THE JUMPING NONLINEARITY PROBLEM REVISITED: AN ABSTRACT APPROACH. David G. Costa Hossein Tehrani. 1. Introduction

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1 Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 1, 003, 49 7 THE JUMPING NONLINEARITY PROBLEM REVISITED: AN ABSTRACT APPROACH David G. Costa Hossein Tehrani Abstract. We consider a class of nonlinear problems of the form Lu + g(x, u) = f, where L is an unbounded self-adjoint operator on a Hilbert space H of L (Ω)-functions, Ω R N an arbitrary domain, and g: Ω R R is a jumping nonlinearity in the sense that the limits g(x, s) g(x, s) lim = a and lim = b s s s s exist and jump over an eigenvalue of the operator L. Under rather general conditions on the operator L and for suitable a < b, we show that a solution to our problem exists for any f H. Applications are given to the beam equation, the wave equation, and elliptic equations in the whole space R N. 1. Introduction The so-called jumping nonlinearity problem has a long and rich history starting with the pioneering paper by Ambrosetti Prodi [] in In the early years following the appearance of [] a number of authors contributed to the study of such problems, notably Berger and Podolak ([7]), Kazdan and Warner ([14]), 000 Mathematics Subject Classification. 35J0, 35J5. Key words and phrases. Jumping nonlinearity, critical point theory, beam equation, wave equation, Schrödinger equation. c 003 Juliusz Schauder Center for Nonlinear Studies 49

2 50 D. G. Costa H. Tehrani Amann and Hess ([1]), Dancer ([9]) and Fučik ([11]) who coined the term jumping nonlinearity in the case of ordinary differential equations. During the 1980 s, we could cite the contributions of de Figueiredo ([10]), Gallouet and Kavian ([13]), Lazer and McKenna ([15]), Ruf ([19]), and Solimini ([]), among others. The interested reader will find a more complete bibliography up to 1990 in [17]. In its simplest form, the jumping nonlinearity problem consists in studying the question of existence of solution for the Dirichlet problem (1.1) u + g(u) = f(x), u(0) = u(π) = 0, where f(x) is a given function in L (0, π) and the nonlinearity g(s) is a C 1 function crossing some eigenvalue λ k = k (k = 1,,... ) of the problem u +λu = 0, u(0) = u(π) = 0, in the sense that (1.) 0 < a = lim s g (s) < λ k < lim s g (s) = b. Of particular importance here is the problem at infinity, (F ) a,b, given by u + bu + au = 0, u(0) = u(π) = 0, and its so-called Fučik spectrum = {(a, b) R (F ) a,b has a nonzero solution}. In fact, since the Fučik spectrum is completely known in the above ODE case, it can be shown by topological arguments ([1]) that, if (a, b) / then, depending on the particular connected component of R \ where the point (a, b) given in (1.) is located, problem (1.) (which could be considered a non-resonant problem) (I) either has at least one solution for any given f, (II) or else may have a solution for certain f s and no solution for others. Our main goal in the present work is to obtain an existence result that holds for any given right hand side f and that can be applied to general classes of operators. In order to motivate our main theorem, we recall a few basic results in the case of the Laplace operator on a bounded domain. In fact, since the appearance of [] many authors have considered existence results for the problem (1.3) u + g(u) = f in Ω, u = 0 on Ω, where Ω R N is a bounded smooth domain and g satisfies (1.), with 0 < λ 1 <... < λ k <... denoting the spectrum of the Laplace operator with Dirichlet boundary condition. It is worth noting that, in this case, the principal eigenvalue λ 1 plays a special role. Namely, alternative (II) above occurs whenever k = 1 in (1.). Therefore, a general existence result for any right hand side can only be expected in the case a > λ 1. Although there are a number of existence and multiplicity theorems when a > λ 1, we would like to mention the following result of Lazer and McKenna (Theorems.4 and.5 in [15]):

3 The Jumping Nonlinearity Problem Revisited 51 Theorem 1 ([15]). Assume (1.) with b (λ k, λ k+1 ), for some k, and that g (s) b 1 < λ k+1 for all s R (where b 1 > b). There exists λ k 1 < α = α(b) < λ k such that, if a (α, λ k ), then problem u + g(u) = tφ 1 + h in Ω, u = 0 on Ω, has at least three solutions for t large negative and a unique solution for t large positive (here, φ 1 > 0 denotes a normalized λ 1 -eigenfunction). In this paper we will prove an existence result in the spirit of Theorem 1 under reasonably weak conditions so that it is applicable to a large class of problems on bounded and unbounded domains (clearly, the uniqueness assertion in Theorem 1 precludes multiplicity for an arbitrary right hand side f). More precisely, we consider a class of nonlinear problems of the form (P) Lu + g(x, u) = f, where L is an unbounded self-adjoint operator on a Hilbert space H of L (Ω)- functions, Ω R N an arbitrary domain, and g: Ω R R is a jumping nonlinearity in the sense that the limits lim s g(x, s)/s = a, lim s g(x, s)/s = b exist and cross a (possibly multiple) eigenvalue of the operator L according to the following assumptions (where σ(l) denotes the spectrum of L, σ e (L) the essential spectrum of L and s + = max{s, 0}, s = s + s): (L 1 ) λ > 0 is an isolated point of σ( L). (L ) 1 dim ker(l + λ) < and every 0 u ker(l + λ) changes sign. (L 3 ) Lu + bu + λu = 0 has no nonzero solution, where λ < b. (L 4 ) (, b) σ e (L) = (so L has a discrete point spectrum in (, b)). (G 1 ) g: Ω R R is a Carathéodory function such that g(x, s) 0 < lim = a < b = lim s s s g(x, s). s (G ) Set g 0 (x, s) := g(x, s) (bs + as ). Then, for every ε > 0 there exists 0 b ε (x) L (Ω) such that g 0 (x, s) b ε (x) + ε s a.e. x Ω, s R. (G 3 ) g 0 (x, ) is nondecreasing for a.a. x Ω. Furthermore, we write λ (resp. λ + ) to denote the closest point in σ( L) to the left (resp. right) of λ. Our main result in this paper is the following theorem: Theorem. Assume (L 1 ) (L 4 ) for some λ < b < λ +. There exists λ < α = α(b, L) < λ such that, if a (α, λ] and (G 1 ) (G 3 ) are satisfied, then problem (P) possesses a solution for any f H. A few comments about the hypotheses are in order.

4 5 D. G. Costa H. Tehrani Remark. (a) Hypotheses (L 1 ) and (L ) say that the closed interval [a, b] contains a single eigenvalue of L (of any finite multiplicity) which cannot be a principal eigenvalue (as every 0 u ker(l + λ) changes sign). (b) Hypothesis (L 3 ) says that the point (b, λ) does not belong to the Fučik Spectrum of L (see Remark 3.8). (c) Hypothesis (L 4 ) allows applications to problems with selfadjoint operators which may have continuous spectrum or eigenvalues of infinity multiplicity. (d) Hypothesis (G 3 ) is needed for the proof of our result in the present generality. Nevertheless, in some situations (G 3 ) can be removed. Theorem will be applied to the question of finding time-periodic solutions of the beam equation and the vibrating string equation, without any symmetry assumption. We will also consider consider existence of H -solutions of the Schrödinger equation in the whole space R N. In particular, in this case (G 3 ) is not needed and we can prove the following result: Theorem 3. Assume that V : R N R satisfies the conditions: (V 1 ) V C(R N ) and V (x) 0 as x. (V ) There exists ϕ C 0 (R N ) such that ( ϕ + V (x)ϕ ) < 0. (V 3 ) λ > 0 is an isolated point of σ( V (x)) with σ( V (x)) [a, b] = { λ}. Let λ 0 < λ 1 <... < 0 denote the distinct eigenvalues of L = + V (x) in H (R N ). Then, given b with λ = λ k < b < λ k 1, for some k 1, there exists α with λ k+1 < α < λ k (or 0 < α < λ k if λ k is the largest negative eigenvalue of L) such that, for a (α, λ k ), and g: R N R R satisfying conditions (G 1 ), (G ) above, equation u + V (x)u + g(x, u) = f(x), u H (R N ) has a solution for any given f L (R N ). The organization of this paper is as follows. In Section we provide the abstract framework for problem (P) while Theorem is proved in Section 3. Applications and proofs of the corresponding theorems are given in Section 4. We thank the referee for informing us of the recent papers [5], [6] and asking for a comparison of the results. In [6] the authors give a description of the Fučik spectrum near ( λ, λ), where λ is an isolated point of the spectrum σ(l) of a general selfadjoint operator L defined on a Hilbert space H of L (Ω), with Ω a bounded subset of R N (their work complements or extends other results in this direction, see e.g. [3], [18], [0] and references therein). Such a description enables them to apply topological arguments (in particular, degree computations) and prove some existence results for solutions of (P). However, we remark that both their approach and existence results are of a different nature from ours.

5 The Jumping Nonlinearity Problem Revisited 53. The abstract framework Let H be a Hilbert space of L (Ω)-functions, with innerproduct and norm denoted by (, ) and, respectively. Let L: D(L) H H be an unbounded self-adjoint operator. In view of the spectral theorem, we recall that, for any continuous function ϕ: R R, we can define the self-adjoint operator ϕ(l) = ϕ(λ) de(λ) = ϕ(λ) de(λ), where {E(λ) λ R} is a spectral family for L, σ(l) is the spectrum of L and { } (.1) D(ϕ(L)) = u H ϕ(λ) d(e(λ)u, u) <, (.) (ϕ(l)u, v) = In particular, we have that where (.3) (.4) D(L) = (Lu, v) = σ(l) ϕ(λ) d(e(λ)u, v), for all u D(ϕ(L)), v H. { u H We note that (.4) implies that is, (.5) (.6) Lu = L = λde(λ), } λ d(e(λ)u, u) <, λ d(e(λ)u, v) for all u D(L), v H. λ d(e(λ)u, u) for all u D(L). Next, we pick ϕ(λ) = λ 1/ and define the unbounded self-adjoint operator (Au, v) = D(A) = A = L 1/ = { u H λ 1/ de(λ), } λ d(e(λ)u, u) <, λ 1/ d(e(λ)u, v) for all u D(A), v H. As is well-known, the closedness of the operator A implies that E := D(A) is a Hilbert space with the graph-innerproduct (u, v) E := (u, v) + (Au, Av)

6 54 D. G. Costa H. Tehrani and corresponding graph-norm ( ) 1/ u E = ( u + Au ) 1/ = (1 + λ ) d(e(λ)u, u). Lemma.1 (Poincaré-type inequality). Assume that 0 / σ(l) and define ( ) 1/ u := λ d(e(λ)u, u) = Au. Then, there exists δ 0 > 0 such that u δ 0 u for all u E. In particular, the norm is equivalent to the graph-norm E on E. Proof. It follows at once by letting δ 0 = dist(0, σ(l)) and noticing that λ d(e(λ)u, u) = λ d(e(λ)u, u) δ 0 u. σ(l) From now on we assume that 0 / σ(l). It follows that D(L) E. We shall consider the Hilbert space E equipped with the norm defined above, which comes from the innerproduct (.7) u, v = λ d(e(λ)u, v) = (Au, Av). Let us write R = R + R = {λ λ 0} {λ λ < 0} and denote (.8) P + = E(R + ) = χ R+ (L), P = E(R ) = χ R (L), where χ S denotes the characteristic function of a set S R. Then, P + and P are orthogonal projections such that P + P = I. Correspondingly, we have the orthogonal decomposition (.9) E = E + E, where E + = P + (E), E = P (E). Also note that the spectral theorem implies that A commutes with both P + and P : in other words, if u D(A) then P ± u D(A) and P ± Au = AP ± u. Moreover, from (.1), (.), (.6) and (.8), it follows that D(L) = D(A ) and, for any u D(L), v E, (.10) ((P + P )Au, Av) = ((P + P )A u, v) = (Lu, v). Next, let Ω R N be an arbitrary domain and let g: Ω R R be a Carathéodory function (i.e. g(, s) is measurable for all s R and g(x, ) is continuous for almost all x Ω) satisfying (.11) g(x, s) A(x) + B s, a.a. x Ω, s R,

7 The Jumping Nonlinearity Problem Revisited 55 where 0 A(x) L (Ω) and B 0. From this, it follows that the Nemytskiĭ operator u(x) g(x, u(x)) is well-defined and continuous from L (Ω) to L (Ω) (see [3]). Given f H and an unbounded self-adjoint operator L: D(L) H H (as in the beginning of this section), we consider the equation (P) Lu + g(x, u) = f, u D(L). A solution of (P) is a function u D(L) that satisfies the equation in H. We now verify that equation (P) has a natural variational structure in the sense that its solutions correspond exactly to the critical points of a related functional I: E R. Indeed, consider the quadratic form Q: E R defined by (.1) Q(u) = 1 ( P +u P u ), u E. (Note that (.10) implies Q(u) = (1/)(Lu, u) whenever u D(L) E). Next, let G(x, s) = s g(x, t) dt, and consider the functional 0 (.1) I(u) = Q(u) + G(x, u) dx fu dx. Then, the following result holds true. Ω Lemma. (Variational structure and regularity). The functional I: E R is well-defined and of class C 1 on E. In addition, its critical points u E are precisely the solutions u D(L) of equation (P). Proof. Under assumption (.11) it follows that G(x, s) satisfies the estimate G(x, s) A(x) s + 1 B s, for a.a. x Ω, for all s R, so that the Nemytskiĭ operator u(x) G(x, u(x)) is well-defined and continuous from L (Ω) to L 1 (Ω), and it is easy to see that the functional I: E R is also well-defined. In fact, it is not hard to show in this case that I is of class C 1 with Ω I (u) h = P + u, h P u, h + (g(x, u), h) (f, h) for all u, h E or I (u) h = ((P + P )Au, Ah) + (g(x, u) f, h) for all u, h E. Now, let û E be a critical point of I, that is, I (û) h = 0 for all h E. From the above expression we obtain (Ah, (P + P )Aû) = (h, f g(x, û)) for all h E = D(A).

8 56 D. G. Costa H. Tehrani This shows that (P + P )Aû D(A ) and A (P + P )Aû = f g(x, û). Using the fact that A = A, we conclude that (P + P )û D(A ) = D(L), that is, û D(L), and so that, in view of (.10), (P + P )A û = f g(x, û) Lû = f g(x, û). We have shown that a critical point û E of the functional I belongs in fact to D(L) and is a solution of equation (P). On the other hand, it is straightforward to see that a solution û D(L) of equation (P) is a critical point of I. 3. Proof of main result In this section we prove our main abstract result on existence of solution for (P) Lu + g(x, u) = f, u D(L). Keeping in mind the abstract framework developed in the previous section, we recall the following assumptions that we are making on the unbounded selfadjoint operator L: D(L) H H and on the function g: Ω R R: (L 1 ) λ > 0 is an isolated point of σ( L), (L ) 1 dim ker(l + λ) < and every 0 u ker(l + λ) changes sign, (L 3 ) Lu + bu + λu = 0 has no nonzero solution, where λ < b, (L 4 ) (, b) σ e (L) = (so L has a discrete point spectrum in (, b)). (G 1 ) g: Ω R R is a Carathéodory function such that g(x, s) 0 < lim = a < b = lim s s s g(x, s), s (G ) Set g 0 (x, s) := g(x, s) (bs + as ). Then, for every ε > 0 there exists 0 b ε (x) L (Ω) such that g 0 (x, s) b ε (x) + ε s a.e. x Ω, s R, (G 3 ) g 0 (x, ) is nondecreasing for a.a. x Ω. Furthermore, we write λ (resp. λ + ) to denote the closest point in σ( L) to the left (resp. right) of λ. Theorem. Assume (L 1 ) (L 4 ) for some λ < b < λ +. There exists λ < α = α(b, L) < λ such that, if a (α, λ] and (G 1 ) (G 3 ) are satisfied, then problem (P) possesses a solution for any f H. The proof of Theorem will follow from a saddle-point type theorem due to Silva [1] (cf. also Brezis and Nirenberg [8]). We start by recalling a Palais Smale type condition.

9 The Jumping Nonlinearity Problem Revisited 57 Let a C 1 functional I: E R be given on the Hilbert space E. Let E = V W be an orthogonal decomposition of the space E, P V : E V, P W : E W the corresponding orthogonal projections, and assume that W 1 W... W is a sequence of finite-dimensional subspaces of W such that n=1 W n is dense in W. A sequence (u n ) E is called a (PS) c sequence w.r.t. {W n } at the level c R if and only if there exists k n such that P W (u n ) W kn, I(u n ) c, and I (u n ) V Wkn 0 as n. And we say that the functional I satisfies condition (PS) c w.r.t. {W n } if and only if any (PS) c sequence w.r.t. {W n } possesses a subsequence that converges to a critical point of I. Theorem 3.1 ([1], [8]). Let E = V W be an orthogonal decomposition of the Hilbert space E. Assume that I: E R is a C 1 functional such that (i) sup u W I(u) := I <, (ii) inf u V I(u) := I >, (iii) I satisfies condition (PS) c w.r.t. {W n } for all I c I. Then I has a critical point û E. Remark 3.. We remark that, as stated, the above result is a slight variation of Theorem 7 in [8]. Moreover, similarly to [8], condition (ii) can be replaced by the weaker requirement that I is bounded below on finite-dimensional subspaces of V, as long as the Palais Smale condition (iii) is strengthened. We start by pointing out that, without loss of generality, we may assume that 0 / σ(l) and λ is the least positive eigenvalue of L. Indeed, by (L 1 ), we have that ( λ, λ + δ] σ(l) = for some δ > 0 and, by letting Lu := Lu + ( λ δ)u, g(x, s) := (b λ + δ)s + (a λ + δ)s + g 0 (x, s), we obtain the equivalent equation Lu + g(x, u) = f, where 0 / σ( L), λ := δ is the least positive eigenvalue of L, and assumptions corresponding to (G 1 ) (G 3 ), (L 1 ) (L 4 ) hold for g and L. Therefore, we will prove all the results that follow from now on under the assumption that 0 / σ(l) and λ is the least positive eigenvalue of L. Then, for λ < b < λ + given, λ is the only point of σ(l) in the interval [ b, 0]. We also let (3.1) (3.) P 1 = χ (, b) (L), E 1 = P 1 (E), P = χ [ b,0] (L), E = P (E),

10 58 D. G. Costa H. Tehrani so that E = P (E) = E 1 E and E = ker(l + λ). In what follows we will consider the orthogonal decomposition where V := E + E and W := E 1. E = V W, Finally, we consider 0 < a λ and recall the definition of the functional I: E R whose critical points are the solutions u D(L) of equation (P): I(u) = 1 ( P+ u P u ) + G(x, u) fu. From now on, unless otherwise stated, all integrals are taken over Ω. In the next Lemmas we verify the geometric conditions (i), (ii) of Theorem 3.1 Lemma 3.3. sup u W I(u) := I <. Proof. In view of (G ) we have G(x, s) = (b/) s + +(a/) s +G 0 (x, s), where G 0 (x, s) b ε (x) s + ε s. Therefore, using the fact that 0 < a < b and u W = E 1 E, we obtain (3.3) I(u) = 1 u + b u + + a u + G 0 (x, u) fu 1 u + b u + ε u + C 1 (ε) f + C (ε). On the other hand, applying Lemma.1 with L E1 and noticing that dist(0, σ(l E1 )) = b + δ, for some δ > 0, we have the Poincaré-type inequality (3.4) u = P 1 u (b + δ) u. Taking ε = δ/4 in (3.3) and using (3.4), it follows that δ I(u) 4(b + δ) u + C for some 0 < C <. Next, we will estimate I(u) from below on the subspace V = E + E and show that inf u V I(u) := I >. We need the following lemmas:

11 The Jumping Nonlinearity Problem Revisited 59 Lemma 3.4. Let B > 0, assume condition (L ), 1 dim ker(l + λ) <, and that every nonzero element ψ ker(l+ λ) changes sign. If (ψ n ) ker(l+ λ), (w n ) E + and (δ n ) R + are sequences such that (i) ψ n as n, (ii) w n 1 for all n 1, (iii) 0 δ n δ 0 < µb/(µ + 1), where ψ + µ = inf ψ 0 ψ > 0, ψ ker(l + λ), then [ lim δ n ψ n + B n (w n + ψ n ) + ] =. Proof. Let s n = ψ n and ϕ n = ψ n / ψ n. Then [ ] δ n ψ n + B (w n + ψ n ) + lim n [ = lim n s n δ n ϕ n + B ( wn s n + ϕ n ) + Since dim ker(l + λ) <, we have (by passing to a subsequence, if necessary) that ϕ n ϕ in E for some 0 ϕ ker(l + λ): in particular ϕ n ϕ in L. Also, we have from (ii) that w n /s n 0 in E (hence in L ), and we may assume that δ n δ, in view of (iii). Therefore, lim n s n [ ( ) + δ n ϕ n wn ] + B + ϕ n s n = lim n s n[ δ ϕ + B ϕ + + o(1)] = lim n s n[(b δ) ϕ + δ ϕ + o(1)] lim n s n[((b δ)µ δ) ϕ + o(1)] =, ]. since (B δ)µ δ = µb (µ + 1) δ > 0, in view of (iii). Lemma 3.5. Consider the function f(ρ, ϕ, w) = 1/ ρ ϕ +B (w+ϕ) + defined for ρ R +, ϕ ker(l + λ), w S + := {u E + u = 1}. Then, there exists ρ 0 = ρ 0 (B, L) > 0 such that m ρ := inf ϕ,w f(ρ, ϕ, w) > 0, 0 < ρ ρ 0. Proof. Indeed, suppose there exists ρ n > 0 such that ρ n 0 and inf f(ρ n, ϕ, w) 0. ϕ,w

12 60 D. G. Costa H. Tehrani Then, there exist sequences ϕ n ker(l + λ) and w n such that w n = 1 and f(ρ n, ϕ n, w n ) 1 n, that is, 1 ρ n ϕ n + B (w n + ϕ n ) + 1 n. In view of Lemma 3.4, there must exist M > 0 such that ϕ n M for all n 1. But then, passing to the limit in the above expression leads to a contradition. Therefore, it follows that inf f(ρ, ϕ, w) > 0, 0 < ρ ρ 0. ϕ,w for some ρ 0 > 0. Lemma 3.6. There exists 0 < α 1 = α 1 (b, L) < λ such that if a (α 1, λ] then inf u V I(u) := I >. hence Proof. Let u = w + ψ E + ker(l + λ) = V. We have I(u) = 1 w 1 ψ + b u + + a u + G 0 (x, u) fu = 1 w 1 ψ + a u + b a u + + G 0 (x, u) fu 1 w λ ψ + a u + b a (w + ψ) + ε u C(ε), I(u) 1 w + a ε w + a λ ε ψ + b a (w + ψ) + C(ε), which gives (3.5) I(u) 1 w λ a + ε ψ + b λ (w + ψ) + C(ε). We note that the above estimate is of the form I(u) 1 w A ψ + B (w + ψ) + C(ε), u = w + ψ V. where A = ( λ a + ε)/ > 0, B = (b λ)/ > 0, w E + and ψ ker(l + λ). We will consider two cases: Case 1. w = 0. In this case we have I(u) = I(ψ) A ψ + B ψ + C(ε),

13 The Jumping Nonlinearity Problem Revisited 61 and, if we take A < µb/(µ + 1), then Lemma 3.4 implies that I(ψ) as ψ. In particular, I(ψ) is bounded from below. In other words, if λ a + ε (3.6) < µ ( ) b λ µ + 1 then, I(ψ) is bounded from below. Case. w 0. In this case we can write [ 1 I(u) w A ψ w and, with ρ 0 > 0 given in Lemma 3.5, if (3.7) ( w + B w + λ a + ε < ρ 0 ψ ) + ] C(ε). w then Lemma 3.6 implies that I(u) m ρ w C(ε) C(ε), hence I(u) is again bounded from below. Combining Cases 1 and and in view of (3.6), (3.7), we define { } (µ + 1) λ µb (3.8) α 1 := max, µ + 1 λ ρ 0, 0. If a (α 1, λ] then, by picking ε = ε > 0 suitably small, it follows that inf I(u) := I >. u V Next, we will consider the compactnesss condition (PS) c of Theorem 3.1. Lemma 3.7. There exists 0 < α = α (b, L) < λ such that if a (α, λ] then (3.9) Lu + bu + au = 0, u D(L) has only the trivial solution u = 0. Proof. For fixed b we consider the functional J a (u) = 1 ( P +u P u ) + 1 (b u + + a u ), u E whose critical points are the solutions of (3.9) (cf. Lemma.). By negation, let us assume there exist sequences (u n ), (a n ) with u n 0, a n λ, a n < λ and (3.10) Lu n + bu + n a n u n = 0, that is, J a n (u n ) = 0. Then, writing u n = P + u n +P u n E + E, and recalling that E = E 1 E = [χ (, b) (L)](E) [χ [ b,0) (L)](E), E + = [χ [0, ) (L)](E)

14 6 D. G. Costa H. Tehrani where E = [χ [ b,0) (L)](E) = ker(l + λ), we obtain from (3.10) that (3.11) (P + P )u n, h + b u + n h a n u n h = 0 for all h E, (3.1) P + u n P u n + b u + n + a n u n = 0. From (3.1) it follows that P + u n P u n, so that P u n = 0 since u n 0. Let z n = u n / P u n. Then 1 z n and (passing to a subsequence, if necessary) we obtain that z n z weakly in E. Next, we note that the embedding E H is compact in view of hypotheses (L ), (L 4 ) and the comments following Remark 3.. Therefore, (3.13) P z n P z strongly in H. Now, dividing (3.11) by P u n and letting h = P z n P z, we obtain P z n + P z n, P z + b z n + (P z n P z) a n zn (P z n P z) = 0, and, by taking limits, it follows that lim P z n = P z, hence (3.14) P z n P z strongly in E. Next, we note that J a (u) = ( P + u P u )/ + Ψ a (u), where Ψ a (u) = (1/) (b u + + a u ) is a convex functional. Therefore, recalling that 0 < a n < λ and a n λ, we obtain Ψ bλ (z) Ψ bλ (z n ) Ψ an (z) Ψ an (z n ) a n λ z n Ψ a n (z n ) (z z n ) a n λ z n = J a n (z n ) (z z n ) (P + P )z n, z z n a n λ z n = P + z n P + z n, z P z n + P z n, z a n λ z n. Taking lim sup in the above estimate and using (3.14) together with the fact that Ψ bλ is weakly lower semicontinuous, we get 0 Ψ bλ (z) lim inf Ψ bλ (z n ) lim sup P + z n P + z which implies lim P + z n = P + z, hence P + z n P + z strongly in E. Finally, dividing (3.11) by P u n and passing to the limit gives (P + P )z, h + b z + h λ z h = 0 for all h E. As in the proof of Lemma., this shows that z D(L) and Lz + bz + λz = 0.

15 The Jumping Nonlinearity Problem Revisited 63 By assumption it follows that z = 0, hence z n 0 strongly in E, which contradicts the fact that 1 z n. The proof is complete. Remark 3.8. Given the selfadjoing operator L: D(L) H H, let us call the Fučik spectrum of L the subset Σ F (L) R given by Σ F (L) = {(a, b) R Lz + bz + az = 0 has a nonzero solution}. An inspection of the proof of Lemma 3.7 shows that, under conditions (L 1 ) (L 4 ), for any given b > 0 the horizontal slice (R {b}) Σ F (L) is a closed set. In fact, our next lemma shows that, under some weaker assumptions, the compactness condition (PS) c is intimately connected to the Fučik spectrum. Lemma Let L: D(L) H H be a selfadjoint operator satisfying conditions (L 1 ), (L ), (L 4 ), and assume (G 1 ) (G 3 ). If (a, b) / Σ F (L) then the functional I(u) = 1 ( P+ u P u ) + G(x, u) fu, u E, satifies (PS) c w.r.t. {W n } for all c R. Proof. Let (u n ) be a (PS) c sequence w.r.t. {W n }. As before, we write u n = P + u n + P u n = u n,1 + u n, E + E. We recall that ( V = E + E, W = E 1 = cl W n ). By assumption, there exists k n such that n=1 P W (u n ) W kn, I(u n ) c and I (u n ) V Wkn 0 as n. And we must show that (u n ) possesses a subsequence that converges to a critical point of I. For that, we first show that (u n ) is bounded. Indeed, note that from condition I (u n ) V Wkn 0 we have I (u n ) u n = u n,1 u n, + b u + n + a u n + g 0 (x, u n )u n fu n = o(1) u n and, in view of hypothesis (G ), it follows that u n,1 + u + n + u n C(1 + u n, ), hence (3.15) u n C(1 + u n, ). Therefore, in order to show that (u n ) is bounded, it suffices to show that (u n, ) is bounded.

16 64 D. G. Costa H. Tehrani Arguing by contradiction, assume that t n = u n, and define z n = u n / u n,. Then, (3.15) implies that z n is bounded, so that (passing to a subsequence, if necessary) we obtain z n z weakly in E. As in the previous lemma, we first show that P z n P z. For that, let us introduce the two families of projections We have P k : E E W k, P k : E E + E W k (k 1). I (u n ) u n, (P z n P kn z) = P z n, P z n P kn z + b z n + (P z n P kn z) a g0 (x, u n ) + u n, (P z n P kn z) f u n, (P z n P kn z). z n (P z n P kn z) Since P kn z P z strongly in E, if we take limits as n and use (G ), we obtain lim P z n = P z, hence P z n P z strongly in E. Again, using the fact that the functional Ψ(u) = (1/) (b u + + a u ) is convex, we get Ψ(P kn z) Ψ(z n ) Ψ (z n ) (P kn z z n ) = I (u n ) u n, (P kn z z n ) (P + P )z n, P kn z z n g0 (x, u n ) u n, (P f kn z z n ) + u n, (P kn z z n ) P + z n P + z n, P kn z + P z n, P kn z P z n + o(1). Since P kn z z then, by taking lim sup in the above expression as before, we obtain lim P + z n = P + z, hence z n z strongly in E. It follows that z satisfies (P + P )z, φ + b z + φ a As before, this yields z D(L) and Lz + bz + az = 0. z φ = 0 for all φ E. Since (a, b) / Σ F (L) by assumption, it follows that z = 0, so that z n = u n 0 strongly in E. u n, which contradicts the fact that z n 1. We have shown that any (PS) c sequence w.r.t. {W n } is necessarily bounded, so that we may assume that u n u weakly in E. The rest of the proof follows by first showing that P u n P u in E. Then, using the fact that the functional u G(x, u)dx is convex by (G 3 ), we argue

17 The Jumping Nonlinearity Problem Revisited 65 as before to conclude that P + u n P + u in E, and finally that u n u in E. This completes the proof of Lemma Remark It is worth noticing that the above proof of boundedness of (PS) c sequences did not use the convexity assumption (G 3 ). In fact, as the following applications show, in some situations the strong convergence of bounded (PS) c sequences follows from (G 1 ), (G ) alone. Proof of Theorem. Lemmas 3.3, 3.6, 3.7 and 3.9 show that the conditions of Theorem 3.1 are satisfied for our functional I(u) = I a (u) provided that a (α, λ], where α := min{α 1, α }. Therefore the functional I(u) has a critical point û E. And, by Lemma., it follows that û D(L) and it is a solution of problem (P). 4. Applications 4.1. The beam equation. Given continuous functions g(x, t, u), f(x, t) which are π-periodic in t, we consider the following problem for the nonlinear beam equation (NBE) u tt + u xxxx + g(x, t, u) = f(x, t) for (x, t) (0, π) R, u(0, t) = u(π, t) = u xx (0, t) = u xx (π, t) = 0 for t R, u(x, t + π) = u(x, t) for (x, t) (0, π) R, that is, we seek time-periodic solutions with period π for the above nonlinear beam equation in the interval (0, π) under Navier boundary conditions. The corresponding eigenvalue problem for the beam operator L = t + 4 x, Lu = λu for (x, t) (0, π) R, u(0, t) = u(π, t) = u xx (0, t) = u xx (π, t) = 0, u(x, t + π) = u(x, t), has infinitely many eigenvalues λ mn and eigenfunctions ϕ mn, ψ mn given by λ mn = m 4 n, m N, n {0} N, { ϕmn (x, t) = sin mx sin nt for m, n N, ψ mn (x, t) = sin mx cos nt for m N, n {0} N. (Similarly to ϕ mn, ψ mn, note that the eigenfunctions ϕ m n, ψ m n correspond to the eigenvalue λ mn whenever m 4 n = m 4 n.) Letting Ω = (0, π) (0, π), we also note that {ϕ mn, ψ mn } is a complete orthogonal system for H = L (Ω),

18 66 D. G. Costa H. Tehrani and the operator L: D(L) H H defined by { D(L) := u = (c mn ϕ mn + d mn ψ mn ) H Lu := λ mn (c mn ϕ mn + d mn ψ mn ) λmn (c mn ϕ mn + d mn ψ mn ) H is a selfadjoint operator with pure point spectrum σ(l) = {λ mn }. Moreover, except for λ = 0, which is an eigenvalue of infinite multiplicity, all other eigenvalues λ mn 0 have finite multiplicity. In particular, it follows that assumption (L 4 ) holds true for any b > 0. Next, let us assume that g(x, t, s): [0, π] [0, π] R R is a continuous function satisfying g(x, t, s) g(x, t, s) (g 1 ) 0 < lim = a < b = lim uniformly for (x, t) Ω. s s s s (g ) g(x, t, ) is nondecreasing for (x, t) Ω. When applying Theorem 1. to (NBE) we plan to use a symmetry result of Lazer McKenna [16]. For that, we must restrict ourselves to crossing a positive eigenvalue of L whose corresponding eigenfunctions ϕ mn (x, t), ψ mn (x, t) (for each fixed t 0) do not change sign for x (0, π), a situation that only occurs when m = 1. Therefore, we shall assume that (l 1 ) σ( L) [ λ 1k, b] = λ 1k = k 1 for some k, and the equation m 4 n = 1 k has the unique solution (m, n) = (1, k). Then, it follows that λ mn λ 1k for (m, n) (1, k), hence ker(l λ 1k ) = span{sin x cos kt, sin x sin kt}. Let us denote by µ 1 the largest eigenvalue of L that is smaller than λ 1k = k 1, and by µ the smallest eigenvalue of L that is larger than λ 1k : clearly, we have µ 1 < k 1 < b < µ. Now, by a symmetry result of Lazer McKenna ([16]), the solutions of the PDE (4.1) Lu + bu + (k 1)u = 0 are of the form u(x, t) = (sin x)y(t) where y(t) is a π-periodic solution of the ODE (4.) y + (b + 1)y + k y = 0. However, (4.) has a π-periodic solution if and only if (see [11]) (4.3) π b π k = π l },

19 The Jumping Nonlinearity Problem Revisited 67 for some l = 1,,..., which gives b = k l /(k l) 1. Since k 1 < b, it follows that k < l < k. Therefore, the smallest value of b for which we have a solution of (4.3) is attained when l = k + 1, that is, (k + 1) b = k (k 1) 1. Now, it is not difficult to see that µ < k (k + 1) /(k 1) 1 (indeed, we have µ λ 1,k+1 = (k + 1) 1, and a simple calculation shows that (k + 1) 1 < k (k + 1) /(k 1) 1 for any k ). Therefore, as long as b (k 1, µ ), the equation Lu + bu + (k 1)u = 0 has no nontrivial solution, so that L satisfies assumption (L 3 ) with λ = k 1. Furthermore, since ψ 10 = sin x is the only eigenfunction that does not change sign, and the corresponding eigenvalue is λ 10 = 1 (a simple eigenvalue of L), it is clear that L satisfies assumption (L ) for any eigenvalue λ mn 1, in particular for λ 1k = 1 k, k. We can now state: Theorem 4. Assume condition (l 1 ) with λ 1k = k 1 < b < µ. Then there exists α = α(b) with µ 1 < α < k 1 such that, for a (α, k 1] and g(x, t, s)satisfying (g 1 ), (g ), equation (NBE) possesses a solution for any given f L (Ω). Remark 4.1. (a) We exhibit below points of σ(l) in the interval [ 0, 1]... < λ 6 = 0 < λ 3 10 = 19 < λ 14 = 15 < λ 3 = 9 < λ 13 = 8 < λ 1 = 3 < λ 11 = 0 < λ 10 = 1 < λ 3 = 7 < λ = 1 <... (b) Simple calculations show that, for example, λ 1 = 3, λ 13 = 8, λ 14 = 15, λ 15 = 4, λ 16 = 35 all satisfy the conditions of the above theorem. 4.. The wave equation. Next we look for time-periodic solutions with period π for the nonlinear wave equation in the interval (0, π) under Dirichlet boundary conditions: u tt u xx + g(x, t, u) = f(x, t) for (x, t) (0, π) R, (NWE) u(0, t) = u(π, t) = 0 for t R, u(x, t + π) = u(x, t) for (x, t) (0, π) R, The corresponding eigenvalue problem for the wave operator L = t x has infinitely many eigenvalues λ mn and eigenfunctions ϕ mn, ψ mn given by λ mn = m n, m N, n {0} N, { ϕmn (x, t) = sin mx sin nt for m, n N, ψ mn (x, t) = sin mx cos nt for m N, n {0} N.

20 68 D. G. Costa H. Tehrani Similarly to the beam equation, letting Ω = (0, π) (0, π), we note that {ϕ mn, ψ mn } is a complete orthogonal system for H = L (Ω), and the operator L: D(L) H H defined by { D(L) := u = (c mn ϕ mn + d mn ψ mn ) H } λmn (c mn ϕ mn + d mn ψ mn ) H, Lu := λ mn (c mn ϕ mn + d mn ψ mn ) is a selfadjoint operator with pure point spectrum σ(l) = {λ mn }. Again, except for λ = 0, which is an eigenvalue of infinite multiplicity, all other eigenvalues λ mn 0 have finite multiplicity. Using similar arguments to those for the beam equation, we can now state Theorem 5. Assume that k is an integer for which the only solution (m, n) of the equation m n = 1 k, (m, n) N ({0} N) is (m, n) = (1, k). In addition, assume that λ 1k = k 1 < b < µ. Then there exists α = α(b) with µ 1 < α < k 1 such that, for a (α, k 1] and g(x, t, s) satisfying conditions (g 1 ), (g ), equation (NWE) possesses a solution for any given f L (Ω). Remark 4.. For other recent results on asymptotically linear wave and beam equations we refer the reader to [4] and references therein The Schrödinger equation. For our last application, we consider the nonlinear Schrödinger equation (NSE) u + V (x)u + g(x, u) = f(x), x R N. Since we would like to give a simple application, we are not going to consider the most possible general situation. We assume that g: R N R R satisfies conditions (G 1 ), (G ) in the introduction, and that V : R N R satisfies: (V 1 ) V C(R N ) and V (x) 0 as x, (V ) there exists ϕ C 0 (R N ) such that ( ϕ + V (x)ϕ ) < 0, (V 3 ) λ > 0 is an isolated point of σ( V (x)) with σ( V (x)) [a, b] = { λ}. Remark 4.3. We will recall a few basic result in the theory of Schrödinger operators which are relevant to our discussion: (a) Given V L (R N ), let the operator L: D(L) L (R N ) L (R N ) be defined by Lu = u + V (x)u, u D(L) = H (R N ).

21 The Jumping Nonlinearity Problem Revisited 69 In addition, assume that lim inf x V (x) γ for some γ R. Then one has σ ess (L) [γ, ). In particular, if lim x V (x) = 0 then σ ess (L) = σ ess ( ) = [0, ). (b) The bottom of the spectrum σ(l) of the operator L is given by (Lu, u) u + V (x)u λ 0 = inf 0 u H (R N ) u = inf. 0 u H (R N ) u Furthermore, if (V ) is satisfied then λ 0 < 0 and, using the Concentration Compactness Principle of P. L. Lions, one shows that λ 0 is the principal eigenvalue of L with a positive eigenfunction Φ 0 : { LΦ = λ0 Φ 0, Φ 0 H (R N ), Φ 0 > 0. In fact, by elliptic regularity theory, it follows that Φ 0 C (R N ). (c) The spectrum of L in (, γ), namely σ(l) (, γ) is at most a countable set, which we denote by λ 0 < λ 1 λ... where each λ k < γ is an isolated eigenvalue of L of finite multiplicity (counted as often as its multiplicity) and characterized by the minimax formula λ k = inf sup F F k 0 u F (Lu, u) u, where F k denotes the collection of all k dimensional subspaces of H (R N ). So, if the potential V (x) verifies (V 1 ) and (V ) then the selfadjoint operator L: D(L) L (R N ) L (R N ) given by Lu = u + V (x)u, u D(L) = H (R N ), satisfies conditions (L 1 ), (L ) and (L 4 ) in Theorem with 0 < a λ = λ k0 < b, for some k 0 1, where we are denoting by λ 0 < λ 1... < 0 the eigenvalues of L which make up the spectrum of L in (, 0) (cf. remark above). In fact, if we denote by µ 1 the largest eigenvalue of L that is smaller than λ k0, and by µ the smallest eigenvalue of L that is larger than λ k0, then we have µ 1 < a < λ k0 < b < µ and we can prove that the following result holds: Lemma 4.4. If b ( λ k0, µ ) then the equation (4.4) u + V (x)u + bu + λ k0 u = 0, u H (R N ) has no nonzero solution. Proof. If (4.4) has a nonzero solution u H (R N ), then u must necessarily change sign in R N. Indeed, since b / σ( u V (x)) we cannot have u positive and, since λ = λ k0, k 0 1, we also cannot have u negative. Therefore, u must change sign and we can rewrite (4.4) as (4.5) u + [V (x) + λ k0 + (b λ k0 )χ Ω ]u = 0, 0 u H (R N ),

22 70 D. G. Costa H. Tehrani where = Ω = {x R N u(x) > 0} R N and χ Ω denotes the characteristic function of the set Ω. Now, if we set Ṽ (x) = V (x) + λ k 0 + (b λ k0 )χ Ω then (4.5) shows that 0 σ p ( L), where L = + Ṽ (x). Clearly, we have Ṽ (x) L (R N ) and lim inf x Ṽ (x) λ k0 = λ > 0. So, if we denote by ξ 0 < ξ 1... < 0 the eigenvalues of L which make up the spectrum of L in (, λ), then, by using the minimax characterization of the eigenvalues λ k, ξ k of L and L, respectively, we can write ( u λ k = inf sup + V (x)u ), u ξ k = dim(f )=k 0 u F inf sup dim(f )=k 0 u F ( u + Ṽ (x)u ). u But then, since V (x) + λ k0 Ṽ (x) V (x) + b, with both inequalities being strict on sets of positive measure, we conclude that λ k λ k0 < ξ k < λ k + b, k = 0, 1,... Without loss of generality we assume λ k0 1 < λ k0. Then, we have ξ k0 1 < λ k0 1 + b = λ k0 1 + b < 0 = λ k0 + λ k0 < ξ k0, hence ξ k0 1 < 0 < ξ k0, which shows that 0 / σ p ( L), a contradiction. We can now state: Theorem 6. Assume V (x) satisfies conditions (V 1 ) (V 3 ) and let L: D(L) L (R N ) L (R N ) be the selfadjoint operator given by Lu = u + V (x)u, u D(L) = H (R N ), so that σ(l) = {λ 0, λ 1,... } [0, ), where λ j < λ j+1 < 0 denote the distinct eigenvalues of L. Let λ := λ k = λ k > 0 for some k 1 and λ k < b < λ k 1 be given. Then there exists α = α(b) with λ k+1 < α < λ k (or 0 < α < λ k if λ k is the largest negative eigenvalue of L) such that, for a (α, λ k ] and g: R N R R satisfying conditions (G 1 ), (G ), equation u + V (x)u + g(x, u) = f(x), u H (R N ) has a solution for any given f L (R N ). Proof. In order to apply Theorem 1, since we do not have the convexity assumption (G 3 ), we must show that bounded (PS) c sequences are actually precompact (see Remark 3.11). In fact, if (u n ) is our bounded (PS) c sequence with u n u weakly in E = H 1 (R N ), then the proof of Lemma 3.10 shows that P u n P u strongly in H 1 (R N ). Then, using the fact

23 The Jumping Nonlinearity Problem Revisited 71 that the nonlinearity g(x, ) is nondecreasing at infinity, so that the functional Ψ(u) = (1/) (b u + + a u ) is convex, we have Ψ(u) Ψ(u n ) Ψ (u n ) (u u n ) = I (u n ) (u u n ) (P + P )u n, u u n g 0 (x, u n )(u u n ) + f(u u n ) P + u n P + u n, P + u g 0 (x, u n )(u u n ) + o(1). Therefore, in order to show that P + u n P + u, it is enough to verify that g0 (x, u n )(u u n ) = o(1). In the present case, this follows from the estimate (G ) and the fact that the embedding Hloc 1 (RN ) L loc (RN ) is compact. The proof of Theorem 4 is complete. References [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), [] A. Ambrosetti and G. Prodi, On the inversion of some differential mappings with singularities between banach spaces, Ann. Mat. Pura Appl. 93 (1973), [3] M. Arias, J. Campos, M. Cuesta and J.-P. Gossez, Asymmetric elliptic problems with indefinite weights, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear). [4] T. Bartsch and Y. H. Ding, Critical point theory with applications to asymptotically linear wave and beam equations, Differential Integral Equations 13 (000), [5] A. K. Ben-Naoum, C. Fabry and D. Smets, Resonance with respect to the Fučik spectrum, Electron. J. Differential Equations 37 (000), 1 1. [6], Structure of the Fučik spectrum and existence of solutions for equations with asymmetric nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 131 (001), [7] M. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 4 (1975), [8] H. Brézis and L. Nirenberg, Nonlinear Anal. (to appear). [9] E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), [10] D. G. de Figueiredo, On the superlinear Ambrosetti Prodi problem, Nonlinear Anal. 8 (1984), [11] S. Fučik, Boundary value problems with jumping nonlinearities, Cas. Pest. Mat. 101 (1976), [1], Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel Publishing Company, Dordrecht, Holland, [13] T. Gallouet and O. Kavian, Résultats d Existence et de Non Existence pour Certains Problèmes Demi-linéaires à l Infini, Ann. Fac. Sci. Toulouse Math. 3 (1981), [14] J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 8 (1975), [15] A. C. Lazer and P. J. McKenna, Critical point theory and Boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. Partial Differential Equations 10 (1985),

24 7 D. G. Costa H. Tehrani [16], A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations 7 (1988), [17], Large-amplitude Periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 3 (1990), [18] C. A. Magalhães, Semilinear elliptic problems with crossing of multiple eigenvalues, Comm. Partial Differential Equations 15 (1990), [19] B. Ruf, On nolinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4) CXXVIII (1981), [0] M. Schechter, The Fučik spectrum, Indiana Univ. Math. J. 43 (1994), [1] E. A. Silva, Critical point theorems and applications to differential equations, Ph.D. Thesis, University of Wisconsin, Madison (1988). [] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire (1985), [3] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden Day, San Francisco, Manuscript received October, 00 David G. Costa and Hossein Tehrani Department of Mathematical Sciences University of Nevada, Las Vegas Las Vegas, NV , USA address: costa@unlv.edu, tehranih@unlv.edu TMNA : Volume N o