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3; = +1 when N = 1; 2 : (1.4) Equation (1.1) with (1.2) or (1.3) arises in many branches of the applied sciences. For example, it can be viewed as a steady-state equation for the shadow system of the Gierer-Meinhardt system in biological pattern formation ( 13], 29], 35]) or of parabolic equations in chemotaxis, population dynamics and phase transitions ( 2], 3], 23], 27]). Without loss of generality, we may assume that b = 1. Associated with (1.1) is the energy functional J de ned by Z 2 1 jruj2 + 2 u2 ? F (u) dx for u 2 H 1( ); J u] := 2 (1.5) R u f (s)ds. where F (u) = 0 It is known that any solution u of (1.1) is a critical point of J and vice versa. In this paper, we restrict ourselves to families of solutions fu g0< < 0 of (1.1) with nite energy, i.e. ?N J u ] < +1 for 0 < < : (1.6) 0 It can be proved that for su ciently small, any family of solutions of (1.1) satisfying (1.6) can have at most a nite number of local maximum points (see 24]). Let the local maximum points be fP1 ; :::; PK g . If Pj 2 @ ; j = 1; :::; K , we call u a K ?boundary spike solution. If K = 1, we call u a single boundary spike solution.
1991 Mathematics Subject Classi cation. Primary 35B40, 35B45; Secondary 35J25. Key words and phrases. Higher-Order Energy Expansions, Singularly Perturbed Neumann Problem, Spike Locations, Ricci Curvature. 1
HIGHER-ORDER ENERGY EXPANSIONS AND SPIKE LOCATIONS
JUNCHENG WEI AND MATTHIAS WINTER
Abstract. We consider the following singularly perturbed semilinear elliptic problem: ( 2 u ? u + f (u) = 0 in ; (I ) u > 0 in and @u = 0 on @ ; @ N where is a bounded domain in R with smooth boundary @ , > 0 is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional J de ned by Z 2 2 1 2 1 J u] := 2 jruj + 2 u ? F (u) dx for u 2 H ( ); R where F (u) = 0u f (s)ds. Ni and Takagi ( 24], 25]) proved that for a single boundary spike solution u , the following asymptotic expansion holds: # " 1 I w] ? c H (P ) + o( ) ; N J u ]= 1 2
ቤተ መጻሕፍቲ ባይዱ
where c1 > 0 is a generic constant, P is the unique local maximum point of u and H (P ) is the boundary mean curvature function at P 2 @ . In this paper, we obtain a higher-order expansion of J u ] : " # 1 I w] ? c H (P ) + 2 c (H (P ))2 + c R(P )] + o( 2 ) J u ]= N 1 2 3 2 where c2 ; c3 are generic constants and R(P ) is the Ricci scalar curvature at P . In particular c3 > 0. Some applications of this expansion are given.
2
JUNCHENG WEI AND MATTHIAS WINTER
where is a bounded domain in RN with smooth boundary @ , > 0 is P 2 a small constant, := N=1 @x@@x denotes the Laplace operator in RN , j stands for the unit outer normal to @ and @=@ for the normal derivative, 2 b > 0 is a positive constant and f (t) is a function in C 1+ (R) \ Cloc (0; +1) such that f (0) = f 0 (0) = 0. Typical examples of the function ?bu + f (u) are ? bu + f (u) = ?u + up with u+ = max(0; u); b = 1; (1.2) +
+
where 1<p< N +2 N ?2
1 ? bu + f (u) = u(u ? a)(1 ? u) with 0 < a < 2 ; b = a; = N + 2 when N N ?2
(1.3)
HIGHER-ORDER ENERGY EXPANSIONS
3
In the pioneering papers 23], 24] and 25], Lin, Ni and Takagi established the existence of least-energy solutions and showed that for su ciently small the least-energy solution is a single boundary spike solution and has only one local maximum point P with P 2 @ . Moreover, H (P ) ! maxP 2@ H (P ) as ! 0, where H (P ) is the mean curvature of @ at P . Since then many works have been devoted to nding solutions with multiple spikes for the Neumann problem as well as the Dirichlet problem. See 1], 2], 3], 4], 6], 7], 8], 9], 10], 11], 12], 15], 16], 17], 18], 19], 21], 22], 24], 25], 26], 27], 28], 31], 32], 36], 37], and the references therein. Recent surveys can be found in 29], 35]. A common tool for proving the existence of spike solutions is the energy expansion: In 24] and 25], Ni and Takagi proved, among others, that for a single boundary spike solution u , the following asymptotic expansion for J u ] holds: # " 1 I w] ? c H (P ) + o( ) ; (1.7) J u]= N 2 1 where c1 > 0 is a generic constant, P is the unique local maximum point of u , H (P ) is the mean curvature function at P 2 @ , w is the unique solution of the following ground-state problem: ( w ? w + f (w) = 0; w > 0 in RN ; (1.8) w(0) = maxy2R w(y); limjyj!+1 w(y) = 0 and I w] is the ground-state energy 1 Z jrwj2 dy + 1 Z w2 dy ? Z F (w) dy: I w] = 2 (1.9) 2 R R R (Note that Ni and Takagi ( 24], 25]) proved (1.7) for least-energy solutions. But it is easy to see that it also holds for any single boundary spike solution.) Based on (1.7), Ni and Takagi 25] showed that the least energy solution must concentrate at a maximum point of the mean curvature function. If H (P ) has more than one maximum points on @ , the asymptotic expansion (1.7) is no longer su cient to derive the spike location and the next order term in (1.7) becomes important. This is exactly the purpose of this paper.
1. Introduction We consider the following singularly perturbed semilinear elliptic problem: ( 2 u ? bu + f (u) = 0 in ; (1.1) u > 0 in and @u = 0 on @ ; @
3; = +1 when N = 1; 2 : (1.4) Equation (1.1) with (1.2) or (1.3) arises in many branches of the applied sciences. For example, it can be viewed as a steady-state equation for the shadow system of the Gierer-Meinhardt system in biological pattern formation ( 13], 29], 35]) or of parabolic equations in chemotaxis, population dynamics and phase transitions ( 2], 3], 23], 27]). Without loss of generality, we may assume that b = 1. Associated with (1.1) is the energy functional J de ned by Z 2 1 jruj2 + 2 u2 ? F (u) dx for u 2 H 1( ); J u] := 2 (1.5) R u f (s)ds. where F (u) = 0 It is known that any solution u of (1.1) is a critical point of J and vice versa. In this paper, we restrict ourselves to families of solutions fu g0< < 0 of (1.1) with nite energy, i.e. ?N J u ] < +1 for 0 < < : (1.6) 0 It can be proved that for su ciently small, any family of solutions of (1.1) satisfying (1.6) can have at most a nite number of local maximum points (see 24]). Let the local maximum points be fP1 ; :::; PK g . If Pj 2 @ ; j = 1; :::; K , we call u a K ?boundary spike solution. If K = 1, we call u a single boundary spike solution.
1991 Mathematics Subject Classi cation. Primary 35B40, 35B45; Secondary 35J25. Key words and phrases. Higher-Order Energy Expansions, Singularly Perturbed Neumann Problem, Spike Locations, Ricci Curvature. 1
HIGHER-ORDER ENERGY EXPANSIONS AND SPIKE LOCATIONS
JUNCHENG WEI AND MATTHIAS WINTER
Abstract. We consider the following singularly perturbed semilinear elliptic problem: ( 2 u ? u + f (u) = 0 in ; (I ) u > 0 in and @u = 0 on @ ; @ N where is a bounded domain in R with smooth boundary @ , > 0 is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional J de ned by Z 2 2 1 2 1 J u] := 2 jruj + 2 u ? F (u) dx for u 2 H ( ); R where F (u) = 0u f (s)ds. Ni and Takagi ( 24], 25]) proved that for a single boundary spike solution u , the following asymptotic expansion holds: # " 1 I w] ? c H (P ) + o( ) ; N J u ]= 1 2
ቤተ መጻሕፍቲ ባይዱ
where c1 > 0 is a generic constant, P is the unique local maximum point of u and H (P ) is the boundary mean curvature function at P 2 @ . In this paper, we obtain a higher-order expansion of J u ] : " # 1 I w] ? c H (P ) + 2 c (H (P ))2 + c R(P )] + o( 2 ) J u ]= N 1 2 3 2 where c2 ; c3 are generic constants and R(P ) is the Ricci scalar curvature at P . In particular c3 > 0. Some applications of this expansion are given.
2
JUNCHENG WEI AND MATTHIAS WINTER
where is a bounded domain in RN with smooth boundary @ , > 0 is P 2 a small constant, := N=1 @x@@x denotes the Laplace operator in RN , j stands for the unit outer normal to @ and @=@ for the normal derivative, 2 b > 0 is a positive constant and f (t) is a function in C 1+ (R) \ Cloc (0; +1) such that f (0) = f 0 (0) = 0. Typical examples of the function ?bu + f (u) are ? bu + f (u) = ?u + up with u+ = max(0; u); b = 1; (1.2) +
+
where 1<p< N +2 N ?2
1 ? bu + f (u) = u(u ? a)(1 ? u) with 0 < a < 2 ; b = a; = N + 2 when N N ?2
(1.3)
HIGHER-ORDER ENERGY EXPANSIONS
3
In the pioneering papers 23], 24] and 25], Lin, Ni and Takagi established the existence of least-energy solutions and showed that for su ciently small the least-energy solution is a single boundary spike solution and has only one local maximum point P with P 2 @ . Moreover, H (P ) ! maxP 2@ H (P ) as ! 0, where H (P ) is the mean curvature of @ at P . Since then many works have been devoted to nding solutions with multiple spikes for the Neumann problem as well as the Dirichlet problem. See 1], 2], 3], 4], 6], 7], 8], 9], 10], 11], 12], 15], 16], 17], 18], 19], 21], 22], 24], 25], 26], 27], 28], 31], 32], 36], 37], and the references therein. Recent surveys can be found in 29], 35]. A common tool for proving the existence of spike solutions is the energy expansion: In 24] and 25], Ni and Takagi proved, among others, that for a single boundary spike solution u , the following asymptotic expansion for J u ] holds: # " 1 I w] ? c H (P ) + o( ) ; (1.7) J u]= N 2 1 where c1 > 0 is a generic constant, P is the unique local maximum point of u , H (P ) is the mean curvature function at P 2 @ , w is the unique solution of the following ground-state problem: ( w ? w + f (w) = 0; w > 0 in RN ; (1.8) w(0) = maxy2R w(y); limjyj!+1 w(y) = 0 and I w] is the ground-state energy 1 Z jrwj2 dy + 1 Z w2 dy ? Z F (w) dy: I w] = 2 (1.9) 2 R R R (Note that Ni and Takagi ( 24], 25]) proved (1.7) for least-energy solutions. But it is easy to see that it also holds for any single boundary spike solution.) Based on (1.7), Ni and Takagi 25] showed that the least energy solution must concentrate at a maximum point of the mean curvature function. If H (P ) has more than one maximum points on @ , the asymptotic expansion (1.7) is no longer su cient to derive the spike location and the next order term in (1.7) becomes important. This is exactly the purpose of this paper.
1. Introduction We consider the following singularly perturbed semilinear elliptic problem: ( 2 u ? bu + f (u) = 0 in ; (1.1) u > 0 in and @u = 0 on @ ; @