MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
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1 Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp ISSN: URL: or ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN GUOWEI LIN, XIONGJUN ZHENG Abstract. This article is devoted to the study of the nonlocal fractional equation involving critical nonlinearities ( ) / u = u + u u in, u = 0 on, where is a smooth bounded domain of R N, N, (0, ), (0, 1 ) and = N N is critical exponent. We show the existence of at least cat () nontrivial solutions for this problem. 1. Introduction This article concerns the critical elliptic problem with the fractional Laplacian ( ) / u = u + u u in, u = 0 on, (1.1) where is a smooth bounded domain of R N with N >, (0, ) is fixed and = N N is the critical Sobolev exponent. In a bounded domain R N, the operator ( ) / can be defined as in [3, 6] as follows. Let {( k, ϕ k )} k=1 be the eigenvalues and corresponding eigenfunctions of the Laplacian in with zero Dirichlet boundary values on normalized by ϕ k L () = 1, i.e. ϕ k = k ϕ k in ; ϕ k = 0 on. We define the space H / 0 () by H / 0 () = {u = u k ϕ k L () : u k k < }, which is equipped with the norm k=1 k=1 k=1 ( u / H 0 () = ) u k 1 k. 010 Mathematics Subject Classification. 35J60, 35J61, 35J70, 35J99. Key words and phrases. Fractional Laplacian, critical exponent, multiple solutions, category. c 016 Texas State University. Submitted January 11, 016. Published April 14,
2 G. LIN, X. ZHENG EJDE-016/97 For u H / 0 (), the fractional Laplacian ( ) / is defined by ( ) / u = u k / k ϕ k. k=1 Problem (1.1) is the Brézis-Nirenberg type problem with the fractional Laplacian. Brézis and Nirenberg [4] considered the existence of positive solutions for problem (1.1) with =. Such a problem involves the critical Sobolev exponent = N N for N 3, and it is well known that the Sobolev embedding H 1 0 () L () is not compact even if is bounded. Hence, the associated functional of problem (1.1) does not satisfy the Palais-Smale condition, and critical point theory cannot be applied directly to find solutions of the problem. However, it is found in [4] that the functional satisfies the (P S) c condition for c (0, 1 N SN/ ), where S is the best Sobolev constant and 1 N SN/ is the least level at which the Palais-Smale condition fails. So a positive solution can be found if the mountain pass value corresponding to problem (1.1) is strictly less than 1 N SN/. Problems with the fractional Laplacian have been extensively studied, see for example [, 3, 5, 6, 7, 9, 10, 1, 13] and the references therein. In particular, the Brézis-Nirenberg type problem was discussed in [1] for the special case = 1, and in [] for the general case, 0 < <, where existence of one positive solution was proved. To use the idea in [4] to prove the existence of one positive solution for the fractional Laplacian, the authors in [, 1] used the following results in [10] (see also [3]): for any u H 0 (), the solution v H 1 0,L (C ) of the problem satisfies div(y 1 v) = 0, v = 0, in C = (0, ), on L C = (0, ), v = u, on {0}, lim k 1 v y y 0 + y = ( ) u, (1.) where we use (x, y) = (x 1,..., x N, y) R N+1, and H0,L(C 1 ) = { w L (C ) : w = 0 on L C, y 1 w dx dy < }. (1.3) C Therefore, the nonlocal problem (1.1) can be reformulated as the local problem w lim y1 y 0 + ν div(y 1 w) = 0, in C, v = 0, on L C, = w(x, 0) w(x, 0) + w(x, 0), on {0}, (1.4) where ν is the outward normal derivative of C. Hence, critical points of the functional J(w) = 1 y 1 w dx dy 1 C w(x, 0) dx {0} (1.5) w(x, 0) dx {0}
3 EJDE-016/97 CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN 3 defined on H 1 0,L (C ) correspond to solutions of (1.4), and the trace u = tr w of w is a solution of (1.1). A critical point of the functional J(u) at the mountain pass level was found in [, 1]. On the other hand, it can be shown by using the Pohozaev type identity that the problem ( ) / u = u p 1 u in, u = 0 on has no nontrivial solution if p + 1 N N and is star-shaped, see for example [3] and [1]. It is well-known that if has a rich topology, (1.1) with =, = 0 has a solution, see [1, 8, 14] etc. In this paper, we assume 0 < < 1, where 1 is the first eigenvalue of the fractional Laplacian ( ) /. We investigate the existence of multiple solutions of problem (1.1). Let A be a closed subset of a topology space X. The category of A is the least integer n such that there exist n closed subsets A 1,..., A n of X satisfying A = n j=1 A j and A 1,..., A n are contractible in X. Our main result is as follows. Theorem 1.1. If is a smooth bounded domain of R N, N 4, 0 < < and 0 < < 1, problem (1.4) has at least cat () nontrivial solutions. Equivalently, (1.1) possesses at least cat () positive solutions. We say that w H0,L 1 (C ) is a solution to (1.4) if for every function ϕ H0,L 1 (C ), we have k y C 1 w, ϕ dx dy = (w + w N+ N )ϕ dx. (1.6) We will find solutions of J at energy levels below a value related to the best Sobolev constant S,N, where k C S,N = inf y 1 w dx dy w H0,L 1 (C),w 0 ( w(x, 0) dx), (1.7) /( ) which is not achieved in any bounded domain and is indeed achieved in the case = R N+1 +. We know from [] that the trace u ɛ (x) = w ɛ (x, 0) of the family of minimizers w ɛ of S,N takes the form u(x) = u ɛ (x) = ɛ N ( x + ɛ ) N, (1.8) with ɛ > 0. Using this property, we are able to find critical values of J in a right range. In section, we prove the (P S) c condition and the main result is shown in section 3.. Palais-Smale condition In this section, we show that the functional J(w) satisfies (P S) c condition for c in certain interval. By a (P S) c condition for the functional J(w) we mean that a sequence {w n } H 1 0,L (C ) such that J(w n ) c, J (w n ) 0 contains a convergent subsequence.
4 4 G. LIN, X. ZHENG EJDE-016/97 Define on the space H0,L 1 (C ) the functionals ψ(w) = (w + (x, 0)) dx, ϕ (w) = k y C 1 w dx dy w(x, 0) dx. We may verify as in [14] that on the manifold V = {w H 1 0,L(C ) : ψ(w) = 1}, ψ (w) 0 for every w V. Hence, the tangent space of V at v is given by T v V := {w H 1 0,L(C ) : ψ (v), w = 0}, and the norm of the derivative of ϕ (w) at v restricted to V is defined by It is well known that ϕ (v) = sup ϕ (v), w. w T vv, w =1 ϕ (w) = min µ R ϕ (w) µψ (w). A critical point v V of ϕ is a point such that ϕ (v) = 0. Since 1 is the first eigenvalue of the fractional Laplacian ( ) /, it can be characterized as 1 = k inf w H0,L 1 (C),w 0 If 0 < < 1, we see that w 1 := (k y 1 w dx dy C is an equivalent norm on H 1 0,L (C ). C y 1 w dx dy w(x,. 0) dx Lemma.1. Any sequence {v n } H 1 0,L (C ) such that ) 1/ w (x, 0) dx d := sup J(v n ) < C := n N S N,N, J (v n ) 0 in H 1 0,L (C ) contains a convergent subsequence. Proof. It is easy to show from the assumptions that d v n 1 J(v n ) 1 J (v n ), v n = ( 1 1 ( ) k y 1 v n dx dy C = ( 1 1 ) v n 1; that is, v n 1 is bounded. We may assume that v n (x, y) v(x, y) in H 1 0,L(C ), v n (x, 0) v(x, 0) in L (), v n (x, 0) v(x, 0) a.e. in. ) v n dx
5 EJDE-016/97 CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN 5 Therefore, for every ϕ H 1 0,L (C ), J (v n ), ϕ J (v), ϕ = 0 as n +. We also have that J(v) 0. By Brézis-Lieb s lemma, J(v) + 1 v n v 1 1 (v n v) + dx = J(v n ) + o(1) = C + o(1). Since J (v n ), v n 0, we obtain v n v 1 (v n v) + dx = v n 1 v 1 ( ) (vn ) + v + dx + o(1) = v 1 + v + dx = J (v), v = 0. Hence, there exist a constant b such that v n v 1 b, (v n ) + dx b, as n +. It follows by v n v in L () that The trace inequality C k y 1 (v n v) dx dy b. C k y 1 (v n v) dx dy S,N (v n v)(x, 0) L () implies b S,N b. Hence, either b = 0 or b S N,N. If b = 0, then v n v in H0,L 1 (C ), and the proof is complete. If b S N,N, we deduce that C = N S/N,N which is a contradiction. ( 1 1 )b = ( 1 1 ) v n v 1 + o(1) J(v) + 1 v n v 1 1 v n v 1 + o(1) = J(v) + 1 v n v 1 1 (v n v) + + o(1) = C d < C, Alternatively, we have the following result. Lemma.. Every sequence {w n } V satisfying ϕ (w n ) c < S,N and ϕ (w n) 0, as n +, contains a convergent subsequence.
6 6 G. LIN, X. ZHENG EJDE-016/97 Proof. Since ϕ (w n ) = min µ R ϕ (w n ) µψ (w n ), there exists a sequence { n } R such that ϕ (w n) n ψ (w n ) 0. It follows that for every h H0,L 1 (C ), k y C 1 w n h dx dy w n h dx µ n (w n + ) 1 h dx 0, (.1) where µ n = n. Choosing h = w n in (.1) and using the fact ψ(w n + ) = 1, we obtain ϕ (w n ) µ n = k y C 1 w n w n µ n (w n + ) 0. Whence by ϕ (w n ) c, µ n c as n +. Setting v n := µ N n ( J(v n ) = 1 µ N n = 1 µ N n C k y 1 w n dx dy ϕ (w n ) N N µ N n. Thus, J(v n ) N c N < N S N. In the same way, for every h H0,L 1 (C ), by (.1), J (v n ), h = µ N n (k y 1 w n h dx dy C Now, the assertion follows by Lemma.1. Let us define Q = inf w V ϕ (w) = inf w V w n, we obtain ) w n dx 1 µ N n (w n + ) w n h µ n { k y 1 w dx dy C ) (w n + ) 1 h dx w(x, 0) dx }. 0. Denote by η 0 (t) C (R + ) a cut-off function, which is non-increasing and satisfies { 1 if 0 t 1 η 0 (t) =, 0 if t 1. Assume 0, for fixed ρ > 0 small enough such that B ρ C, we define the function η(x, y) = η ρ (x, y) = η 0 ( (x,y) ρ ). Then ηw ɛ H0,L 1 (C ). It is standard to establish the following estimates, see [] for details. Lemma.3. The family {ηw ɛ } H 1 0,L (C ) and its trace on y = 0 satisfy If N >, If N =, for ɛ > 0 small enough and some C > 0. ηw ɛ = w ɛ + O(ɛ N ), (.) ηw ɛ L () = Cɛ + O(ɛ N ), (.3) ηw ɛ L () = Cɛ log( 1 ɛ ) + O(ɛ ) (.4)
7 EJDE-016/97 CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN 7 Lemma.4. Assume N, 0 < < 1, then Moreover, there exists u V such that ϕ (u) = Q. Q = inf w V ϕ (w) < S,N. (.5) Proof. We first show that (.5) holds if N and 0 < < 1. Since ɛ N u ɛ NN dx = ( x + ɛ dx ) N ρ N ɛn, x > ρ { x ρ } we have ηu ɛ dx u ɛ dx = uɛ u { x ρ } L () ɛ dx { x ρ } u ɛ L () + O(ɛN ). By Lemma.3, for N >, we have k C y 1 (ηw ɛ ) dx dy ηu ɛ dx ( ηu ɛ dx) k C y 1 w ɛ dx dy Cɛ + O(ɛ N ) u ɛ + L () O(ɛN ) S,N Cɛ u ɛ L () + O(ɛ N ) < S,N. Similarly, for N =, we find for ɛ small enough such that Q S,N Cɛ log( 1 ɛ ) u ɛ L () + O(ɛ ) < S,N. Consequently, inequality (.5) holds. Next, we show that Q is achieved if 0 < < 1. Obviously, Q > 0. Now, let {w n } H 1 0,L (C ) be a minimizing sequence of Q > 0 such that w n 0 and w n (x, 0) L () = 1. The boundedness of {w n} implies that where 1 q. Since by the Brezis-Lieb Lemma, w n w n (x, 0) L () w n (x, y) w(x, y) in H 1 0,L(C ), w n (x, 0) w(x, 0) in L q (), w n (x, 0) w(x, 0) a.e. in, w n = w n w + w + o(1), = w n w + w w n (x, 0) L () + o(1) S,N w n (x, 0) w(x, 0) L () + Q w(x, 0) L () + o(1) (S,N Q ) w n (x, 0) w(x, 0) L () + Q w n (x, 0) L () + o(1) = (S,N Q ) w n (x, 0) w(x, 0) L () + Q + o(1).
8 8 G. LIN, X. ZHENG EJDE-016/97 Hence, we obtain o(1) + Q (S,N Q ) w n (x, 0) w(x, 0) L () + Q + o(1). The S,N > Q implies w n (x, 0) w(x, 0) in L () and w(x, 0) L () = 1. This yields Q w w(x, 0) L () that is, w is a minimizer for Q. lim ( w n w n (x, 0) n + L () ) Q ; 3. Proof of main theorem Taking into account the concentration-compactness principle in [11], we may derive the following result, its proof can be found in []. Lemma 3.1. Suppose w n w in H0,L 1 (C ), and the sequence {y 1 w n } is tight, i.e. for any η > 0 there exists ρ 0 > 0 such that for all n, y 1 w n dx dy < η. {y>ρ 0} Let u n = t r w n and u = t r w and let µ, ν be two non negative measures such that y 1 w n µ and u n ν (3.1) in the sense of measures as n. Then, there exist an at most countable set I and points x i with i I such that (1) ν = u + Σk I ν k δ xk, ν k > 0, () µ = y 1 w + Σ k I µ k δ xk, µ k > 0, (3) µ k S,N ν k. On the manifold V, we define the mapping β : V R N by β(w) := x(w + (x, 0)) dx, which has the following properties. Lemma 3.. Let {w n } V be a sequence such that w n H 1 0,L (C) = C k y 1 w n dx dy S,N as n, then dist(β(w n ), ) 0, as n. Proof. Suppose by contradiction that dist(β(w n ), ) 0 as n. We may verify that {w n } is tight. By Lemma 3.1, there exist sequences {µ k } and {ν k } such that S,N = lim w n = k y 1 w dx dy + Σ k I µ k, (3.) n C 1 = lim u n = u dx + Σk I ν k. (3.3) n By the Sobolev inequality and Lemma 3.1, from (3.) we deduce that S,N = w H 1 0,L (C) + Σ k Iµ k S,N u L () + S,N(Σ k I ν k ).
9 EJDE-016/97 CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN 9 Hence, u L () + (Σ k Iν k ) 1. (3.4) Equations (3.3) and (3.4) imply either Σ k I ν k = 0 or u = 0. L () If Σ k I ν k = 0, that is u () = 1, the lower semi-continuity of norms yields L S,N w H0,L 1 (C) = k C y 1 w dx dy (. u dx) While by the Sobolev trace inequality, S,N k C y 1 w dx dy (, u dx) it then implies that S,N is achieved, which is a contradiction to the fact that S,N is not achieved unless C = R N+1 +. Thus, u () 1. Consequently, L Σ k I ν k = 1 and u = 0. Furthermore, by the uniqueness of the extension of u, we have w = 0. Now, it is standard to show that ν is concentrated at a single x 0 of. So we have β(w n ) x dν(x) = x 0, this is a contradiction. Since is a smooth bounded domain of R N, we choose r > 0 small enough so that + r = {x R N : dist(x, )) < r} and r = {x : dist(x, ) > r} are homotopically equivalent to. Moreover we assume that the ball B r (0), and then C Br(0) := B r (0) (0, + ) C. We define V 0 := {w H0,L(C 1 Br(0)) : + (x, 0) dx = 1} V as well as C Br (0) w Q 0 = inf w V 0 ϕ (w). Denote by ϕ Q0 := {w V : ϕ (w) < Q 0 } the level set below Q 0. We may verify as in Lemma 3. that Q 0 < S,N. Lemma 3.3. There exists a, 0 < < 1 such that for 0 < <, if w ϕ Q0, then β(w) + r. Proof. By Hölder s inequality, for every w V, ( ) w(x, 0) dx w(x, 0) dx /N = /N. ɛ Let =. If 0 < < and w ϕ Q0 /N, we have w w(x, 0) dx + Q 0 /N + S,N = S,N + ɛ. Therefore, we conclude by Lemma 3. that β(w) + r.
10 10 G. LIN, X. ZHENG EJDE-016/97 Now, we establish the relation of category between the domain and the level set ϕ Q0. Lemma 3.4. If N and 0 < <, then we have cat ϕ Q 0 ϕ Q0 cat (). Proof. Let w 0 H0,L 1 (C B(0,r)) be a minimizer of Q 0. Hence, we may assume that w 0 > 0 is cylinder symmetric and w 0 L (B = 1, r(0)) Q 0 = k y 1 w 0 dx dy w 0 (x, 0) dx. C Br (0) B r(0) For z r, we define γ : r ϕ Q0 by { w 0 (x z, y), (x, y) B r (z) (0, + ), γ(z) = 0, (x, y) / B r (z) (0, + ). Since w 0 (x, 0) is a radial function, β γ(z) = x(w 0 ) + (x z, 0)) dx = B r(z) B r(0) x(w 0 ) + (x, 0) dx + z = z. Hence, β γ = id. Assume that ϕ Q0 = A 1 A A n, where A j, j = 1,... n, is closed and contractible in ϕ Q0, i.e. there exists h j C([0, 1] A j, ϕ Q0 ) such that, for every u, v A j, h j (0, u) = u, h j (1, u) = h j (1, v). Let B j := γ 1 (A j ), 1 j n. The sets B j are closed and r = B 1 B B n. By Lemma 3.3, we know β(h j (t, γ(x))) + r. Using the deformation g j (t, x) = β(h j (t, γ(x))), we see that B j is contractible in + r. Indeed, for every x, y B j, there exist γ(x), γ(y) A j such that It follows that cat ϕ Q 0 g j (0, x) = β(h j (0, γ(x))) = β(γ(x)) = x, g j (1, x) = β(h j (1, γ(x))) = β(h j (1, γ(y))) = g j (1, y). ϕ Q0 cat + r ( r ) = cat (). Lemma 3.5. If ϕ V is bounded from below and satisfies the (P S) c condition for any c [ inf ϕ, Q 0 ], w V then ϕ V has a minimum and level set ϕ Q0 contains at least cat Q ϕ 0 ϕ Q0 critical points of ϕ V. The proof of the above lemma can be found in [14]. Proof of Theorem 1.1. By Lemma 3.5, for 0 < <, the level set ϕ Q0 contains at least m := cat Q ϕ 0 ϕ 0 critical points w 1, w,..., w m of ϕ V. For j = 1,,..., m, there exist µ j R such that, for h H0,L 1 (C ), k y C 1 w j h dx dy wh dx µ j (w + j ) 1 h dx = 0.
11 EJDE-016/97 CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN 11 Choosing h = w j, we have 0 = k y C 1 w j dx dy w j dx. Since 0 < < 1, it implies w j = 0 and k y C 1 w j dx dy w j dx µ j (w + j ) dx = 0. Therefore, µ j = ϕ (w j ) and v j := µ N j w j is a positive solution of (1.4), tr (v j ) is a solution of (1.1). By Lemma 3.4, problems (1.4) and (1.1) have at least cat () positive solutions. References [1] A. Bahri, J. M. Coron; On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), [] B. Barrios, E. Colorado, A. de Pablo, U. Sánchez; On some critical problems for the fractional Laplacian oprator, J. Differential Equations, 5 (01), [3] C. Brãndle, E. Colorado, A. de Pablo; A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh, Sect. 143A (013), [4] H. Brezis, L. Nirenberg; Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), [5] X. Cabré, Y. Sire; Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linaire, 31 (014), [6] X. Cabré, J. Tan; Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math., 4 (010), [7] A. Capella, J. Dávila, L. Dupaigne, Y. Sire; Regularity of radial extremal solutions for some non local semilinear equations, Comm. in Part. Diff. Equa., 36 (011) [8] G. Cerami, D. Passaseo; Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with rich topology, Nonlinear Analysis: TMA, 18(199), [9] L. A. Caffarelli, S. Salsa, L. Silvestre; Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math., 171 (008), no., [10] L. Caffarelli, L. Silvestre; An extention problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 3 (007), [11] P. L. Lions; The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana 1 (1985), 45 11, [1] J. Tan; The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differential Equations, 4 (011), [13] J. Tan, J. Xiong; A Harnack inequality for fractional Laplace equations with lower order terms. Discrete Contin. Dyn. Syst., 31 (011), [14] M. Willem; Minimax Theorems, Birkhauser, Basel, Guowei Lin Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 3300, China address: lgw008@sina.cn Xiongjun Zheng Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 3300, China address: xjzh1985@16.com
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