Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity

Abstract: In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth.

A typical model for K is given by the singular kernel K(x) = |x| −(N+2s) which coincides with the fractional Laplace operator −(− ) s of the following fractional Laplacian equations in Ω, where λ is a positive real parameter.
In problem (1.1) and problem (1.2), the set Ω ⊂ R N (N > 2s) is an open bounded with Lipschitz boundary and 2 * := 2N N−2s is the fractional critical Sobolev exponent. The operator (− ) s can be seen as the infinitesimal generators of Lévy stable diffusion Processes, see [1] for example. It is easy to see that the integro-differential operator L K is a generalization of the fractional Laplace operator −(− ) s . Elliptic equations involving nonlocal integro-differential operators appear frequently in many different areas of research and find many applications in engineering and finance, including statistical mechanics, fluid flow, pricing of financial instruments, and portfolio optimization, see [2][3][4]. In the past few years, a great deal of attention has been devoted to nonlocal operators of elliptic type, both for their interesting theoretical structure and in view of concrete applications, see [5][6][7][8][9][10][11][12] and the references therein. By minimax method, invariant sets of descending flow method or constrained minimization method, many authors obtained the existence results of sign-changing solutions of some nonlinear elliptic equations, see [13][14][15][16][17][18][19][20]. To show their results, the authors always assumed the nonlinearity f (x, t) is subcritical and/or f (x, t) satisfies (AR) condition and/or f (x, t) is differentiable with respect to t. The existence of nontrivial solutions, positive solutions, negative solutions and sign-changing solutions, for nonlocal elliptic problem (1.1) has been investigated by using variational method, fixed point index theory and critical point theorems, see [2][3][4]12,20].
Motivated by the papers mentioned above, the main purpose of this paper is to establish the existence of sign-changing solution for problem (1.1) and problem (1.2) under much weaker conditions.
We define the sets X and E as and E = {g | g ∈ X and g = 0 a.e. in R N \ Ω}, where u | Ω represents the restriction to Ω of function u(x), O = (R N \ Ω) × (R N \ Ω). We note that E is non-empty and is a norm on E, equivalent to the standard one (see [2,3]). Also, (E, · ) is a Hilbert space and For any λ > 0 fixed, we define the the energy functional J λ : E −→ R by Under the condition ( f 1 ), by standard argument, it is easy to obtain that J λ ∈ C 1 (E, R) and It is easy to see that for u ∈ E Obviously, the critical points of J λ are equivalent to the weak solutions of problem (1.1). Furthermore, if u ∈ E is a solutions of (1.1) and u ± 0 in R N , then u is a sign-changing solution of (1.1), where u + (x) := max{u(x), 0}, u − (x) := min{u(x), 0}.

Preliminaries
In this section we collect some preliminary lemmas which will be used in the next section to prove the existence of sign-changing solutions of problem (1.1) and problem (1.2). Let is the first eigenvalue of the operator −L K with homogeneous Dirichlet boundary data. By using lemma 1.1 and (2.2), we have Since 2 < q < 2 * , we can obtain that there exists . By ( f 1 )-( f 3 ) and the second integral mean value theorem, we easily deduce that and From the definition of φ u (t, s), it follows that (t,s) is a critical point of φ u (t, s) if and only iftu + +su − is a weak solution of (1.1).
Let t, s ≥ 0, then, by (1.5) and (2.4) we have φ u (0, 0) = 0 and Without loss of generality, we can suppose that t ≥ s > 0. Thus we can get that Since 2 * > 2, we can infer that lim By using the continuity of φ u we can deduce the existence of (t,ŝ) ∈ R 2 + that is a global maximum point of φ u , i.e., φ u (t,ŝ) = max (t,s)∈R 2 + φ u (t, s).

ρ}.
For each u ∈ M, there is t 1 > 0 such that t 1 u ∈ S ρ . By Lemma 2.1 and Lemma 2.2, we have .
(2) For each u ∈ M, we have J λ (u), u ± = 0, so which together with (2.2) and Lemma 1.1 gives There exists λ * > 0 such that for λ > λ * , where S K is the best fractional critical Sobolev constant, namely Proof. By Lemma 2.1, for each λ > 0 and u ∈ E with u ± 0, there exists a pair (t λ , s λ ) of positive numbers such that t λ u + + s λ u − ∈ M. Namely (2.8) By using (2.4), we have which imply that {t λ } and {s λ } are bounded. Then, for the sequence λ n → ∞ as n → ∞, there exist t 0 ≥ 0, s 0 ≥ 0 such that t λ n → t 0 and s λ n → s 0 . If we assume that t 0 > 0, it follows from (2.4) that which leads to a contradiction with (2.7). Thus, t 0 = 0. Similarly we can deduce that s 0 = 0. Namely, By (1.5), (2.4) and Lemma 2.3, for C λ we have that choosing u ∈ E with u ± 0, By (2.9), it is easy to see that there exists λ * > 0 such that for λ > λ * , 0 < C λ < s N (S K ) N 2s .

Proofs of the main results
Proof of Theorem 1.1 For any fixed λ > λ * , by Lemma 2.4, J λ is bounded below over M. By Ekeland's variational principle, there exists {w n } ⊂ M such that Similar to the proof of Proposition 2 in [4], one can prove that {w n } is bounded in E. By (1.7), {w + n } and {w − n } are both bounded in E. we can assume that We can claim that w ± n → w ± in E. In the following, by contradiction, we assume that w + n w + or w − n w − in E. Let v ± n := w ± n − w ± , then it follows from the Brezis-lieb theorem that On the other hand, it follows from (2.6) that Similar to the proof of Proposition 2 in [4], we can deduce that J λ (w) ≥ 0. Thus, which contradicts with C λ < s N (S K ) N 2s . Hence we have w ± n → w ± in E. Therefore, by Lemma 1.1, we have Therefore, we have proved that J λ (tw + + sw − ) = C λ and t = s = 1, that is, w = w + + w − ∈ M and J λ (w + + w − ) = C λ . Finally we prove that w is a critical point of J λ for λ > λ * . If w is not a critical point of J λ for λ > λ * , then there are α 0 < 0 and v 0 ∈ E such that J λ (w), v 0 = 2α 0 . So there is δ ∈ (0, 1 2 ) such that  Then Q ∈ C(D, E) and H ∈ C(D, R 2 ). If |t − 1| = δ or |s − 1| = δ, η(t, s) = 0, then H(t, s) = ( J λ (tw + + sw − ), tw + , J λ (tw + + sw − ), sw − ) (0, 0) in view of (t, s) (1, 1). As a consequence, the Brouwer's degree deg(H, int(D), (0, 0)) is well defined. By using the homotopy invariance and the normalization, we have deg(H, int(D), (0, 0)) = 1. Thus there exists a pair (t,s) ∈ int(D) such that H(t,s) = (0, 0), so Q(t,s) ∈ M and J λ (Q(t,s)) ≥ C λ .
Proof of Theorem 1.2 We take K(x) = |x| −(N+2s) , then it is obvious that K(x) satisfies the conditions (K 1 ), (K 2 ) and problem (1.1) turns into problem (1.2). By using Lemma 5 in [2], we can obtain that E ⊆ H s (R N ). Thus, the assertion of Theorem 1.2 follows from Theorem 1.1.

Conclusions
We have established the existence theorems of sign-changing solution for problem (1.1) and problem (1.2) under much weaker conditions (Theorem 1.1 and Theorem 1.2). In comparison with previous works, this paper has several new features. Firstly, we consider the more general nonlinear term without (AR) condition. Secondly, the nonlinear term involves critical growth. Thirdly, we do not require the continuous differentiability of the nonlinear term with respect to the second argument. Finally, the existence of a least energy sign-changing solution is obtained by using constrained minimization method and topological degree theory. Therefore the previous related results in [19,20] are improved and generalized. There have been no previous studies considering the existence of sign-changing solutions for problem (1.1) and problem (1.2) involving critical growth to the best of our knowledge.