞ = Ið|u | Þ, we may assume that u n ≥ 0. By applying the Mountain Pass Theorem without ðPSÞ condition [24], we construct a sequence fu n g ⊂ H 1 0 ðΩÞ satisfying Iðu n Þ ⟶ c * and I ′ðu n Þ ⟶ 0 for where From easy calculations, we get This and the definition of c * yield that 0 < c * < a 2 /4b. Thus, we complete the proof of Lemma 3. ☐ Proposition 4. Assume a, b > 0, 0 < λ < aλ 1 and f ðxÞ ∈ L ∞ ðΩÞ is a positive function. Then, problem (1) admits a pair of nontrivial solutions if jf j ∞ is small enough.
Proof. By Lemma 3, we obtain a sequence fu n g ⊂ H 1 0 ðΩÞ such that u n ≥ 0, Iðu n Þ ⟶ c * and I ′ ðu n Þ ⟶ 0, provided j f j ∞ is small enough. It then follows from Lemma 2 that there exists u ∈ H 1 0 ðΩÞ such that u n ⟶ u in H 1 0 ðΩÞ with IðuÞ = c * and I ′ðuÞ = 0, which implies that u is a nontrivial nonnegative solution of (1). By the symmetry of functional I, we further deduce that −u is a nontrivial nonpositive solution of (1). This completes the proof. Since λ ≥ aλ 1 , the method used in the previous section does not work here. Indeed, we shall apply the following Linking Theorem [25] to establish the existence of a pair of nontrivial solutions for problem (1) when λ ≥ aλ 1 .