LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR NONLINEAR PROBLEMS INVOLVING FRACTIONAL LAPLACIAN

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1 Electronic Journal of Differential Equations, Vol. 016 (016), No. 38, pp ISSN: URL: or LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR NONLINEAR PROBLEMS INVOLVING FRACTIONAL LAPLACIAN ZU GAO, XIANHUA TANG, WEN ZHANG Abstract. In this article, we study the existence of least energy sign-changing solutions for nonlinear problems involving fractional Laplacian. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution. 1. Introduction This article concerns the nonlinear problem involving fractional Laplacian ( ) α u = f(x, u), x, u = 0, x R N \, (1.1) where R N is a bounded domain with smooth boundary, 0 < α < 1, N > α, ( ) α is the fractional Laplacian of order α, f C( R, R). To prove our results, we use the following assumptions: (A1) lim s 0 f(x, s)/s = 0, uniformly in x ; (A) lim s f(x, s)/s α 1 = 0, uniformly in x, where α = N N α ; (A3) lim s f(x, s)/ s = + for a.e. x ; (A4) f(x, s)/ s is increasing in s on R\{0} for every x. In recent years, nonlinear problems involving fractional Laplacian have been investigated extensively. Indeed, they have impressive applications in many fields, such as thin obstacle problem, optimization, finance, phase transitions, anomalous diffusion and so on. For previous related results see [1, 6, 8, 9, 11, 14, 15, 16, 17, 18, 5, 7, 9, 40, 41] and the references therein. Precisely, under the assumption that the nonlinearity satisfies the Ambrosetti-Rabinowitz condition or is indeed of perturbative type, the author proved some existence results of solutions for fractional Schrödinger equations in [5]. Using mountain pass theorem, Raffaella and Servadei studied the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions in [6]. In fact, by the extension theorem in [7] Caffarelli and Silvestrein made greatest achievement in overcoming the difficulty, which is the nonlocality of fractional Laplacian ( ) α in the fractional Schrödinger equation. Moreover, a great deal of progress 010 Mathematics Subject Classification. 35R11, 58E30. Key words and phrases. Nonlinear problems; sign-changing solutions; nonlocal term. c 016 Texas State University. Submitted June 1, 016. Published August 31,

2 Z. GAO, X. TANG, W. ZHANG EJDE-016/?? has been made to the fractional Laplacian equations after the work [7]. We refer to [10, 1, 30, 31, 37, 43] for the existence results and multiplicity results of solutions, and to [4, 5] for the regularity results, maximum principle, uniqueness result and other properties. As we know, a great attention has been devoted to the existence and multiplicity of positive and nodal solutions of elliptic problems in recent years, see for example [, 3, 13,, 3, 33, 4] and the references therein. Actually, with the descended flow method and harmonic extension techniques, Chang and Wang studied the existence and multiplicity of sign-changing solutions in [1]. Via costrained minimization method, Tang [34, 35, 36, 19, 0] obtained the existence of Nehari-type ground state positive solutions. By combing minimax method with invariant sets of descending flow, some results about nodal solutions have been obtained in [1]. Motivated by papers above, and we especially borrow some ideas from [19]. What is more, we are interested in Problem (1.1) with constraint variational method and quantitative deformation lemma, and study the existence of a least energy signchanging solution. For any measurable function u : R N R with respect to the Gagliardo norm ( RN u(x) u(y) ) 1/ [u] α = α+n dxdy. We introduce the fractional Sobolev space H α (R N ) = {u L (R N ) : [u] α < + }, which is a Hilbert space. A complete introduction to fractional Sobolev spaces can be found in [4]. We also define a closed subspace X() = {u H α (R N ) : u = 0 a.e. in R N \}. Then, by [5], X() is a Hilbert space with the inner product (u(x) u(y))(v(x) v(y)) (u, v) = α+n dxdy, u, v X(), and the corresponding norm X = [ ] α. For u X(), set Φ(u) = 1 u X F (x, u)dx, (1.) where F (x, u) = u 0 f(x, t)dt. Then Φ C1 (X(), R) and Φ (u(x) u(y))(v(x) v(y)) (u), v = α+n dxdy f(x, u)vdx, (1.3) for all u, v X(). Obviously, its critical points are weak solutions of Problem (1.1). Furthermore, if u X() is a solution of (1.1) with u ± 0, then u is a sign-changing solution, where We set u + (x) := max{u(x), 0} and u (x) =: min{u(x), 0}. M := {u X() : u ± 0, Φ (u), u + = Φ (u), u = 0}, and define m = inf Φ(u). u M Throughout this paper, p denotes the usual norm in L p ().

3 EJDE-016/38 LEAST ENERGY SIGN-CHANGING SOLUTIONS 3 Theorem 1.1. Assume that conditions (A1) (A4) hold. Then (1.1) possesses one least energy sign-changing solution u M such that inf u M Φ(u) = m > 0. The rest of this article is organized as follows. In Section, we prove several lemmas, which are crucial to investigate our main result. The proof of Theorem 1.1 is given in Section 3.. Preliminary results Lemma.1 ([38, Lemma.1]). For any a, b R, we have (i) (ka) ± = ka ±, for all k 0, a ± b ± a b ; (ii) (a b)(a + b + ) (a + b + ) and (a b)(a b ) (a b ) ; (iii) (a + b + )(a b ) 0. By simple computations from the above lemma, we obtain the following lemma. Lemma.. Under assumptions (A1) and (A), for any u X(), the following facts hold: (i) u ± X u X ; (ii) (u, u ± ) = (u ±, u ± ) = (u ±, u ± ) (iii) Φ (u), u ± = Φ (u ± ), u ± = Φ (u ± ), u ± u + (x)u (y) dxdy α+n u + (x)u (y) dxdy; α+n In what follows, we denote B(u) := It is obvious that B(u) 0. u (x)u + (y) dxdy α+n u + (x)u (y) u (x)u + (y) dxdy dxdy α+n α+n u + (x)u (y) dxdy. α+n u + (x)u (y) dxdy. α+n Lemma.3. Assume (A1) and (A), and let {u n } be a bounded sequence in X(). Then up to a subsequence, still denoted by {u n }, there exists u X() such that (i) (ii) (iii) (iv) lim u ± n p dx = u ± p dx, p [, α); lim u n f(x, u n )dx = uf(x, u)dx; lim F (x, u n )dx = F (x, u)dx; lim inf Φ (u n ), u ± n Φ (u), u ±.

4 4 Z. GAO, X. TANG, W. ZHANG EJDE-016/?? Proof. (i) (iii) are easily proved; so we omit their proofs. (iv)from (ii), Fatou s Lemma and (iii) of Lemma., it follows that Φ (u), u ± = Φ (u ± ), u ± u + (x)u (y) [u ± (x) u ± (y)] = α+n dxdy { lim inf lim α+n dxdy [u ± n (x) u ± n (y)] α+n dxdy u ± n f(x, u ± n )dx = lim inf Φ (u n ), u ± n. This shows that (iv) holds. u + (x)u (y) dxdy α+n u + n (x)u n (y) } α+n dxdy u ± f(x, u ± )dx Lemma.4. Under assumptions (A1) and (A), if {u n } is a bounded sequence in M and q (, α), we have lim inf u ± n q dx > 0. Proof. From (A1) and (A), for any ε > 0 and fixed τ [, α), there exists C ε > 0 such that sf(x, s) ε s + C ε s τ + ε s α, x, s R. (.1) For u n M, we have Φ (u n ), u ± n = 0. From (iii) of Lemma., we have Φ (u ± n ), u ± u + n (x)u n (y) n dxdy = 0, α+n which, together with Sobolev embedding and (.1), for q (, α), yields u ± n X u ± n f(x, u ± n )dx ε u ± n dx + C ε u ± n q dx + ε u ± n α dx εγ u± n X + C ε γq u ± n X u ± n q q + εγ α α u± n α X, (.) where γ s := inf u s=1 u X, s α. From the boundedness of {u n }, there is M such that u ± n α X M. From (.), taking ε = min{γ /4, γ α α /4M}, C 0 C ε, it follows that Then 1 C 0γq u ± n q q. ( γ ) 1 lim inf u ± n q q q dx > 0. C 0

5 EJDE-016/38 LEAST ENERGY SIGN-CHANGING SOLUTIONS 5 Lemma.5. Under assumptions (A1), (A), (A4), for any u X() with u ± 0, s, t 0 and (s 1) + (t 1) 0, we have Φ(u) > Φ(su + + tu ) + 1 s Proof. For τ 0, (A4) yields It follows that 1 θ τf(x, τ) > Thus, we deduce that Φ(u) Φ(su + + tu ) f(x, s) < f(x, s) > τ θτ Φ (u), u t Φ (u), u + B(u)(s t). f(x, τ) s, τ f(x, τ) s, τ s < τ ; s > τ. f(x, s)ds, x, τ 0, θ 0 and θ 1. = 1 s Φ (u), u t Φ (u), u [1 s + f(x, u + )u + ( F (x, u + ) F (x, su + ) ) ] dx [1 t + f(x, u )u ( F (x, u ) F (x, tu ) ) ] dx + B(u)(s t) = 1 s Φ (u), u t Φ (u), u [1 s + f(x, u + )u + u + f(x, ξ)dξ ] [ 1 t u ] dx + f(x, u )u f(x, ξ)dξ dx su + tu + B(u)(s t) > 1 s Φ (u), u t Φ (u), u + B(u)(s t), for all s, t 0, (s 1) + (t 1) 0. From Lemma.5, we have the following two corollaries. Corollary.6. Under assumptions (A1), (A), (A4), we have Φ(u) Φ(tu) + 1 t Φ (u), u, u X(), t 0. (.3) Corollary.7. Under assumptions (A1),(A), (A4), we have Φ(u) Φ(su + + tu ), u M, s, t 0. (.4) Lemma.8. Assume (A1) (A4) hold; if u X() with u ± 0, then there exists a unique pair (s u, t u ) of positive numbers such that s u u + + t u u M. Proof. Let g 1 (s, t) = Φ (su + + tu ), su + = s u + f(x, su + )su + dx + B(u)st, (.5)

6 6 Z. GAO, X. TANG, W. ZHANG EJDE-016/?? g (s, t) = Φ (su + + tu ), tu = t u f(x, tu )tu dx + B(u)st. (.6) From (A1), (A) and (A3), a straightforward computation yields that there are r > 0 small enough and R > 0 large enough such that g 1 (r, r) > 0, g (r, r) > 0, g 1 (R, R) < 0, g (R, R) < 0. Notice that for any fixed s > 0, g 1 (s, t) is increasing in t on [0, + ), then g 1 (r, t) g 1 (r, r) > 0, g 1 (R, t) g 1 (R, R) < 0, t [r, R], t [r, R]. Analogously, for g (s, t), one has g (s, r) g (r, r) > 0, g (s, R) g (R, R) < 0, s [r, R], s [r, R]. The above inequalities and the Miranda theorem [3] imply that there is a pair (s u, t u ) (r, R) (r, R) such that g 1 (s u, t u ) = g (s u, t u ) = 0, and then, s u u + + t u u M. Next, we prove the uniqueness. Let (ŝ 1, ˆt 1 ) and (ŝ, ˆt ) such that ŝ i u + + ˆt i u M, i = 1,. We assume that ( ŝ ŝ 1 1) + ( ˆt ˆt 1) 0, then Lemma.5 implies 1 Φ(ŝ 1 u + + ˆt 1 u ) > Φ(ŝ u + + ˆt u ), Φ(ŝ u + + ˆt u ) > Φ(ŝ 1 u + + ˆt 1 u ). This contradiction shows (ŝ 1, ˆt 1 ) = (ŝ, ˆt ), this completes the proof. Corollary.9. Under assumptions (A1) (A4), m := inf u M Φ(u) = inf u X(),u ± 0 max s,t 0 Φ(su+ + tu ). Lemma.10. Assume that (A1) (A4) hold. If u 0 M, and Φ(u 0 ) = m, then u 0 is a critical point of Φ. Proof. Arguing by contradiction, Φ(u 0 ) = m and Φ (u 0 ) 0. Therefore, there exist δ > 0 and ρ > 0 such that v X(), v u 0 3δ Φ (v) ρ. Let D = ( 1, 3 ) ( 1, 3 ). It follows from Lemma.5 that m := max (s,t) D Φ(su+ 0 + tu 0 ) < m. For ε := min{(m m)/3, 1, ρδ/8}, S := B(u 0, δ), [39, Lemma.3] yields a deformation η C([0, 1] X(), X()) such that (i) η(1, u) = u if Φ(u) < m ε or Φ(u) > m + ε; (ii) η(1, Φ m+ε B(u 0, δ)) Φ m ε ; (iii) Φ(η(1, u)) Φ(u), for all u X().

7 EJDE-016/38 LEAST ENERGY SIGN-CHANGING SOLUTIONS 7 By Corollary.7, Φ(su tu 0 ) Φ(u 0) = m, for s, t 0, then from (ii) it follows that Φ(η(1, su tu 0 )) m ε, s, t 0, s 1 + t 1 < u 0. (.7) X On the other hand, by (iii) and Lemma.5, one has Φ(η(1, su tu 0 )) Φ(su+ 0 + tu 0 ) < Φ(u 0) = m, (.8) for all s, t 0, s 1 + t 1 δ u 0. Combining (.7) and (.8), we have X max Φ(η(1, su + (s,t) D 0 + tu 0 )) < m. By the similar method in [8], we can prove that η(1, su tu 0 ) M for some (s, t) D, which contradicts the definition of m. 3. Proof of main result Proof of Theorem 1.1. We shall show that m > 0 can be achieved to get a critical point of Φ. Let u n be a sequence in M such that lim Φ(u n) = m. First of all, we claim that {u n } is bounded in X(). To this end, suppose by contradiction that u n X, and set v n = un u. Since v n n X = 1, passing to a subsequence, there exists v X() such that v n v in X(), v n v in L p (), for p < α, and v n (x) v(x) a.e. on. If v = 0, then we have v n 0 in L p (), for p < α. Fix τ [, α) and R = (m + 1). By (A1) and (A), given ε > 0, there exists C ε > 0, such that F (x, s) ε s + C ε s τ + ε s α, x, s R. (3.1) By (3.1), Corollary.6 and Lebesgue s dominated convergence theorem, it follows that m = Φ(u n ) + o(1) ( 1 ) Φ(Rv n ) + R u n Φ u n ), u n + o(1) = R F (x, Rv n )dx + o(1) R F (x, Rv n ) dx + o(1) R [ ε Rvn + C ε Rv n τ ] + ε Rv n α dx + o(1) = m + 1 { ε [ R v n + R α vn α α ] + Cε R τ v n τ τ } + o(1) m + 1 C 1 ε + o(1), the contradiction is obvious due to the arbitrariness of ε. Thus, v 0. Denote A = {x : v(x) 0}. Then for x A, we have lim u n (x) =. By (A3), (A4) and Fatou s Lemma 0 = lim Φ(u n ) u n δ

8 8 Z. GAO, X. TANG, W. ZHANG EJDE-016/?? [ 1 = lim 1 lim inf 1 A A A lim inf F (x, u n ) u n ] vndx F (x, u n ) vndx u n F (x, u n ) u vndx =. n The contradiction shows that {u n } is bounded in X(). Passing to a subsequence, there exists u X() such that u n u in X(), u n u in L p (), for p < α, and u n (x) u(x) a.e. on. Next, we show that m > 0 is attained. From Lemma.4, it follows that u ± 0. Then by Lemma.8, there are s, t > 0 such that su + + tu M. By Lemmas.3 and.5, we have m Φ(su + + tu ) Φ(u) 1 s Φ (u), u + 1 t Φ (u), u = Φ(u) 1 Φ (u), u + s Φ (u), u + + t Φ (u), u [1 lim f(x, u n) F (x, u n ) ] dx {s + lim inf Φ (u n ), u + n + t Φ (u n ), u n } [ = lim Φ(un ) 1 Φ (u n ), u n ] = m, which implies Φ(su + + tu ) = m. From Lemma.10, Φ (su + + tu ) = 0, and then su + + tu is a sign-changing solution of (1.1). Acknowledgments. This work is partially supported by grants from the NNSF (Nos: , , ). References [1] G. Autuori, P. Pucci; Elliptic problems involving the fractional Laplacian in R N, J. Differ. Equ. 55 (013), [] T. Bartsch, K. C. Chang, Z. Q. Wang; On the Morse indices of sign-changing solutions for nonlinear elliptic problems, Math. Z. 33 (000), [3] T. Bartsch, Z. Q. Wang; On the existence of sign-changing solutions for semilinear Dirichlet problems, Topl. Methods Nonlinear Anal. 7 (1996), [4] X. Cabre, Y. Sire; Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincare Anal. Non Lineaire 31 (014), [5] X. Cabre, Y. Sire; Nonlinear equations for fractional Laplacians, II: existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367 (015), [6] L. Caffarelli; Surfaces minimizing nonlocal energies, AttiAccad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 0 (009), [7] L. Caffarelli, L. Silvestre; An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equations 3 (007), [8] L. Caffarelli, J. L. Vazquez; Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal. 0 (011), [9] M. Cheng; Bound state for the fractional Schrödinger equation with unbounded potentials, J. Math. Phys. 53 (01),

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10 10 Z. GAO, X. TANG, W. ZHANG EJDE-016/?? [40] R. Yang; Optimal regularity and nondegeneracy of a free boundary problem related to the fractioal Laplacian, Arch. Ration. Mech. Anal. 08 (013,) [41] W. Zhang, X. H. Tang, J. Zhang; Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl. 71 (016), [4] J. Zhang, X. H. Tang, W. Zhang; Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl. 40 (014), [43] H. Zhang, J. X. Xu, F. B. Zhang; Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in R N, J. Math. Phys. 56 (015), Zu Gao School of Mathematics and Statistics, Central South University, Changsha, Hunan, China address: gaozu7@163.com Xianhua Tang School of Mathematics and Statistics, Central South University, Changsha, Hunan, China address: tangxh@mail.csu.edu.cn Wen Zhang School of Mathematics and Statistics, Hunan University of Commerce, Changsha, Hunan, China address: zwmath011@163.com