Electronic Journal of Qualitative Theory of Differential Equations 009, No. 18, 1-9; htt://www.math.u-szeged.hu/ejqtde/ A generalized Fucik tye eigenvalue roblem for -Lalacian Yuanji Cheng School of Technology Malmö University, SE-05 06 Malmö, Sweden Email: yuanji.cheng@mah.se Abstract In this aer we study the generalized Fucik tye eigenvalue for the boundary value roblem of one dimensional Lalace tye differential equations (ϕ(u )) = ψ(u), T < x < T; ( ) u( T) = 0, u(t) = 0 where ϕ(s) = αs 1 + βs 1,ψ(s) = s 1 + s 1, > 1. We obtain a exlicit characterization of Fucik sectrum (α,β,,), i.e., for which the (*) has a nontrivial solution. (1991) AMS Subject Classification: 35J65, 34B15, 49K0. 1 Introduction In the study of nonhomogeneous semilinear boundary roblem u = f(u) + g(x), in Ω u = 0, on Ω it has been discovered in [3, 7] that the solvability of another boundary value roblem u = u+ u, in Ω u = 0, on Ω where u + = maxu, 0}, u = max u, 0} lays an imortant role. Since then there are many works devoted to this subject [5, 13, 14, references therein], and the study has also been extended to the -Lalacian u = u 1 + u 1, in Ω u = 0, on Ω EJQTDE, 009 No. 18,. 1
where =: div u u}, > 1 [, 4, 6, 10, 1] and even associated trigonometrical sine and cosine functions have been studied [9]. In this aer, we are interested in generalization of such Fucik sectrum and will consider one dimensional boundary value roblem (α(u ) 1 + β(u ) 1 )) = u 1 + u 1, T < x < T (1.1) u( T) = 0, u(t) = 0 where α, β,, > 0 are arameters, and call (α, β,, ) the generalized Fucik sectrum, if (1.1) has a non-trivial solution. The roblem is motivated by the study of two-oint boundary value roblem (ϕ(u ))) = ψ(x, u), T < x < T (1.) u( T) = 0, u(t) = 0 and to our knowledge it is always assumed in the literature that ϕ is an odd function. Thus a natural question arises: what would haen, if the function ϕ is merely a homeomorhism, not necessarily odd function on R? Here we shall first investigate the autonomous eigenvalue tye roblem and in the forthcoming treat non-resonance roblem. By a solution of (1.) we mean that u(x) is of C 1 such that ϕ(u (x)) is differentiable and the equation (1.) is satisfied ointwise almost everywhere. The main results of this aer are comlete characterization of Fucik tye eigenvalues, their associated eigenfunctions and observations of changes of frequency, amlitude of solutions, when they ass the mini- and maximum oints resective change their signs (see the figures below and (3.8) in details). Let π = π sin(π/), then we have Theorem 1 (α, β,, ) belongs to the generalized Fucik sectrum of (1.1), if and only if for some integer k 0 1) ( α + β)((k + 1) 1 + k 1 )π = T and corresonding eigenfunction u is initially ositive and has recisely k nodes. ) ( α + β)(k 1 + (k + 1) 1 )π = T and the corresonding eigenfunction u is initially negative and has also k nodes 3) (k+1)( α+ β)( 1 + 1 )π = T and the corresonding eigenfunction u 1, u has exact k + 1 nodes and u 1 is initially ositive and u is negative. Moreover, the eigenfunctions are iecewise sine functions (see art for definitions). 1.5 1 Fucik ositive eigenfunction, α = 1, β = 4, = 9 Fucik eigenfunction Classical eigenfunction 1.5 1 0.5 0 Nodal eigenfunction α = 1, β = 4, = 81, = 81/4 Switch of frequency /amlitude 0.5 0.5 1 Classical eigenfunction 1.5 Fucik eigenfunction 0 1.5 1 0.5 0 0.5 1 1.5.5 1.5 1 0.5 0 0.5 1 1.5 EJQTDE, 009 No. 18,.
Review on 1D Lalacian We shall review some basic results about eigenvalues and associated eigenfunctions for one dimensional Lalacian. Eigenfunctions are also called sine and -cosine functions, sin (x), cos (x), which have been discussed in details in [9, 11], but for our urose we adot the version in [1]. Let π = sin (x), cos (x) are defined via x = sin(x) 0 π sin π, the sine, cosine functions dt 1 t, 0 x π / (.1) and extended to [π /, π ] by sin (π /+x) = sin (π / x) and to [ π, 0] by sin (x) = sin ( x) then finally extended to a π eriodic function on the whole real line; Then cosine function is defined as cos x = d dx (sin x) and they have the roerties: sin 0 = 0, sin π / = 1; cos 0 = 1, cos π / = 0. They share several remarkable relations as ordinary trigonometric functions, for instance sin x + cos x = 1. But d dx (cos x) = tan x sin x sin x, where tan x = sin x/ cos x. The eigenvalues of one dimensional -Lalace oerator ( u (x) u (x)) = u(x) u(x), 0 x π u(0) = 0, u(π ) = 0 are 1,, 3, and the corresonding eigenfunctions are recisely sin (x), sin (x), sin (3x), and therefore we have another relation between cosine and sine functions ( cos (kx) cos (kx)) = k sin (kx) sin (kx), k = 1,, (.) EJQTDE, 009 No. 18,. 3
Here we note that function sin (x) is a solution to the following roblem ( u (x) u (x)) = u(x) u(x), 0 < x < π / u(0) = 0, u (π /) = 0 (.3) and any solution is of form C sin (x) for some constant C. If the Lalacian is associated to another interval [a, b], different from [0, π ], then by change of variables we see that eigenvalues and the associated eigenfunctions are ( kπ b a ), sin (k x a b a π ), k = 1,, 3,. (.4) 3 Proof of Theorem 1 To understand the generalized Fucik sectrum of (1.1), we need to examine the following Dirichlet-Neumann boundary value roblem (α(u ) 1 + β(u ) 1 )) = u 1 + u 1, a < x < b u(a) = 0, u (b) = 0 (3.1) We shall focus only on the constant sign solutions of (3.1) and note that there exist essentially only two solutions, one ositive and another negative, due to the ositive homogeneity of (3.1). If u(x) is a ositive solution of (3.1), then u must be increasing on [a, b), because the equation (3.1) says that the function g(x) =: ϕ(u (x)) is decreasing on (a, b] and satisfies g(b) = 0, thus g(x) is ositive for all x [a, b) and thus u (x) has to be ositive, due to strict monotonicity of function ϕ(s) and ϕ(0) = 0. It follows that u satisfies ( u (x) u (x)) = α u(x) u(x), a < x < b u(a) = 0, u (3.) (b) = 0 It follows from (.3) that C sin ( x a ) and α, satisfy π α α/ = b a. Likely if u is negative solution to (3.1), then ( u (x) u (x)) = β u(x) u(x), a < x < b u(a) = 0, u (b) = 0 C sin ( x a ) and β, satisfy π β β/ = b a. In analogy we see that for the following boundary value roblem (3.3) (α(u ) 1 + β(u ) 1 )) = u 1 + u 1, a < x < b u (a) = 0, u(b) = 0 (3.4) the ositive and negative solutions are D sin ( b x D sin ( b x b a b a π ) resectively π ) and α, β,, satisfy π β/ = b a or π α/ = b a. EJQTDE, 009 No. 18,. 4
It follows from the above analysis that if u is a ositive solution to (1.1) and u(x 0 ) = maxu(x) := C, then x 0 is determined by α/π = x 0 + T and α, β,, satisfy Furthermore the solution u(x) is given by C sin ( x+t α ), C sin ( T x ), β L + := ( α/ + β/)π = T. and for the negative solution u of (1.1), then it holds L := ( α/ + β/)π = T D sin ( x+t D sin ( T x T x T + π α T π β x T (3.5) ), T x T + π β β ), T π α α x T (3.5 ) It is clear from (3.5) that u(x) changes its frequency, when it asses its maximum oint and is not symmetric anymore, which is in contrast to the symmetry rincile of Gidas, Ni and Nirenberg [8]. For an initially ositive nodal solution u to (1.1) with only one node at T 1, we get that T 1, α, β,, ν satisfy ( α/ + β/)π = T 1 + T. ( α + β)( 1 + 1 )π = T (3.6) If C = max x [ T,T1 ] u(x), D = max x [T,T] u(x), then in view of identity α (u (x)) + + β (u (x)) + + (u(x)) + + (u (x)) = constant, x [ T, T] (3.7) we deduce C = D, C = t, D = t, for some t > 0. Using the ositive homogeneity of (1.1) we derive that the solution u(x) is given in by t sin ( x+t ), T x T + π α α t sin ( T 1 x ), T β β 1 π x T 1 t sin ( x T 1 ), T (3.8) β β 1 x T 1 + π t sin ( T x ), T π α α x T EJQTDE, 009 No. 18,. 5
It follows from (3.8) that the differences between α and β are reflected by change of amlitudes between ositive and negative waves at the switch between lus and minus. For the switch from negative wave to ositive wave, it holds sin ( x+t ), β sin ( T 1 x ), α sin ( x T 1 ), α sin ( T 1 x ), β T x T + π β T 1 π α x T 1 T 1 x T 1 + π α T π β x T (3.8 ) where T 1 = T +( α/+ β/)π. In view of (3.8) and (3.8 ) we see that u ( T) = u (T) and therefore can extend u(x) to a T-eriodic function on the whole real line R. For any given integer k 1. If u is a solution of (1.1) with (k + 1) nodes, then it must have equal number of ositive and negative (k+1) waves. Let T 1 < T < < T k+1 be the nodal oints of u, it follows from (3.7) that all ositive waves of u have same height and so are the same for negative waves. Moreover similarly as deriving (3.6) we get ( α + β)( 1 + 1 )π = T i+ T i, i = 1,, k 1. Thereby α, β,, satisfy (k + 1)( α + β)( 1 + 1 )π = T (3.6 ) and the nodes are T 1+i = T+(1+i)L + +il, T i = T+i(L + +L ), i = 0, 1,,, k. Furthermore let T 0 = T, T k+ = T then the initially ositive solution u(x) on [T i, T i+ ], i = 0, 1,, k, is given by t sin ( x T i t sin ( T i+1 x t sin ( x T 1 t sin ( T i+ x ), α ), β ), β ), α T i x T i + π α T β i+1 π x T i+1 T i+1 x T i+1 + π T i+ π α β x T i+ (3.9) and the initially negative solution u by t sin ( x T i ), β t sin ( T i+1 x ), α t sin ( x T i+1 ), α t sin ( T i+ x ), β T i x T i + π T i+1 π α β x T i+1 T i+1 x T i+1 + π α T β i+ π x T i+ (3.9 ) where T 1+i = T + (1 + i)l + il +, T i = T + i(l + + L ), i = 0, 1,,, k. EJQTDE, 009 No. 18,. 6
If u has k nodes, then there are two ossibilities 1) (k + 1) ositive waves and k negative waves, ) k ositive waves and (k + 1) negative waves. In 1) the solution should be initially ositive and be initially negative in ). It follows then 1) ( α + β)((k + 1) 1 + k 1 )π = T (3.10) and the solution u on [ T, T L ] is (T L )/k-eriodic and is given by (3.9) on [T i, T i+ ], i = 0, 1,, k, and on [T k, T], u(x) is given by t sin ( x T k ), T α k x T k + π α t sin ( T x ), T π β β x T (3.11) ) For an initially negative solution with k nodes, then ( α + β)(k 1 + (k + 1) 1 )π = T (3.1) and on [T i, T i+ ] the solution u is given by (3.9 ) for i = 0, 1,, k, and on [T k, T], u(x) is given by t sin ( x T k β ), T β k x T k + π t sin ( T x ), T π α α x T. (3.11 ) So the roof is comlete. 4 A final remark In the study of nontrivial solutions to one dimensional nonlinear differential equation (ϕ(x, u )) = f(x, u) (4.1) one usually adot the notation of solution by (4.1) being satisfied ointwise, which in turn ensures er definition only C 1 smoothness of solution. Of course, one exects higher order smoothness of the solutions. Here we shall examine this question for a very secial case, namely ϕ(x, s) = α(x)s 1 + β(x)s q 1,, q > 1 e.g., (α(x)(u ) 1 + β(x)(u ) q 1 ) = f(x, u) (4.) If we assume that the equation (4.) is satisfied ointwise and moreover α, β > 0 are also C 1, then any solution u is obviously C for any oint x where u (x) 0. So in order to get differentiability of u we need a closer examination at those oints where u (x) = 0. Let x 0 be a critical oint of u(x), u 0 = u(x 0 ), if = q =, then we easily deduce from the equation (4.) that for small δ > 0 u (x 0 δ) = f(x 0, u 0 )/α(x 0 )(δ + o(δ)); u (x 0 + δ) = f(x 0, u 0 )/β(x 0 )(δ + o(δ)) EJQTDE, 009 No. 18,. 7
thereafter u (x) has a jum at x 0 since α β and thus u C 1,1, but not C. In general, for any critical oint x = x 0 of u(x), we have the following asymtotic as δ 0 u (x 0 δ) = C 1 δ 1/( 1) (1 + o(1)) u (x 0 + δ) = C δ 1/(q 1) (1 + o(1)) where C 1 = 1 f(x 0, u 0 )/α(x 0 ), C = q 1 f(x 0, u 0 )/β(x 0 ). In view of the above estimates, we deduce that 1. If 1 <, q < then u C. If max, q} = then u C 1,1 3. If <, q then u C 1,ε, ε = min 1 1 1 q 1 Acknowledgement This work is artially suorted by the roject Nr 0-06/10 for romotion of research at Malmö University and also the G S Magnusons fond, the Swedish Royal Academy of Science. The author thanks the referee for careful reading and suggestions which lead to imrovement of the resentation. References [1] Bennewitz C. and Saito Y. An embedding norm and the Lindqvist trigonometric functions. Electronic J. Diff. Equa. (00) No. 86, 1-6. [] Cuesta M., de Figueriredo D., Gossez, J.-P., The beginning of the Fucik sektrum for the Lalacian. J. Diff. Equa., 159: (1999) 1-38. [3] Dancer N., On the Dirichlet roblem for weakly nonlinear ellitic differential equations. Proc. Royal Soci. Edinburgh, 76 (1977) 83-300. [4] Dancer, E. N., Remarks on juming nonlinearities. Toics in nonlinear analysis, Birkhäuser, Basel, 1999,. 101-116 [5] de Figueiredo D., Gossez J.-P., On the first curve of the Fucik sectrum of an ellitic oerator. Diff. and Integral Equa. 7:5 (1994), 185 130. [6] Drábek P., Solvability and bifurcations of nonlinear equations. Pitman Research Notes in Math. vol 64 Longman, New York, 199. [7] Fucik S., Boundary value roblems with juming nonlinearities, Casois Pest. Mat. 101 (1976) 69 87. [8] Gidas B., Ni N. and Nirenberg L., Symmetry and related roerties via the maximum rincile. Comm. Math. Phys. 68 (1979) 09-43. [9] Lindqvist, P. Some remarkable sine and cosine functions, Ricerche di Matematica, XLIV (1995) 69-90. EJQTDE, 009 No. 18,. 8
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