The multiplicity of solutions and geometry in a wave equation

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 2, Number 2, June 2003 Website: http://AIMsciences.org pp. 159–170 THE MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN A WAVE EQUATION Q-Heung Choi Department of Mathematics Inha University Incheon 402-751, Korea Changbum Chun School of Liberal Arts & Education Korea University of Technology and Education Cheonan 330-708, Korea Tacksun Jung Department of Mathematics Kunsan National University Kunsan 573-701, Korea Abstract. We investigate multiplicity of solutions of the nonlinear one dimensional wave equation with Dirichlet boundary condition on the interval (− π2 , π2 ) and periodic condition on the variable t. Our concern is to investigate a relation between multiplicity of solutions and source terms of the equation when the nonlinearity −(bu+ − au− ) crosses an eigenvalue λ10 and the source term f is generated by three eigenfunctions. 1. Introduction. We investigate multiplicity of solutions of the nonlinear onedimensional wave equation with Dirichlet boundary condition on the interval (− π2 , π2 ) and periodic condition on the variable t, π π utt − uxx + g(u) = f (x, t) in (− , ) × R, (1.1) 2 2 π (1.2) u(± , t) = 0, 2 u is π − periodic in t and even in x and t, (1.3) where we assume that the semilinear term g(u) = bu+ − au− . Here the source term f is generated by the eigenfunctions of the one-dimensional wave operator utt − uxx under Dirichlet boundary condition. We let L the wave operator, Lu = utt − uxx . Then the eigenvalue problem for u(x, t) π π Lu = λu in (− , ) × R 2 2 with (1.2) and (1.3), has infinitely many eigenvalues λmn = (2n + 1)2 − 4m2 , m, n = 0, 1, 2, · · · . 1991 Mathematics Subject Classification. 35B10, 35L20. Key words and phrases. Multiplicity of solutions, eigenvalue, eigenfunction, Dirichlet boundary condition. 159 160 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG and corresponding normalized eigenfunctions φmn (m, n ≥ 0) given by √ 2 cos(2n + 1)x f or n ≥ 0, φ0n = π 2 φmn = cos 2mt cos(2n + 1)x f or m > 0, n ≥ 0. π We note that all eigenvalues in the interval (−9, 9) are given by λ21 = −7 < λ10 = −3 < λ00 = 1 < λ11 = 5. Let Q be the square [− π2 , π2 ] × [− π2 , π2 ] and H the Hilbert space defined by © ª H0 = u ∈ L2 (Q) : u is even in x and t . Then the set of eigenfunctions P {φmn } is an orthonormal base in H. Let us denote an element u, in H0 , as u = hmn φmn and we define a subspace H of H0 as X H = {u ∈ H0 : |λmn |h2mn < ∞}. P 1 Then this is a complete normed space with a norm kuk = ( |λmn |h2mn ) 2 . If f ∈ H0 and a, b are not eigenvalues of L, then every solution in H0 of Lu + bu+ − au− = f belongs to H (cf. [2]). Hence equation (1.1) with (1.2) and (1.3) is equivalent to Lu + bu+ − au− = f in H. (1.4) In [2] the authors investigate multiplicity of solutions of (1.4) when the nonlinearity −(bu+ − au− ) crosses the eigenvalue λ10 , the source term f is a multiple of the positive eigenfunction φ00 , and the condition (1.3) is replaced by u(x, t+π) = u(x, t). Our concern is to investigate a relation between multiplicity of solutions and source terms of jumping problem (1.4) when the nonlinearity −(bu+ − au− ) crosses an eigenvalue and the source term f is generated by three eigenfunctions. In Section 2, we suppose that the nonlinearity −(bu+ −au− ) crosses an eigenvalue λ10 and the source term f is generated by φ00 and φ10 and we investigate the properties of the reduced map Φ (see equation (2.6)). In Section 3, we reveal a relation between multiplicity of solutions and source terms in equation (1.4) when f belongs to the two dimensional space V spanned by φ00 and φ10 . In Section 4, we reveal a relation between multiplicity of solutions and source terms in equation (1.4) when f belongs to the three dimensional space spanned by φ00 , φ10 , and φmn (φmn 6= φ00 , φ10 ). 2. A Variational Reduction Method. In this section, we investigate multiplicity of solutions u(x, t) for a piecewise linear perturbation −(bu+ − au− ) of the one-dimensional wave operator utt − uxx with the nonlinearity −(bu+ − au− ) crossing the eigenvalue λ10 . We suppose that −1 < a < 3 and 3 < b < 7. Under this assumption, we have a concern with a relation between multiplicity of solutions and source terms of a nonlinear wave equation Lu + bu+ − au− = f in H. (2.1) Here we suppose that f is generated by two eigenfunctions φ00 and φ10 . We shall use the contraction mapping theorem to reduce the problem from an infinite dimensional one in H to a finite dimensional one. We investigate multiplicity of solutions and source terms of equation (2.1). MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN WAVE EQUATION 161 Let V be the two dimensional subspace of H spanned by {φ00 , φ10 } and W be the orthogonal complement of V in H. Let P be an orthogonal projection H onto V . Then every element u ∈ H is expressed by u = v + w, where v = P u, w = (I − P )u. Hence equation (2.1) is equivalent to a system Lw + (I − P )(b(v + w)+ − a(v + w)− ) = 0, (2.2) Lv + P (b(v + w)+ − a(v + w)− ) = s1 φ00 + s2 φ10 . (2.3) Lemma 2.1. For fixed v ∈ V , Furthermore, θ(v) is Lipschitz continuous (with respect to L2 norm) in terms of v. The proof of the lemma is similar to that of Lemma 2.1 of [3]. By Lemma 2.1, the study of multiplicity of solutions of (2.1) is reduced to the study of multiplicity of solutions of an equivalent problem Lv + P (b(v + θ(v))+ − a(v + θ(v))− ) = s1 φ00 + s2 φ10 (2.4) defined on the two dimensional subspace V spanned by {φ00 , φ10 }. If v ≥ 0 or v ≤ 0, then θ(v) ≡ 0. For example, let us take v ≥ 0 and θ(v) = 0. Then equation (2.2) reduces to L0 + (I − P )(bv + − av − ) = 0 which is satisfied because v + = v, v − = 0 and (I − P )v = 0, since v ∈ V . Since the subspace V is spanned by {φ00 , φ10 } and φ00 (x, t) > 0 in Q, there exists a cone C1 defined by n c1 o C1 = v = c1 φ00 + c2 φ10 : c1 ≥ 0, |c2 | ≤ √ 2 so that v ≥ 0 for all v ∈ C1 and a cone C3 defined by n |c1 | o C3 = v = c1 φ00 + c2 φ10 : c1 ≤ 0, |c2 | ≤ √ 2 so that v ≤ 0 for all v ∈ C3 . We define a map Φ : V → V given by Φ(v) = Lv + P (b(v + θ(v))+ − a(v + θ(v))− ), v ∈ V. (2.5) Then Φ is continuous on V, since θ is continuous on V and we have the following lemma (cf. Lemma 2.2 of [3]). Lemma 2.2. Φ(cv) = cΦ(v) for c ≥ 0 and v ∈ V . Lemma 2.2 implies that Φ maps a cone with vertex 0 onto a cone with vertex 0. Let Ci (1 ≤ i ≤ 4) be the same cones of V as in Section 1. We investigate the images of the cones C1 and C3 under Φ. First we consider the image of the cone C1 . If v = c1 φ00 + c2 φ10 ≥ 0, we have Φ(v) = L(v) + P (b(v + θ(v))+ − a(v + θ(v))− ) = c1 λ00 φ00 + c2 λ10 φ10 + b(c1 φ00 + c2 φ10 ) = c1 (b + λ00 )φ00 + c2 (b + λ10 )φ10 . 162 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG Thus the images of the rays c1 φ00 ± and they are √1 c1 φ10 (c1 2 ≥ 0) can be explicitly calculated 1 c1 (b + λ00 )φ00 ± √ c1 (b + λ10 )φ10 , c1 ≥ 0. 2 Therefore Φ maps C1 onto the cone n 1 ³ b + λ10 ´ o R1 = d1 φ00 + d2 φ10 : d1 ≥ 0, |d2 | ≤ √ d1 . 2 b + λ00 The cone R1 is in the right half-plane of V and the restriction Φ|C1 : C1 → R1 is bijective. We determine the image of the cone C3 . If v = −c1 φ00 + c2 φ10 ≤ 0, we have Φ(v) = = = L(v) + P (b(v + θ(v))+ − a(v + θ(v))− ) Lv + P (av) −c1 (λ00 + a)φ00 + c2 (λ10 + a)φ10 . Thus the images of the rays −c1 φ00 ± √12 c1 φ10 (c1 ≥ 0) can be explicitly calculated and they are 1 −c1 (λ00 + a)φ00 ± √ c1 (λ10 + a)φ10 c1 ≥ 0. 2 Thus Φ maps the cone C3 onto the cone n o 1 ¯¯ λ10 + a ¯¯ R3 = d1 φ00 + d2 φ10 : d1 ≤ 0, d2 ≤ √ ¯ ¯|d1 | . 2 λ00 + a The cone R3 is in the left half-plane of V and the restriction Φ|C3 : C3 → R3 is bijective. We note that R1 is in the right half plane and R3 is in the left half plane. Theorem 2.3. (i) If f belongs to R1 , then equation (2.1) has a positive solution and no negative solution. (ii) If f belongs to R3 , then equation (2.1) has a negative solution and no positive solution. The cones C2 , C4 are as follows o n 1 C2 = c1 φ00 + c2 φ10 : c2 ≥ 0, c2 ≥ √ |c1 | , 2 n o 1 C4 = c1 φ00 + c2 φ10 : c2 ≤ 0, c2 ≤ − √ |c1 | . 2 Then the union of four cones Ci (1 ≤ i ≤ 4) is the space V. Lemma 2.2 means that the images Φ(C2 ) and Φ(C4 ) are the cones in the plane V. Before we investigate the images Φ(C2 ) and Φ(C4 ), we set n √ ¯¯ b + λ00 ¯¯ o √ ¯¯ λ00 + a ¯¯ R2′ = d1 φ00 + d2 φ10 : d2 ≥ 0, − 2¯ ¯d2 ≤ d1 ≤ 2¯ ¯d2 , λ10 + a b + λ10 o n √ ³ b + λ00 ´ √ ³ λ00 + a ´ |d2 | . d2 ≤ d 1 ≤ 2 R4′ = d1 φ00 + d2 φ10 : d2 ≤ 0, 2 λ10 + a b + λ10 Then the union of four cones R1 , R2′ , R3 , R4′ is also the space V. To investigate a relation between multiplicity of solutions and source terms in the nonlinear wave equation Lu + bu+ − au− = f in H, (2.6) MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN WAVE EQUATION 163 we consider the restrictions Φ|Ci (1 ≤ i ≤ 4) of Φ to the cones Ci . Let Φi = Φ|Ci , i.e., Φi : Ci → V. For i = 1, 3, the image of Φi is Ri and Φi : Ci → Ri is bijective. From now on, our goal is to find the image of Ci under Φi for i = 2, 4. Suppose that γ is a simple path in C2 without meeting the origin, and end points (initial and terminal) of γ lie on the boundary ray of C2 and they are on each other boundary ray. Then the image of one end point of γ under Φ is on the ray c1 (b + λ00 )φ00 + √1 c1 (b + λ10 )φ10 , c1 ≥ 0 (a boundary ray of R1 ) and the image of the other end 2 point of γ under Φ is on the ray −c1 (λ00 + a)φ00 + √12 c1 (λ10 + a)φ10 , c1 ≥ 0 (a boundary ray of R3 ). Since Φ is continuous, Φ(γ) is a path in V. By Lemma 1.2, Φ(γ) does not meet the origin. Hence the path Φ(γ) meets all rays (starting from the origin) in R1 ∪ R4′ or all rays (starting from the origin) in R2′ ∪ R3 . Therefore it follows from Lemma 1.2 that the image Φ(C2 ) of C2 contains one of sets R1 ∪ R4′ and R2′ ∪ R3 . Similarly, we have that the image Φ(C4 ) of C4 contains one of sets R1 ∪ R2′ and ′ R4 ∪ R 3 . 3. Multiplicity for Source Terms in Two Dimensional Space. In this section we reveal the relation between multiplicity of solutions and source terms in the nonlinear wave equation (2.1). Now we remember the map Φ : V → V given by Φ(v) = Lv + P (b(v + θ(v))+ − a(v + θ(v))− ), v ∈ V, where −1 < a < 3 < b < 7, θ(v) is a solution of (2.2), and V is the two-dimensional subspace of H0 spanned by two eigenfunctions φ00 , φ10 . The map Φ is continuous on V, since θ is continuous on V. For f ∈ V, we establish an a priori bound for solutions of Lv + P (b(v + θ(v))+ − a(v + θ(v))− ) = f √1 b+1 in V. (3.1) √1 a+1 = 1}. Let k(≥ 16) be fixed and f ∈ V + Lemma 3.1. Let C = {(a, b) : with kf k = k. Let α, β, ǫ > 0 be given. Let 3 + α < b < 7 − α, −1 + β < a < 3 − β 1 1 + √a+1 6= 1 and dist((a, b), C) ≥ ǫ. Then there exists satisfy the condition √b+1 R0 > 0 (depending only on k and α, β, ǫ) such that the solutions of (2.7) satisfy kvk < R0 . Proof. Let −1 < a < 3 < b < 7, f ∈ V . Let v ∈ V be given. Then there exists a unique solution z ∈ W of the equation Lz + (I − P )[b(v + z)+ − a(v + z)− − f ] = 0 in W. If z = θ(v), then θ is continuous on V . In particular θ(v) satisfies a uniform Lipschitz in v with respect to the L2 norm (cf. [3]). Suppose the lemma does not hold. Then there is a sequence (bn , an , vn ) such that bn ∈ [−1 + α, 7 − α], an ∈ [−1 + β, 3 − β] satisfy dist((an , bn ), C) ≥ ǫ, kvn k → +∞, and vn = L−1 (f − P (bn (vn + θ(vn ))+ − an (vn + θ(vn ))− ) in V. Let un = vn + θ(vn ). Then the sequence (bn , an , un ) with bn ∈ [−1 + α, 7 − α], an ∈ [−1 + β, 3 − β] satisfies kun k → +∞ and − un = L−1 (f − bn u+ n + an u ) in H. 164 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG Put wn = un kun k . Then we have wn = L−1 ( f − bn wn+ + an wn− ). kun k The operator L−1 is compact. Therefore we may assume that wn → w0 , bn → b0 ∈ (−1, 7), an → a0 ∈ (−1, 3) with (a0 , b0 ) ∈ / C. Since kwn k = 1 for all n, kw0 k = 1 and w0 satisfies w0 = L−1 (−b0 w0+ + aw0− ) in H0 . This contradicts the fact (Theorem 1.2 of [2]) that for −1 < a, b < 7 with the 1 1 + √a+1 condition √b+1 6= 1 the equation Lu + bu+ − au− = 0 has only the trivial solution. Lemma 3.2. Let −1 < a < 3, −1 < b < 7 satisfy 1 1 √ +√ < 1. a+1 b+1 Let k(≥ b + 1) be fixed and f ∈ V with kf k = k. Then we have (3.2) d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR , 0) = 1 for all R ≥ R0 . Proof. Let b = a = 0. Then we have d(v − L−1 (f ), BR , 0) = 1, since the map is simply a translation of the identity and since kL−1 (f )k < R0 by Lemma 3.1. 1 1 + √a+1 In case b, a 6= 0(−1 < a < 3, −1 < b < 7) with √b+1 < 1, the result follows in the usual way by invariance under homotopy, since all solutions are in the open ball BR0 . Lemma 3.3. Let −1 < a < 3 < b < 7 satisfy the condition (3.2) and f ∈ IntR1 with kf k ≥ b + 1. Then equation (3.1) has a positive solution in IntC1 , at least one sign changing solution in IntC2 , and at least one sign changing solution in IntC4 . Proof. First we compute the degree (R > R0 ) d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C1 , 0) = d(v − L−1 (f − bv), BR ∩ C1 , 0) = −1, since v −L−1 (f −bv) = 0 has a unique solution in IntC1 and 1+ λb00 > 0, 1+ λb10 < 0. Since, for f = (b + 1)φ00 , equation (3.1) has no negative solution in IntC3 , d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C3 , 0) = 0. By the domain decomposition lemma, d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ (C2 ∪ C4 ), 0) = 2. Hence equation (3.1) has at least one sign changing solution in Int(C2 ∪ C4 ). Suppose that (3.1) has a solution in Int C2 . Then Φ(C2 )∩R1 6= φ. Let B : V → V be a linear map, where the matrix B is given by   b + a + 2λ b−a 00 √   2 2  b−a b + a + 2λ10  . √ 2 2 2 MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN WAVE EQUATION 165 Then B(C2 ) = R2 = Φ(C2 ) and Bv = Φ(v) for all v ∈ ∂C2 . Now we may assume that the solution of Bv = f is in BR0 . Hence if 0 ≤ t ≤ 1 and R ≥ R0 , then we have tBv + (1 − t)Φ(v) 6= f, v ∈ ∂(BR ∩ C2 ). So we have d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C2 , 0) = d(v − L−1 (f − Bv + Lv), BR ∩ C2 , 0) = 1, since Bv = f has a unique solution in IntC2 and det(L−1 B) > 0. Since d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR , 0) = 1 and d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C3 , 0) = 0, d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C4 , 0) = 1. Therefore (2.7) has at least one solution in IntC4 . Similarly, if we assume that (3.1) has a solution in IntC4 , then d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C4 , 0) = 1 and hence we get d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C2 , 0) = 1. Therefore (3.1) has at least one solution in IntC2 . With Theorem 2.3, Lemma 3.3, we get the following. Lemma 3.4. Let −1 < a < 3 < b < 7 satisfy the condition (3.2). For 1 ≤ i ≤ 4, let Φ(Ci ) = Ri . Then R2 contains R1 ∪ R4′ and R4 contains R1 ∪ R2′ , where R2′ , R4′ are the same cones as in section 2. Proof. We note that the equation Lu + bu+ − au− = f and the equation Lu + bu+ − au− = αf (α > 0) have the same number of solutions in H. It follows from Lemma 3.3 that R2 contains IntR1 . Let f ∈ IntR4′ and kf k ≥ b + 1. Using the same method as in the proof of Lemma 3.3, d(v − L−1 (f − P (b(v + θ(v))+ − a(v + θ(v))− )), BR ∩ C2 , 0) = 1 for R ≥ R0 . Hence R2 contains IntR4′ . Therefore the image R2 of C2 under Φ must contain R1 ∪ R4′ , since R2 contains one of sets R1 ∪ R4′ , R3 ∪ R2′ and is closed in V. Similarly, we have that R4 contains R1 ∪ R2′ . If a solution of (2.4) is in IntC1 , then it is positive. If a solution of (2.4) is in IntC3 , then it is negative. If a solution of (2.4) is in Int(C2 ∪ C4 ), then it has both signs. Therefore we have the main theorem of this section with aid of Lemma 3.4. Theorem 3.5. Let −1 < a < 3 < b < 7 satisfies the condition (3.2). Then we have the followings. (i) If f ∈ Int R1 , then equation (2.1) has a positive solution and at least two sign changing solutions. (ii) If f ∈ ∂R1 , then equation (2.1) has a nonnegative solution and at least one sign changing solution. (iii) If f ∈ Int Ri′ (i = 2, 4), then equation (2.1) has at least one sign changing solution. (iv) If f ∈ Int R3 , then equation (2.1) has only the negative solution. (v) If f ∈ ∂R3 , then equation (2.1) has a nonpositive solution. 166 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG 4. Multiplicity for Source Terms in Three Dimensional Space. In this section, we investigate a relation between multiplicity of solutions and the source terms of a nonlinear wave equation when the source terms belong to the three dimensional space spanned by φ00 , φ10 , and φmn (φmn 6= φ00 , φ10 ). First we consider Lu + bu+ − au− = s1 φ00 + s2 φ10 + ǫφ20 . (4.1) Let V , W and P be the same as in section 2. Then equation (4.1) is equivalent to a system Lw + (I − P )(b(v + w)+ − a(v + w)− ) = ǫφ20 , (4.2) + − Lv + P (b(v + w) − a(v + w) ) = s1 φ00 + s2 φ10 . (4.3) Lemma 4.1. For fixed v ∈ V , (4.2) has a unique solution w = θǫ (v). Furthermore, θǫ (v) is Lipschitz continuous (with respect to L2 norm) in terms of v. Proof. We use the cotraction mapping principle. Let δ = 21 (a + b). We rewrite (4.2) as (−L − δ)w = (I − P )(b(v + w)+ − a(v + w)− δ(v + w) − ǫφ20 ), or equivalently, w = (−L − δ)−1 (I − P )(gv (w) − ǫφ20 ), (4.4) where gv (w) = b(v + w)+ − a(v + w)− − δ(v + w). Since |gv (w1 ) − gv (w2 )| ≤ |b − δ||w1 − w2 |, we have 2 kgv (w1 ) − gv (w2 )k ≤ |b − δ|kw1 − w2 k, where k k is the L norm in H. The operator (−L − δ)−1 (I − P ) is a selfadjoint compact map from (I − P )H into itself. The eigenvalues of (−L − δ)−1 (I − P ) in W are (λmn − δ)−1 , where λmn ≥ 7 or λmn ≤ −1. Therefore its L2 norm is 1 1 max{ 7−δ , 1+δ }. Since |b − δ| < min{7 − δ, 1 + δ}, it follows that for fixed v ∈ V , the right hand side of (4.4) defines a Lipschitz mapping W into itself with Lipschitz constant γ < 1. Hence, by the cotraction mapping principle, for given v ∈ V , there is a unique w ∈ W which satisfies (4.2). Also, it follows, by the standard arguement principle, that θǫ (v) is Lipschitz continuous in terms of v. By Lemma 4.1, the study of multiplicity of solutions of (4.1) is reduced to the study of multiplicity of solutions of an equivalent problem Lv + P (b(v + θǫ (v))+ − a(v + θǫ (v))− ) = s1 φ00 + s2 φ10 (4.4) defined on the two dimensional subspace V . Since the subspace V is spanned by {φ00 , φ10 } and v > 0 in Q for all v ∈ C1 , there exists a convex subset C1ǫ of C1 defined by C1ǫ = {v = c1 φ00 + c2 φ10 : v + ǫφ20 > 0 in Q} and a convex subset C3ǫ of C3 defined by C3ǫ = {v = c1 φ00 + c2 φ10 : v + ǫφ20 < 0 in Q} . We define a map J : R × V → V given by J(ǫ, v) = Lv + P (b(v + θǫ (v))+ − a(v + θǫ (v))− ), v ∈ V. (4.6) Then for fixed ǫ, J is continuous on V, since θǫ (v) is continuous on V . Also, it is easily proved that for fixed v, J is continuous on V. MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN WAVE EQUATION 167 Lemma 4.2. For fixed v ∈ V , J is continuous on R. Proof. It is enough to show that for fixed v, θǫ (v) is a continuous function of ǫ. We use the cotraction mapping principle. Let δ = 12 (a + b). Let wi (i = 1, 2) be the unique solution of (−L − δ)w = (I − P )(b(v + w)+ − a(v + w)− δ(v + w) − ǫi φ20 ), or equivalently, w = (−L − δ)−1 (I − P )(gv (w) − ǫi φ20 ), where (4.7) gv (w) = b(v + w)+ − a(v + w)− − δ(v + w). Then we have, by Lemma 4.2, kw1 − w2 k ≤ γkw1 − w2 k + or equivalently, kφ20 k |ǫ1 − ǫ2 |, |λ20 + δ| kφ20 k |ǫ1 − ǫ2 |, |λ20 + δ| which means that for fixed v, θǫ (v) is a continuous function of ǫ, where γ < 1. (1 − γ)kw1 − w2 k ≤ ǫ We note that if v is in C1ǫ , then θǫ (v) = b−15 φ20 . In fact, if v is in C1ǫ , then ǫ ǫ φ20 satisfies v + ǫφ20 > 0 in Q and hence v + b−15 φ20 > 0 in Q. Hence θǫ (v) = b−15 Lθǫ (v) + (I − P )(b(v + θǫ (v))+ − a(v + θǫ (v))− ) = ǫφ20 . ǫ Also, if v is in C3ǫ , then v + ǫφ20 < 0 in Q and hence v + a−15 φ20 < 0 in Q. Hence ǫ θǫ (v) = a−15 φ20 satisfies the above equation. We investigate the images of the convx sets C1ǫ and C3ǫ under J. First we consider the image of the cone C1ǫ . If v = c1 φ00 + c2 φ10 is in C1ǫ , then v + θǫ (v) > 0 in Q and hence we have J(ǫ, v) = L(v) + P (b(v + θǫ (v))+ − a(v + θǫ (v))− ) = c1 λ00 φ00 + c2 λ10 φ10 + b(c1 φ00 + c2 φ10 ) = c1 (b + λ00 )φ00 + c2 (b + λ10 )φ10 . Thus, for fixed ǫ, the image of C1ǫ under J, J(ǫ, C1ǫ ), is a convex subset of n 1 ³ b + λ10 ´ o R1 = d1 φ00 + d2 φ10 : d1 ≥ 0, |d2 | ≤ √ d1 . 2 b + λ00 For fixed ǫ, the restriction J|C1ǫ : C1ǫ → J(ǫ, C1ǫ ) is bijective. We determine the image of the cone C3ǫ . If v = c1 φ00 + c2 φ10 is in C3ǫ , then v + θǫ (v) < 0 in Q and hence we have J(ǫ, v) = L(v) + P (b(v + θǫ (v))+ − a(v + θǫ (v))− ) = Lv + P (av) = −c1 (λ00 + a)φ00 + c2 (λ10 + a)φ10 . Thus, for fixed ǫ, the image of C3ǫ under J, J(ǫ, C3ǫ ), is a convex subset of n o 1 ¯¯ λ10 + a ¯¯ R3 = d1 φ00 + d2 φ10 : d1 ≤ 0, d2 ≤ √ ¯ ¯|d1 | . 2 λ00 + a For fixed ǫ, the restriction J|C3ǫ : C3ǫ → J(ǫ, C3ǫ ) is bijective. 168 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG Let ǫ > 0 be fixed. If v is in C1ǫ , then θǫ (v) = b−15−ǫ b−15 v > 0 in Q. Hence we have the lemma. ǫ b−15 φ20 and ǫ b−15 (v + φ20 ) > 0, ′ ′ ′ Lemma 4.3. Let ǫ > 0 be fixed. Then there are open sets C1ǫ , C3ǫ with C¯1ǫ ⊂ C1ǫ ⊂ ǫ ǫ ′ ′ ¯ C1 , C3ǫ ⊂ C3ǫ ⊂ C3 such that θǫ (v) = b−15 φ20 for all v ∈ C1ǫ , θǫ (v) = a−15 φ20 for ′ all v ∈ C3ǫ Theorem 4.4. Let ǫ > 0 be fixed. Then we have: (i) If f belongs to J(ǫ, C1ǫ ), then equation (4.1) has a positive solution and no negative solution. (ii) If f belongs to J(ǫ, C3ǫ ), then equation (4.1) has a negative solution and no positive solution. We define two sets C2ǫ , C4ǫ as follows C2ǫ = {v = c1 φ00 + c2 φ10 : c2 ≥ 0, v ∈ / IntC1ǫ , v ∈ / IntC3ǫ }, C4ǫ = {v = c1 φ00 + c2 φ10 : c2 ≤ 0, v ∈ / IntC1ǫ , v ∈ / IntC3ǫ }. Then C2 ⊂ C2ǫ , C4 ⊂ C4ǫ and the union of four sets Ciǫ (1 ≤ i ≤ 4) is the space V. For fixed ǫ > 0, we set R1ǫ = J(ǫ, C1ǫ ), R3ǫ = J(ǫ, C3ǫ ), ′ R2ǫ = {v = d1 φ00 + d2 φ10 : d2 ≥ 0, v ∈ / IntR1ǫ , v ∈ / IntR3ǫ } , ′ R4ǫ = {v = d1 φ00 + d2 φ10 : d2 ≤ 0, v ∈ / IntR1ǫ , v ∈ / IntR3ǫ } , Iǫ = {v = c1 φ00 : |c1 | ≤ ǫ}, Iǫδ = {v ∈ V : d(v, Iǫ ) < δ}\(R1ǫ ∪ R3ǫ ), ′ ′ where d(v, Iǫ ) = inf{kv − wk : w ∈ Iǫ }. Then R2′ ⊂ R2ǫ , R4′ ⊂ R4ǫ and the union of ′ ′ four sets R1ǫ , R2ǫ , R3ǫ , R2ǫ is V. ′ We note that Ci = Ci0 (1 ≤ i ≤ 4) and Ri = Ri0 (i = 1, 3), Rj′ = Rj0 (j = 2, 4). Since J is continuous, by Lemma 3.4 we have: Lemma 4.5. Let −1 < a < 3 < b < 7 satisfy the condition (3.2). Then, for small ′ ǫ > 0, there is δ > 0 such that J(ǫ, C2ǫ ) contains (R1ǫ ∪ R4ǫ )\Iǫδ and J(ǫ, C4ǫ ) ′ contains (R1ǫ ∪ R2ǫ )\Iǫδ . Lemma 4.6. Let −1 < a < 3 < b < 7 satisfy the condition (3.2). Then , for small ǫ > 0, the equation Lu + bu+ − au− = s1 φ00 + s2 φ10 + ǫφ20 has at least one solution. Proof. J(ǫ, ·) is continuous in v and homotopic to Φ. J(ǫ, C3ǫ ) = Φ(C3ǫ ) = R3ǫ and J(ǫ, ∂C3ǫ ) = Φ(∂C3ǫ ). Since Φ(V \C3ǫ ) contains V \R3ǫ , J(ǫ, C3ǫ ) contains it, which completes the proof. Remark 4.7. By the modification of the proof of Lemma 3.2, we have: Let −1 < a < 3, −1 < b < 7 satisfy (3.2). Let k(≥ b + 1) be fixed and f ∈ V with kf k = k. Then there is R0 > 0 such that d(v − L−1 (f − P (b(v + θǫ (v))+ − a(v + θǫ (v))− )), BR , 0) = 1 for all R ≥ R0 . If Lu + bu+ − au− = s1 φ00 + s2 φ10 + ǫφ20 has m multiple solutions, then so is Lu + bu+ − au− = k(s1 φ00 + s2 φ10 + ǫφ20 ) (k > 0). The above equation implies Lemma 4.7. With Lemma 4.3 , Lemma 4.5, and , we have the following. MULTIPLICITY OF SOLUTIONS AND GEOMETRY IN WAVE EQUATION 169 Lemma 4.8. Let −1 < a < 3 < b < 7 satisfies the condition (3.2). Let v = s1 φ00 + s2 φ10 and f = v + ǫφ20 . Then , for small ǫ > 0, we have the followings. (i) If v ∈ (Int R1ǫ ), then equation (4.1) has a positive solution and at least two sign changing solutions. (ii) If v ∈ (∂R1ǫ ), then equation (4.1) has a nonnegative solution and at least one sign changing solution. ′ (iii) If v ∈ (Int Riǫ )(i = 2, 4), then equation (4.1) has at least one sign changing solution. (iv) If v ∈ (Int R3ǫ ), then equation (4.1) has a negative solution. (v) If v ∈ (∂R1ǫ ), then equation (4.1) has a nonpositive solution. If Lu + bu+ − au− = s1 φ00 + s2 φ10 + ǫφ20 has m multiple solutions, then so is Lu + bu+ − au− = k(s1 φ00 + s2 φ10 + ǫφ20 ) (k > 0). If α = kǫ(k > 0, ǫ > 0), then C1α = kC1ǫ , C3α = kC3ǫ . We note that C1(−α) = C1α , C3(−α) = C3α . If v + ǫφ20 > 0 in Q (v ∈ V ), then v ∈ C1ǫ and if v + ǫφ20 < 0 in Q (v ∈ V ), then v ∈ C3ǫ . We set : U1 = {v + ǫφ20 : v ∈ R1ǫ , ǫ ∈ R}, U3 = {v + ǫφ20 : v ∈ R3ǫ , ǫ ∈ R}, U2 = span{φ00 , φ10 , φ20 }\(IntU1 ∪ IntU3 ) With the above notations and facts, we have: Theorem 4.9. Let −1 < a < 3 < b < 7 satisfy the condition (3.2). Let f = s1 φ00 + s2 φ10 + ǫφ20 . Then we have the followings. (i) If f ∈ IntU1 , then equation (4.1) has a positive solution and at least two sign changing solutions. (ii) If f ∈ ∂U1 , then equation (4.1) has a nonnegative solution and at least one sign changing solution. (iii) If f ∈ Int U2 , then equation (4.1) has at least one sign changing solution. (iv) If f ∈ U3 , then equation (4.1) has a negative solution. Let φmn be an eigenfunction corresponding to λmn (λmn 6= λ00 , λ10 ). We consider the equation Lu + bu+ − au− = s1 φ00 + s2 φ10 + s3 φmn . (4.8) By the similar method of the proof of Theorem 4.9, we have the following. Theorem 4.10. Let −1 < a < 3 < b < 7 satisfy the condition (3.2). Let f = s1 φ00 + s2 φ10 + s3 φmn (λmn 6= λ00 , λ10 ). Then there are cones U1 , U2 , U3 in span{φ00 , φ10 , φmn } which satisfy he followings. (i) If f ∈ IntU1 , then equation (4.8) has a positive solution and at least two sign changing solutions. (ii) If f ∈ ∂U1 , then equation (4.8) has a nonnegative solution and at least one sign changing solution. (iii) If f ∈ Int U2 , then equation (4.8) has at least one sign changing solution. (iv) If f ∈ U3 , then equation (4.8) has a negative solution. Acknowledgments. This work was supported by grant 2000-2-101-001-3 from the Basic Research Program of the Korea Science and Engineering Foundation. 170 Q-HEUNG CHOI, CHANGBUM CHUN AND TACKSUN JUNG REFERENCES [1] A. Ambrosetti and G. Prodi, A primer of nonlinear analysis, Cambridge, University Press, Cambridge Studies in Advanced Math., 34 (1993). [2] Q.H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations, 117 (1995), 390–410. [3] Q.H. Choi and T. Jung, A nonlinear wave equation with jumping nonlinearity, Discrete and Continuous Dynamical Systems, 4 (2000), 797–802. [4] A.C. Lazer and P.J. McKenna, Some multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl., 107 (1985), 371–395. [5] A.C. Lazer and P.J. McKenna, Critical points theory and boundary value problems with nonlinearities crossing multiple eigenvalues II., Comm.in P.D.E., 11 (1986), 1653–1676. [6] K. Schmitt, oundary value problems with jumping nonlinearities, Rocky Mountain Math. J., 16 (1986), 481–496. Received June 2002; revised January 2003. E-mail address: [email protected]