SIGN-CHANGING SOLUTIONS FOR NON-LOCAL ELLIPTIC EQUATIONS WITH ASYMPTOTICALLY LINEAR TERM

. In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodiﬀerential operator with homogeneous Dirichlet boundary condition where Ω ⊂ R n ( n ≥ 2) is a bounded, smooth domain and f ( x,u ) is asymptotically linear at inﬁnity with respect to u . By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).


(Communicated by Changfeng Gui)
Abstract. In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition where Ω ⊂ R n (n ≥ 2) is a bounded, smooth domain and f (x, u) is asymptotically linear at infinity with respect to u. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).

1.
Introduction. In recent years, the fractional and non-local operators of elliptic type have been widely investigated. This type of operators arise in several areas such as anomalous diffusion, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, quasi-geostrophic flows, multiple scattering and materials science. One can see [4,9,10,11,16,17,24,25,26] and their references.
On the other hand, many papers [3,5,6,7,8,13,15,18,21,22,23] are devoted to the study of the existence of sign-changing solutions of the nonlinear problem where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary ∂Ω, f ∈ C(Ω × R, R). There have been several methods developed in studying sign-changing solutions of problem (2), such as the invariant sets of descending flow method developed by Liu and Sun [5,18,23], the minimax method which is established by Berestycki and Lions in the classical paper [8], Brouwer's degree theory and variational method in [2,15,29]. When f (x, u) is asymptotically linear at infinity with respect to u, the existence of solutions for problems like (2) has been studied in some papers, see Tang [15,19,27,31]. We think the first natural question is whether these methods can be adapted to the fractional analogue of problem (2) under asymptotically linear assumption on f , namely where f (x, u) is asymptotically linear at infinity with respect to u, (−∆) s (0 < s < 1) is the fractional Laplacian operator, which (up to normalization factors) may be defined as We should remark that the Dirichlet datum is given in R n \ Ω and not simply on ∂Ω, consistently with the non-local character of the operator (−∆) s . Furthermore, the second natural problem is the existence of sign-changing solutions for the non-local elliptic problem where the non-local integrodifferential operator L K is defined as follows and K : R n \ {0} → (0, +∞) is a function with the properties that (K 1 ) γK ∈ L 1 (R n ), where γ(x) = min{|x| 2 , 1}; (K 2 ) there exists λ > 0 such that K(x) ≥ λ|x| −(n+2s) for any x ∈ R n \ {0}.
A typical example for K is given by K(x) = |x| −(n+2s) . In this case, L K is the fractional Laplace operator −(−∆) s , see (3). In this article, the work space X introduced by Raffaella Servadei and Enrico Valdinoci [24,25] is defined as a linear space of Lebesgue measurable functions from R n to R, such that, any function u restricted in X belongs to L 2 (Ω) and the map ( The function space X is equipped with the following norm where Q = R 2n \ (Ω c × Ω c ). And the function space X 0 is defined as which is equivalent to the usual one defined in (5). Throughout this paper, we also denote the norm · X0 by · .
Recall that the fractional Sobolev space H s (Ω) is defined as For the basic properties of fractional Sobolev spaces, we refer the interested readers to [12,14,16,26]. Note that even in the model case, where K(x) = |x| −(n+2s) , the norms in (5) and (6) are not the same, because Ω × Ω is strictly contained in Q. This leads that the classical fractional Sobolev space approach is not sufficient for studying problem (3).
To the best of our knowledge, there are no works concerning the least energy sign-changing solutions for problems (3) and (4) with asymptotically linear case at infinity.
As is explained in [8, Remark 9.2], the minimax method of Berestycki and Lions strongly depends on a kind of nodal structure associated with problem (2), which is unknown for problems (3) and (4). The variational methods used in [5,21] heavily rely on the following decomposition, where u + := max{u, 0}, u − := min{u, 0}, and Φ is the energy functional of problem (2) given by But for problem (4) and u ∈ X 0 , we have where I is the energy functional of problem (4) given by In what follows, we denote It is obvious that B(u) ≥ 0. Clearly, the functional I does no longer satisfy the decomposition (7). In present paper, motivated by [1,3,5,6,21,28], we try to get the sign-changing solution for problem (4) by seeking the minimizer of the energy functional I over the following constraint: To obtain our result, we suppose that the function f :Ω × R n → R verifying the following conditions: Choose a piecewise constant function in Π, we can give a upper bound of µ: Now, we give an example to illustrate the feasibility of assumptions ( where α ∈ (0, 2), V ∞ ∈ C(Ω), infΩV ∞ > µ. By elementary computations, we can get that f satisfies (f 1 ) − (f 5 ). Remark 1. In fact, the sign-changing solution u ∈ X 0 of problem (4) given by the above Theorem 1.1 has the least energy among all sign-changing solutions. For problem (2), we can follow the argument of [5] to show that the least energy signchanging solution has exactly two nodal domains. But in problem (4), it seems not easy to get the same result.
The rest of the paper is organized as follows. In Section 2, we prove some lemmas, which are crucial to investigate our main result. The proof of Theorem 1.1 is given in Section 3.

2.
Preliminaries. Firstly, we collect some useful information in the paper. We will denote o(1) by the infinitesimal as j → +∞. For the sake of simplicity, the norm · L p (Ω) will be often written · p .
For the reader's convenience, we review the main embedding results for the space X 0 . Lemma 2.1 ( [9,24,25]). The embedding X 0 → L r (Ω) is continuous for any r ∈ [2, 2 * s ], and compact for any r ∈ [2, 2 * s ), where 2 * s = 2n n−2s . Now, we state some preliminary lemmas which will be used in the last section to prove our main result.
Lemma 2.2. Assume f satisfies (f 1 ) − (f 2 ). Let u j be a sequence such that u j u in X 0 , then, up to a subsequence, Proof. (i) By the compact embedding X 0 → L p (Ω)(2 ≤ p < 2 * s ), taking if necessary a subsequence, we have u j → u in L p (Ω) and u j (x) → u(x) a.e. on R n . By a standard discussion, there exists a function g ∈ L p (Ω) such that .
By (f 2 ) and u ∈ L p (Ω), we have it follows that f (·, u) ∈ L p p−1 (Ω). Since it follows from Lebesgue dominated convergence theorem that By the Hölder inequality, we have (ii) By the mean value theorem, there exists λ ∈ [0, 1] such that
(iii) Recall that u j (x) → u(x) a.e. on R n , then, by Fatou's Lemma, we have

Corollary 1.
Under assumptions (f 1 ) − (f 3 ), we have I(u) ≥ I(tu + + su − ), ∀u ∈ M, s, t ≥ 0. Now, we define the set E 0 as follows |τ | is strictly increasing on (−∞, 0) ∪ (0, +∞), which, together with In view of (f 4 ) and the definition of µ, we deduce that there exists v ∈ Π such that Hence, we have v ∈ E 0 . This shows that E 0 = ∅ because of (f 4 ). Moreover, we can easily verify that for any u ∈ M, This shows that M ⊂ E 0 .
As a consequence, t u u + + s u u − ∈ M.
Then {w j } is bounded in X 0 .
Proof. Arguing by contradiction, suppose that w j → ∞. Let v j = wj wj , then v j = 1. By Lemma 2.1, passing to a subsequence, we may assume that there . By (f 1 ), (f 2 ), for any ε > 0, there exists C ε > 0 such that By (12) Furthermore, we consider the Jacobian matrix of Φ v : And we have the following Lemma.