Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type

In this paper by using the minimal principle and Morse theory, we 
prove the existence of solutions to the following Kirchhoff 
nonlocal fractional equation: 
 
\begin{eqnarray} 
& M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y 
 \right) (- \Delta)^s u 
 = f (x, u (x)),\quad \textrm{in}\;\; \Omega,\\ 
& u = 0, \quad \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, 
\end{eqnarray} 
where $(- \Delta)^s$ is the fractional Laplace operator, $s \in 
(0, 1)$ is a fix, $\Omega$ an open bounded subset of 
$\mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary, $f: \Omega 
\times \mathbb{R} \to \mathbb{R}$ Caratheodory function and $M : 
\mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy 
some suitable conditions.


NEMAT NYAMORADI AND KAIMIN TENG
The homogeneous Dirichlet datum in (1) is given in R n \ Ω and not simply on the boundary ∂Ω, consistent with the nonlocal character of the kernel operator L K .
Inspired by the above articles and very recent articles [6,16], in this paper, we would like to investigate the existence of solutions for problem (1). The technical tool is the minimal principle and Morse theory.
The paper is organized as follows. In Section 2, we give preliminary facts and provide some basic properties which are needed later. Sections 3 and 4 are devoted to our results on existence of one solution by the minimal principle and existence of two nontrivial solutions by Morse theory, respectively.

2.
Preliminaries. In this section, we present some preliminaries and lemmas that are useful to the proof to the main results. For the convenience of the reader, we also present here the necessary definitions.
Let (X, || · || X ) be a Banach space, (X * , || · || X * ) be its topological dual, and ϕ : X → R be a functional. First, we recall the definition of the Palais-Smale condition which plays an important role in our paper. Definition 2.1. We say that ϕ satisfies the Palais-Smale condition if any sequence (u n ) ∈ X for which ϕ(u n ) is bounded and ϕ (u n ) → 0 as n → ∞ possesses a convergent subsequence.
Before proving the main result, some preliminary material on function spaces and norms is needed. In the following we briefly recall the definition of the functional space X 0 , firstly introduce in [13], and we give some notations. We denote Q = R 2n \ O, where O = R n \ Ω × R n \ Ω. We denote the set X by where u| Ω represents the restriction to Ω of function u(x). Also, we denote by X 0 the following linear subspace of X X 0 = {g ∈ X : g = 0 a.e. in R n \ Ω}.
In this paper we will prove the existence of nontrivial weak solutions for problem (1) using variational and topological methods. By a weak solutions of (1) we mean a solution of the following problem We know that X and X 0 are nonempty, since C 2 0 (Ω) ⊆ X 0 by Lemma 11 of [13]. Moreover, the linear space X is endowed with the norm defined as It is easy seen that || · || X is a norm on X (see, for instance, [11] for a proof). By Lmmas 6 and 7 of [11], in the sequel we can take the function as norm on X 0 . Also (X 0 , || · || X0 ) is a Hilbert space, wit scalar product Note that in (7) the integral can be extend to all R n × R n , since v ∈ X 0 and so v = 0 a.e. in R \ Ω.
In what follows, we denote by λ 1 the first eigenvalue of the operator L K with homogeneous Dirichlet boundary data, namely the first eigenvalue of the problem which λ 1 is positive and that can be characterized as follows We refer to Proposition 9 and Appendix A of [12], for the existence and the basic properties of this eigenvalue, where a spectral theory for general integro-differential nonlocal operators was developed.
Let V = span{e 1 } be the one-dimensional eigenspace associated with λ 1 , where Let H s (R n ) be the usual fractional Sobolev space endowed with the norm (the so-called Gagliardo norm) Also, we recall the embedding properties of X 0 into the usual Lebesgue spaces (see Lemma 8 of [11]). The embedding j : for any v ∈ X 0 .
3. Existence solution by the minimal principle. The functional J : X 0 → R corresponding to problem (1) is defined by where In this section, we assume that we assume that f : Ω × R → R is a Carathéodory function with subcritical growth with respect to t, that is: We recall a convergence property for bounded sequences in X 0 (see [11], for this we need a Lipschitz boundary): Lemma 3.1. Let K : R n \ {0} → (0, +∞) satisfy assumptions (K1)-(K3) and let {u n } be a bounded sequence in X 0 . Then, there exists u ∈ L p (R n ) such that, up to a subsequence, u n → u in L p (R n ), as n → ∞, for any p ∈ [1, 2 * ). Now, we can state our main result in this section. Proof. Let (u n ) be a sequence that converges weakly to u in X 0 , so by Lemma 3.1, Therefore, by the weak lower semicontinuous of the norm, one can get Combining this with the continuity and monotonicity of the function M , we have By (F1), (10) and the Hölder inequality, we get , which tends to 0 as n → ∞, where 0 ≤ θ n (x) ≤ 1, for all x ∈ Ω, From (14) and (15), the functional J is weakly lower semicontinuous in X 0 .
On the other hand, by assumptions (H0) and (F1), one can get Since 1 < q < 2α, it follows from (16), that the functional J is coercive. Thus, using the minimal principle, we deduce that the functional J has at least one weak solution and therefore the problem (1) has at least one weak solution. Also, we make the following assumptions: (F2) there exist r > 0, λ ∈ (λ 1 , λ) such that m 1 λ 1 < m 0 λ, and |u| ≤ r implies

NEMAT NYAMORADI AND KAIMIN TENG
Proof. For all u, v ∈ X 0 , we have . So, the operator Φ is strongly monotone, then possesses the property of type (S + ). Under assumptions (m0) and (F1), any bounded sequence {u n } in X 0 such that J (u n ) → 0 in (X 0 ) * as n → ∞ has a convergent subsequence.
Proof. Since (u n ) is bounded in X 0 and X 0 is a reflexive Banach space (X 0 is a Hilbert space) and so by passing to a subsequence (for simplicity denoted gain by {u n }) if necessary, by Lemma 3.1, we may assume that Therefore with ε n → 0. Thus lim sup n→∞ Φ (u n ), u n − u ≤ 0. By (17), it is easy to get lim n→∞ Φ (u), u n − u = 0. Therefore Since Φ is of type (S) + (see Lemma 4.1), so we obtain u n → u as n → ∞ in X 0 . Proof. We divide the proof into two steps.
Step 1. Let (F3) hold. By (F1) and (F3) one can get for small enough ε > 0, there exists a constant C 1 > 0 such that Thus, by definition of λ 1 , for u ∈ X 0 Step 2. If (F4) and (F5) hold, the we can write F (x, t) = m0 2 λ 1 |t| 2 + G(x, t) and So, for any M > 0, there is R M > 0 such that Integrating the equality d dt In a similar way, We suppose that, to the contrary, there exists a sequence {u n } ⊂ X 0 such that u n X0 → ∞ as n → ∞, but J(u n ) ≤ C 2 for some constant C 2 ∈ R. Set v n = un un X 0 then up to a subsequence, we assume that there is some v 0 ∈ X 0 such that

NEMAT NYAMORADI AND KAIMIN TENG
where C 3 is a positive constant, which implies lim sup By the weakly semicontinuous property of u → ||u|| X0 in the weak topology X 0 and the definition of λ 1 , we have By (19) and (20), Hence v 0 = ±e 1 . Take v 0 = e 1 ; then u n → +∞ a.e. on Ω, which implies that G(x, u n (x)) → −∞ by (18). So we have which is a contradiction. So we have that J is coercive in X 0 .
By Lemmas 4.2 and 4.3, it follows that: Now, we recall the definition of local linking to proceed with our proof.
Definition 4.5. We say that a functional ϕ has a local linking to the decomposition of the space X = V ⊕ Y near the origin 0 iff there is a small ball B ρ with the center at 0 and small radius ρ > 0 such that Proof. First, we take u ∈ V ; since V is finite dimensional, we can see that u X0 ≤ ρ implies |u| ≤ r, for all x ∈ Ω and r > 0 small enough. Thus by (F2), for u X0 ≤ ρ, we get On the other hands, we take u ∈ Y ; from (F1), (F2) and the definition of λ, one can get So we can derive that when u ∈ X and 0 < u X0 ≤ ρ and ρ > 0 small, J(u) > 0, which completes the proof.
Remark 1. From the proof of Lemma 4.3, we can get a stronger result: there exists a ρ 0 > 0 such that for any 0 < ρ < ρ 0 , B ρ satisfies all the conditions required by the definition of local linking. From this point of view, we can conclude that 0 ∈ X 0 is the unique critical point of our J in a ball that is small enough.
For an isolated critical point u of a C 1 functional f : E → R, we define a the critical group of f at u as follows: . Let E be a Banach space and f : E → R a C 1 functional satisfying the Palais-Smale condition. Suppose that E has a decomposition E = W ⊕ Z, where W is a finite dimensional subspace, say dimW = m < ∞. Suppose that there exists a small ball B ρ with its center at the origin 0 and small radius ρ > 0 such that Here, since dimW = 1 < ∞, by Lemma 4.6, Remark 1 and Lemma 4.7, we have the following lemma:   For the proof of Theorems 4.9 and 4.10, we present the following theorem from [8].
Lemma 4.11. Let X be a real Banach space and let J ∈ C 1 (X, R) satisfy the Palais-Smale condition and be bounded from below. If J has a critical point that is homologically nontrivial and is not the minimizer of J, then J has at least three critical points.
Proof of Theorems 4.9 and 4.10. By Lemmas 4.3 and 4.4, J is coercive and satisfies the Palais-Smale condition. Thus J is bounded below. By Lemma 4.8, 0 ∈ X 0 is a homologically nontrivial critical point of J but not a minimizer. Then the conclusion follows from Lemma 4.11.