Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 09, pp. 1–8. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.09 NECESSARY AND SUFFICIENT CONDITION FOR EXISTENCE FOR A CASE OF EIGENVALUES OF MULTIPLICITY TWO PHILIP KORMAN Abstract. We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation. 1. Introduction The study of elliptic problems at resonance was initiated by the classical paper of Landesman and Lazer [8]. On a bounded smooth domain D ⊂ Rn consider the Dirichlet problem ∆u+ λku+ g(u) = f(x) for x ∈ D , u = 0 on ∂D . (1.1) Here λk is an eigenvalue of the Laplacian ∆ on D with zero boundary condition, so that the problem is at resonance. The function f(x) ∈ L2(D) is given. For the nonlinear term g(u) it is assumed that the limits g(∞) and g(−∞) exist and g(−∞) < g(u) < g(∞), for all u ∈ (−∞,∞) . (1.2) Let us recall the classical theorem of Landesman and Lazer [8] in the form of S.A. Williams [14] (both necessary and sufficient conditions can be separately general- ized, see [8]). Theorem 1.1 ([8, 14]). Assume that g(u) satisfies (1.2), f(x) ∈ L2(D), while for any w(x) 6= 0 belonging to the eigenspace of λk,∫ D f(x)w(x) dx < g(∞) ∫ w>0 w dx+ g(−∞) ∫ w<0 w dx . (1.3) Then problem (1.1) has a solution u(x) ∈ W 2,2(D) ∩W 1,2 0 (D). Condition (1.3) is also necessary for the existence of solutions. Originally Landesman and Lazer [8] assumed additionally that the eigenvalue λk is simple. Soon, Williams [14] produced the more general statement given above. However, no examples for multiple eigenvalues were known for a while, until we 2020 Mathematics Subject Classification. 35J25. Key words and phrases. Existence of solutions; Landesman-Lazer condition; resonance. ©2024. This work is licensed under a CC BY 4.0 license. Submitted December 22, 2023. Published January 24, 2024. 1 2 P. KORMAN EJDE-2024/09 observed in [5] that another classical result of Lazer and Leach [9] on periodic solu- tions of semilinear harmonic oscillator provides an example to Theorem 1.1 in case of double eigenvalues (giving incidentally another proof of Lazer-Leach theorem, in addition to a number of other known proofs, see e.g., [3, 5]). We showed in [5] that while the necessary condition of Lazer-Leach result is easy to prove, the sufficiency part follows by verifying the condition (1.3), and applying Theorem 1.1. In this article we prove a similar result for a disc in R2, thus providing the first PDE example for Theorem 1.1 in case of a multiple dimensional eigenspace. Even for simple domains the eigenspace of a multiple eigenvalue can be very complicated, and multiplicity of eigenvalues may vary in non-obvious ways. So that verifying the inequality (1.3) for any element w(x) of the eigenspace appears to be very challenging for other domains (the integrals ∫ w>0 w(x) dx and ∫ w<0 w(x) dx are unlikely to remain constant over an eigenspace). Example 1.2. Let D = (0, π)× (0, π) in R2. The eigenvalues of ∆u+ λu = 0 , in D u = 0 on ∂D are λnm = n2 +m2 with positive integers n and m, corresponding to the eigenfunc- tions sinnx sinmy, see e.g., [11]. These eigenfunctions are obtained by separation of variables, and there are no other eigenfunctions since these eigenfunctions form a complete set in L2(D). The principal eigenvalue λ1 = 2 is simple, with the corresponding eigenfunction sinx sin y > 0. The eigenvalue λ2 = 5 = 12 + 22 has multiplicity two, with the eigenspace spanned by sinx sin 2y, sin 2x sin y. The eigenvalue λ3 = 8 = 22 + 22 is simple, with the eigenspace spanned by sin 2x sin 2y. The eigenvalue 50 = 12 + 72 = 52 + 52 is triple, with the eigenspace spanned by sinx sin 7y, sin 7x sin y, sin 5x sin 5y. The eigenvalue 65 = 12 + 82 = 42 + 72 is quadruple, the eigenvalue 325 = 12 + 182 = 62 + 172 = 102 + 152 has multiplicity six, and so on. It is natural to ask if there is an eigenvalue of any multiplicity. In number theoretic terms the possible conjecture is: for any even integer 2m one can find an integer N that can be represented as N = p2 + q2, with integers p 6= q, in exactly m different ways, while for any odd integer 2m+ 1 one can find an integer M that can be represented as M = p2 + q2, with integers p 6= q, in exactly m different ways, and in addition, M = r2 + r2 for some integer r > 0. By contrast, for a disc Da : x2+y2 < a2 in R2, we show that all eigenvalues of the Laplacian have multiplicity two, except for the principal one (which is simple), and that the integrals ∫ w>0 w(x, y) dx dy and ∫ w<0 w(x, y) dx dy remain constant over the entire eigenspaces, and can be explicitly calculated. We present a necessary and sufficient condition for the existence of solutions of the problem (1.1) on Da, for this case of resonance at a double eigenvalue. We prove the necessity part directly, while sufficiency is derived by verifying the conditions of Theorem 1.1. Our result can be seen as a PDE analog of the Lazer-Leach theorem. As an application, we give a condition for unboundness of all solutions of the corresponding semilinear heat equation. By contrast, for a disc Da : x2 + y2 < a2 in R2, we show that all eigenvalues of the Laplacian have multiplicity two, except for the principal one (which is simple), and that the integrals ∫ w>0 w(x, y) dx dy and ∫ w<0 w(x, y) dx dy remain constant over the entire eigenspaces, and can be explicitly calculated. We present a necessary and sufficient condition for the existence of solutions of the problem (1.1) on Da, for this case of resonance at a double eigenvalue. We prove the necessity part directly, while sufficiency is derived by verifying the conditions EJDE-2024/09 NECESSARY AND SUFFICIENT CONDITION 3 of Theorem 1.1. Our result can be seen as a PDE analog of the Lazer-Leach theorem. As an application, we give a condition for unboundness of all solutions of the corresponding semilinear heat equation. Radial solutions on balls in Rn were studied extensively, see e.g., Korman [4] or Ouyang and Shi [10], stimulated by the classical theorem of Gidas, Ni and Nirenberg [2] which asserts that any positive solution of semilinear Dirichlet problem on a ball is necessarily radially symmetric. Our result suggests that ball may be a special domain even when studying sign-changing non-symmetric solutions. Previously, Korman and Schmidt [6] studied resonance at the principal eigenvalue on B. They constructed g(u) for which the problem has infinitely many solutions for any f(x, y) ∈ L2(Da). 2. Resonance for a two-dimensional disc Remarkably, the eigenvalues of Laplacian on a disc Da : x2 + y2 < a2 in two dimensions all have multiplicity two, except for the principal eigenvalue, which is simple. Recall (see e.g. [11], p. 251) that the eigenvalues of the Laplacian on Da with zero boundary condition are λn,m = α2 n,m a2 (n = 0, 1, 2, . . .; m = 1, 2, . . .), with the corresponding eigenfunctions Jn (αn,m a r ) (A cosnθ +B sinnθ) , (2.1) where αn,m is the m-th root of Jn(t), the n-th Bessel function of the first kind, r = √ x2 + y2 (A and B are arbitrary constants; αn,m are all positive). There are no other eigenfunctions, since the ones given above form a complete set in L2(Da). The principal eigenpair is λ1 = α2 0,1 a2 ≈ 5.78 a2 , ϕ1(r) = J0( α0,1 a r). One calculates λ2 = α2 1,1 a2 ≈ 14.62 a2 , with α1,1 ≈ 3.83, and ϕ2 = J1( α1,1 a r)(A cos θ + B sin θ), and so on, see the Example below. The principal eigenvalue is simple, while all other eigenvalues have multiplicity two, because any two Bessel functions with indices different by an integer do not have any roots in common, see G.N. Watson [13, p. 484] for the following result. Proposition 2.1. For any integers n ≥ 0 and m ≥ 1, the functions Jn(t) and Jn+m(t) have no common zeros other than the one at x = 0. This result was apparently once a long standing conjecture (published in 1866), known in the 19-th century as Bourget’s hypothesis (after a 19th-century French mathematician), until it was proved in 1929 by Siegel, see [13], and a very infor- mative Wikipedia article on the Bessel functions. The name “hypothesis” suggests that it was used to prove other results. It immediately implies the following result that we need. Proposition 2.2. For the disc Da, all eigenvalues, other than the principal one, have multiplicity two. Proof. By Proposition 2.1, all αn,m are different, and hence the eigenspace of λn,m is two-dimensional, and is given by (2.1). � It turns out that for any eigenvalue λk, k ≥ 2, both integrals ∫ w>0 w(x, y) dx dy and ∫ w<0 w(x, y) dx dy on Da (appearing in (1.3)) remain constant for all w(x, y) in the eigenspace of λk (with A2+B2 = 1), and both integrals can be easily calculated. 4 P. KORMAN EJDE-2024/09 Let Pn,m denote the subset of (0, a) where Jn( αn,m a r) > 0, and Nn,m the subset of (0, a) where Jn( αn,m a r) < 0. The quantity Jn,m = 2 ∫ Pn,m Jn (αn,m a r ) r dr − 2 ∫ Nn,m Jn (αn,m a r ) r dr . (2.2) can be easily calculated using Mathematica for each pair of n and m. Proposition 2.3. Let w = Jn(αn,mr)(A cosnθ + B sinnθ) be any element of the eigenspace of the eigenvalue α2 n,m/a 2 > λ1, normalized so that A2 +B2 = 1. Then on Da ∫ w>0 w(r, θ) rdr dθ = Jn,m ,∫ w<0 w(r, θ) rdr dθ = −Jn,m . Proof. Write A cosnθ +B sinnθ = √ A2 +B2 cos(nθ − δ) = cos(nθ − δ) , for some δ. Then w = Jn(αn,mr) cos(nθ − δ) . (2.3) Let P denote the set of θ ∈ (0, 2π) where cos(nθ − δ) > 0, and N the set of θ ∈ (0, 2π) where cos(nθ − δ) < 0. It is easy to show that∫ P cos(nθ − δ) dθ = 2 ,∫ N cos(nθ − δ) dθ = −2 . (2.4) Then, in view of (2.3), (2.2) and (2.4)∫ w>0 w(r, θ) rdr dθ = ∫ Pn,m×P w(r, θ) rdr dθ + ∫ Nn,m×N w(r, θ) rdr dθ = ∫ Pn,m Jn( αn,m a r)r dr ∫ P cos(nθ − δ) dθ + ∫ Nn,m Jn( αn,m a r)r dr ∫ N cos(nθ − δ) dθ = Jn,m , and similarly∫ w<0 w(r, θ) rdr dθ = ∫ Pn,m×N w(r, θ) rdr dθ + ∫ Nn,m×P w(r, θ) rdr dθ = −Jn,m , completing the proof. � We now consider the problem (here u = u(x, y)) ∆u+ λku+ g(u) = f(x, y) , for (x, y) ∈ Da , u = 0 on ∂Da , (2.5) EJDE-2024/09 NECESSARY AND SUFFICIENT CONDITION 5 with the eigenvalue λk = α2 n,m a2 > λ1 for some n and m, corresponding to the eigenspace Jn( αn,m a r)(A cosnθ +B sinnθ). Let us denote ϕk = Jn( αn,m a r) cosnθ , ψk = Jn( αn,m a r) sinnθ , Ak(f) = Ak = ∫ Da f(x, y)ϕk dx dy , Bk(f) = Bk = ∫ Da f(x, y)ψk dx dy . (2.6) The numbers Ak and Bk can be easily approximated by Mathematica for any f(x, y) and k. Theorem 2.4. Assume that g(u) satisfies the condition (1.2). Then the condition√ A2 k +B2 k < Jn,m(g(∞)− g(−∞)) (2.7) is both necessary and sufficient for the existence of solution u(x, y) ∈ W 2,2(Da) ∩ W 1,2 0 (Da) of (2.5). (The projection of f(x, y) on the kernel is small enough.) Proof. (i) Necessity. Multiply (2.5) by ϕk and ψk respectively and integrate Ak = ∫ Da g(u)ϕk dx dy , Bk = ∫ Da g(u)ψk dx dy . (2.8) Multiply the first equation in (2.8) by Ak√ A2 k+B 2 k , the second equation by Bk√ A2 k+B 2 k , and add the results. Denoting wk = Jn (αn,m a r )( Ak√ A2 k +B2 k cosnθ + Bk√ A2 k +B2 k sinnθ ) , and using Proposition 2.3, we obtain√ A2 k +B2 k = ∫ B g(u)wk dx dy < g(∞) ∫ wk>0 wk dx dy + g(−∞) ∫ wk<0 wk dx dy = Jn,m(g(∞)− g(−∞)) . (ii) Sufficiency. Assuming that (2.7) holds, we shall verify the condition (1.3) of Theorem 1.1. Assuming that λk = α2 n,m a2 , let w(x, y) = Jn (αn,m a r ) (A cosnθ +B sinnθ) be any element of its eigenspace. By scaling w in (1.3), we may assume that A2 +B2 = 1 . 6 P. KORMAN EJDE-2024/09 In view of Proposition 2.3 and (2.6), the condition (1.3) of Theorem 1.1 that we need to verify takes the form∫ Da f(x, y)w(x, y) dx dy = AAk +BBk < Jn,m(g(∞)− g(−∞)) = g(∞) ∫ w>0 w dxdy + g(−∞) ∫ w<0 w dxdy . Since AAk + BBk ≤ √ A2 k +B2 k by Cauchy-Schwarz, the last inequality holds by (2.7). By Theorem 1.1 the problem (2.5) has a solution. � Example 2.5. Consider the unit disc x2 + y2 < 1, a = 1. Mathematica readily returns zeroes of the Bessel functions α0,1 ≈ 2.40483 , α0,2 ≈ 5.52008 , α0,3 ≈ 8.65373 , . . . α1,1 ≈ 3.83171 , α1,2 ≈ 7.01559 , . . . α2,1 ≈ 5.13562 , α2,2 ≈ 8.41724 , . . . α3,1 ≈ 6.38016 , α3,2 ≈ 9.76102 , . . . . The eigenvalues are λ1 = α2 0,1, λ2 = α2 1,1, λ3 = α2 2,1, λ4 = α2 0,2, λ5 = α2 3,1, λ6 = α2 1,2, and so on. Let us consider a case of resonance at the sixth eigenvalue ∆u+ λ6u+ u√ u2 + 1 = f(x, y) , for x2 + y2 < 1 , u = 0, on x2 + y2 = 1 . (2.9) By above, the eigenspace of λ6 is J1(α1,2r)(A cos θ+B sin θ), with arbitrary numbers A and B. The graph of J1(α 1,2 r) on (0, 1) has one root r0 = α1,1 α1,2 , and it is positive on P1,2 = (0, r0), and negative on N1,2 = (r0, 1), see Figure 1. By (2.2), using Mathematica J1,2 = 2 ∫ r0 0 J1(α1,2r)r dr − 2 ∫ 1 r0 Jn(α1,2r)r dr ≈ 0.260759 . For any f(x, y), Mathematica can also easily compute highly accurate approxima- tion of the integrals A6 = ∫ x2+y2<1 f(x, y)J1( α1,2 a r) cos θ dx dy , B6 = ∫ x2+y2<1 f(x, y)J1( α1,2 a r) sin θ dx dy . (Here x = r cos θ, y = r sin θ, and dx dy = r drdθ.) Since g(∞) = 1 and g(−∞) = −1, it follows by Theorem 2.4 that problem (2.9) has a solution if and only if√ A2 6 +B2 6 < 2J1,2 . We now consider an application to the semilinear heat equation on a disc Da : x2 + y2 < a2 (here u = u(x, y, t)) ut = ∆u+ λku+ g(u)− f(x, y) , for (x, y) ∈ Da , t > 0 u(x, y, t) = 0 , for (x, y) on ∂Da , t > 0 u(x, y, 0) = u0(x, y) , (2.10) EJDE-2024/09 NECESSARY AND SUFFICIENT CONDITION 7 Figure 1. Graph of J1(α 1,2 r) on the interval (0, 1) with given functions f(x, y) and u0(x, y), and g(u) satisfying (1.2). Here λk, k ≥ 2, is a double eigenvalue of Laplacian, as above. Steady states for this equation satisfy the equation (2.5). By Theorem 2.4, no steady states exist if√ A2 k +B2 k > Jn,m(g(∞)− g(−∞)) . (2.11) Denote wk = Ak√ A2 k+B 2 k ϕk + Bk√ A2 k+B 2 k ψk, as above. Recall that Ak = ∫ Da f(x, y)ϕk dx dy, Bk = ∫ Da f(x, y)ψk dx dy. Theorem 2.6. Assume that g(u) satisfies the condition (1.2), and that (2.11) holds. Then solution of (2.10) is unbounded for any initial data u0(x, y). In fact,defining H(t) = ∫ Da u(x, y, t)wk dx dy, one has H(t)→ −∞ as t→∞. Proof. Multiply (2.10) by Ak√ A2 k+B 2 k ϕk and integrate both sides over Da Ak√ A2 k +B2 k ∫ Da utϕk dx dy = Ak√ A2 k +B2 k ∫ Da g(u)ϕk dx dy − A2 k√ A2 k +B2 k . Multiply (2.10) by Ak√ A2 k+B 2 k ψk, and integrate over Ba, Ak√ A2 k +B2 k ∫ Da utψk dx dy = Ak√ A2 k +B2 k ∫ Da g(u)ψk dx dy − B2 k√ A2 k +B2 k . Add the results, to obtain H ′(t) = ∫ Da g(u)wk dx dy − √ A2 k +B2 k < −ε , for some ε > 0, by estimating the integral ∫ Da g(u)wk dx dy as in part (i) of Theorem 1.2, and using (2.11). Then H(t) < H(0)− εt, concluding the proof. � In the ODE context related results on unbounded solutions were given by Seifert [12], Alonso and Ortega [1], and Korman [7]. 8 P. KORMAN EJDE-2024/09 References [1] J. M. Alonso, R. Ortega; Unbounded solutions of semilinear equations at resonance, Nonlin- earity, 9 (1996) no. 5, 1099-1111. [2] B. Gidas, W.-M. Ni, L. Nirenberg; Symmetry and related properties via the maximum prin- ciple, Commun. Math. Phys., 68 (1979), 209-243. [3] S. P. Hastings, J. B. McLeod; Short proofs of results by Landesman, Lazer, and Leach on problems related to resonance, Differential Integral Equations, 24 (2011) no. 5-6, 435-441. [4] P. Korman; Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Hack- ensack, NJ 2012. [5] P. Korman; Nonlinear elliptic equations and systems with linear part at resonance, Electron. J. Differential Equations, 2016 (2016) No. 67. [6] P. Korman, D.S. Schmidt; Infinitely many solutions and asymptotics for resonant oscillatory problems, Special issue in honor of Alan C. Lazer, Electron. J. Diff. Equ., Special Issue 01 (2021), 301-313. [7] P. Korman; Unbounded solutions of periodic systems, arXiv 2303.12919. [8] E. M. Landesman, A. C. Lazer; Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623. [9] A. C. Lazer, D. E. Leach; Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969) (4), 49-68. [10] T. Ouyang, J. Shi; Exact multiplicity of positive solutions for a class of semilinear problems, II, J. Differential Equations 158 (1999), no. 1, 94-151.. [11] Y. Pinchover, J. Rubinstein; An Introduction to Partial Differential Equations. Cambridge University Press, Cambridge, 2005. [12] G. Seifert; Resonance in undamped second-order nonlinear equations with periodic forcing, Quart. Appl. Math. 48 (1990) no. 3, 527-530. [13] G. N. Watson; A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1995. [14] S. A. Williams; A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem, J. Differential Equations, 8 (1970), 580-586. Philip Korman Department of Mathematical Sciences, University of Cincinnati, Cincinnati OH 45221- 0025, USA Email address: kormanp@ucmail.edu 1. Introduction 2. Resonance for a two-dimensional disc References