Multiplicity ofnontrivial solutions for an asymptotically linear nonlocal Tricomi problem

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1 Nonlinear Analysis 46 (001) Multiplicity ofnontrivial solutions for an asymptotically linear nonlocal Tricomi problem Daniela Lupo 1, Kevin R. Payne ; ; Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, Milano, Italy Received 7 December 1998; accepted 16 December 1999 Keywords: Mixed type equations; Dual variational method; Linking theorem 1. Introduction and statement of the result In this work, we are concerned with the multiplicity ofnontrivial solutions for the following nonlocal semilinear Tricomi problem, introduced in [7]: Tu yu xx u yy = Rf(u) in u =0 on AC ; (NST) where T y@ y is the Tricomi operator on R ;Ris the reection operator induced by composition with the map : R R dened by (x; y)=( x; y); f(u) is an asymptotically linear term such that f(0) = 0, and is a symmetric admissible Tricomi domain (cf. Denition :1 oflupo and Payne [7]). That is, is a bounded region in R that is symmetric with respect to the y-axis and has a piecewise C BC, where is a symmetric with respect to the y-axis C arc in the elliptic region y 0, with endpoints on the x-axis at A=( x 0 ; 0) and B=(x 0 ; 0) and AC and BC are two characteristic arcs for the Tricomi operator in the hyperbolic region ofnegative and positive slope issuing from A and B, respectively, which meet at the point C on the y-axis. It should be noted that u 0 is always a solution to (NST). We refer the reader to [7] for the rationale concerning this problem, the interest Corresponding author. Tel.: ; fax: addresses: danlup@mate.polimi.it (D. Lupo), paynek@mate.polimi.it (K.R. Payne). 1 Supported by MURST, Project Metodi Variazionali ed Equazioni Dierenziali Non Lineari. Supported by a CNR grant; program for foreign mathematicians. Current address: Dipartimento di Matematica, Universita di Milano, Via C. Saldini 50, 0133 Milano, Italy X/01/$ - see front matter? 001 Elsevier Science Ltd. All rights reserved. PII: S X(00)

2 59 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) in variational methods for mixed elliptic hyperbolic type equations, and for references to relevant literature on the Tricomi operator and its importance. First of all, let us specify in which sense we are looking for generalized solutions. For, we denote by C () the set ofall smooth functions on such that u 0 on, by W 1 the closure of C () with respect to the W 1; () norm, and by W 1 the dual of W 1. Denition 1.1. One says that u WAC 1 is a generalized solution of(nst) if Tu = Rf(u) in L (); and there exists a sequence {u j } CAC () such that lim u j u W 1 j AC = 0 and lim Tu j Rf(u) W 1 j BC =0: Let T AC be the unique continuous extension of T from CAC () into W AC 1 and note that R induces an isometric isomorphism between WAC 1 and W BC 1. In [7], following ideas in [], it is shown that, given a symmetric admissible domain, the reected Tricomi operator RT AC : D(RT AC )=W A W 1 AC L (); admits a continuous left inverse K =(RT AC ) 1 : L () W A W 1 AC for which K is compact as an operator on L (). Denote by + k the sequence of positive eigenvalues of K decreasing to zero (cf. Sections 3 and of [6] for more details). Suppose that (f1) f C 0 (R; R); fis invertible and f(0)=0. In addition, suppose that a and b are real numbers such that a ((RT AC ) 1 ) and b ( j+1 + ;+ j ) for some j 1. After writing f(s)=s=a+f (s)=s=b+f 0 (s), suppose that the perturbations f ;f 0 C 0 (R; R) satisfy (f) f is bounded and lim s 0 f 0 (s)=s = 0, and (f3) s =b sf 0 (s) 0; s R. In [6] we have proved the following result. Theorem A. Let be a symmetric admissible Tricomi domain and assume that f satises (f1) (f3). Then; if a j + ; the problem (NST) admits at least one nontrivial generalized solution. In this paper we want to show how, ifthe nonlinearity interacts more with the spectrum ofthe inverse operator K =(RT AC ) 1, then one obtains more nontrivial solutions. More precisely, the following result holds. Theorem B. Let be a symmetric admissible Tricomi domain and assume that f satises (f1) (f3). Suppose that a ((RT AC ) 1 ) and that for some j ;

3 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) j+1 + b + j + j 1 a. Then there exists a value b ( j+1 + ;+ j ) such that for every b (b ; j + ) the problem (NST) admits at least two nontrivial solutions. The result will be obtained utilizing the dual variational method, as in [6,7], combined with the following critical point theorem of Marino, Micheletti, and Pistoia on the existence ofdistinct linking critical points. More precisely, consider a Hilbert space H decomposed into H = W Z, with dim Z +. Within Z consider a proper subspace Z 1 and two elements e 1 Z 1 and e (Z Z 1 ) with e 1 = e = 1. Dene, for any Y subspace of H, B (Y ):={u Y u } and (Y ) the boundary of B (Y ) relative to Y. Furthermore, for any e H, consider Q R (Y; e):={u + ae Y Re u Ya 0; u + ae R} and denote R (Y; e) its boundary relative to Y Re. Theorem 1. (cf. Theorem 8:4 ofmarino et al. [8]). Suppose that I C 1 (H; R) satises the (PS) condition.in addition; assume that there exist i ;R i ; for i =1; ; such that 0 i R i and inf I sup R1 (W; e 1 (Z) inf I ; Q R1 (W; e 1) inf sup I; R (W Z 1;e (Z Z 1) inf I : Q R (W Z 1;e ) If R 1 R, then there exist at least three critical levels of I.Moreover; the critical levels satisfy the following inequalities: sup I c 1 B 1 (Z) inf I sup I R1 (W; e 1 (Z) inf I R (W Z 1;e (Z Z 1) I c 3 inf I Q R1 (W; e 1) inf I: Q R (W Z 1;e ) Remark 1.3. (1) Since dim Z +, one has sup B1 (Z) I +. () Theorem :1 of[5] gives (utilizing a result of[4]) the analog to Theorem 1. above in the case dim Z =+ ; (cf. also [3, Theorem :1] for a simpler statement for Z 1 = Re 1 ).. Setting of the problem Following ideas contained in [6,7], we translate the problem ofnding nontrivial solutions of(nst) into the problem ofnding nontrivial critical points ofthe dual

4 594 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) action functional associated to (NST); the rough idea behind the use of a dual variational method is the following. Let f be an invertible function and denote by g = f 1 its inverse. The statement there exists u 0 WAC 1 such that u 0 0 and T AC u 0 = Rf(u 0 )inl () is equivalent to there exists u 0 WAC 1 such that u 0 0 and RT AC u 0 = f(u 0 )inl (): Hence, ifwe are able to nd a v 0 L () such that g(v 0 )=Kv 0 ; where K =(RT AC ) 1 is compact on L () (cf. [Section of 7] for a precise description ofthe linear theory), then u 0 = Kv 0 is a solution ofour problem. Therefore, our goal will be to set-up a variational formulation of the dual problem. To this end, dene J : L () R as J (v)= G(v)dxdy 1 vkv dx dy; (.1) where G(v) = v g(t)dt denotes the primitive of g such that G(0)=0. If g satises 0 the growth condition (for well-known properties of Nemistkii operators cf. [1]) g(t) K 1 + K t (.) it is easy to check that J C 1 (L (); R) and that J (v)[w]= g(v)w dx dy wkv dx dy: (.3) Therefore, critical points of J will be weak solutions ofthe dual problem; let us remark explicitly that the key point in proving (.3) is the symmetry of K. 3. The critical point result We consider the case ofnonlinearity f which is asymptotically linear, both in zero and at innity, where the slope in zero is dierent from the slope at innity. More precisely, rst we recall that since K =(RT AC ) 1 is a compact linear operator on L () which is injective, nonsurjective, and self-adjoint the spectrum of K is {0} { k }, where k is a sequence ofeigenvalues ofnite multiplicity whose only possible accumulation point is zero. Furthermore, it is shown in Proposition :1 of[6] that there exist innitely many positive and innitely many negative eigenvalues, which we will denote by ± k. As pointed out in Remark :3 of[6], it is possible to associate to ± k a complete orthonormal basis in L (). The solutions will be found as the preimage of two distinct linking critical points which are obtained by applying Theorem 1. to the dual action functional J. Hence, we have to show that J satises an appropriate geometrical condition (cf. Lemmas 3. and 3.4) and a suitable compactness condition (cf. Lemma 3.1). To begin with, one can check that the hypotheses (fi); i=1; ; 3, imply that the inverse g = f 1 satises

5 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) (g1) g C 0 (R; R); gis invertible, g(0)=0;g(t)=at + g (t)=bt + g 0 (t), where the perturbations g 0 and g belong to C 0 (R; R) and satisfy (g) g is bounded and lim t 0 g 0 (t)=t =0; and (g3) tg 0 (t) 0; t R. One then notes that (g1) (g3) imply that the primitives G 0 (t)= t 0 g 0()d and G (t)= t 0 g ()d satisfy, for every t R, G (t) M t and 0 G 0 (t) 1 (a b)t + M t ; (3.1) where M = am (denoting by M the positive constant such that f (s) M,for every s R), and G 0 (t)= 1 (a b)t + G (t): (3.) Furthermore, one has G 0 (t) lim t 0 t = 0 and G (t) (b a) t ; t R; (3.3) and G(t) G(t) lim t ± t = a and lim t 0 t = b: (3.4) Let us now consider the dual action functional J dened in (.1) J (v)= G(v)dxdy 1 vkv dx dy = G 0 (v)dxdy + b v dx dy 1 vkv dx dy = G (v)dxdy + a v dx dy 1 vkv dx dy; where the use ofone ofthe last two equivalent decompositions will be made as suits the situation. It is clear that the growth condition on g implies (.) and hence J C 1 (L (); R). As a nal preparation, let L () = V V, where V = j k=1 V k is a direct sum of eigenspaces V k = {v L () Kv = + k v}, while V is the orthogonal complement of V in L (). From now on, we will denote by v = v L (). Keeping in mind the ordering 0 j + + 1, it is easy to see that one has vkv dx dy j + v for every v V; (3.5) vkv dx dy 1 + v for every v V (3.6) and vkv dx dy j+1 + v for every v V : (3.7)

6 596 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) Lemma 3.1. Let be a symmetric admissible Tricomi domain and assume that g satises (g1); (g) and (g3). Then J satises the Palais Smale condition on L (). Proof. Consider a Palais Smale sequence; that is {v n } such that g (v n )+av n Kv n 0 in L () (3.8) and J (v n ) M: One has that {v n } must be bounded in L (). Ifnot, one has that v n +. Then, one has g (v n )= v n 0inL () since (g) yields g (v n )dxdy v n M dx dy v n M v n : Now, since v n = v n is a norm-bounded sequence, by the compactness of K, there is a w L () \{0} such that K(v n = v n ) w in L (), where one passes to another subsequence ifneeded. Then by (3.8), one must have av n = v n w in L (), and thus by the continuity of K we have aw Kw =0 in L (). This gives a contradiction since a ((RT AC ) 1 ). Hence, v n must be bounded. Then, keeping in mind that (3.8) can be written as g(v n ) Kv n 0 in L () and that K is compact, one has that, at least for a subsequence, g(v nk ) z in L () for some z L (). Therefore, f(g(v nk )) = v nk f(z) inl () since f is a continuous Nemitskii operator and hence the result. We now consider the geometrical conditions. Denoting (V )={v V v = } and, letting e j + V j such that e j + =1, R (V ;e j + )={w + e+ j 0 and w + e j + = R} {w V w R}; the following lemmas provide a geometrical structure suitable for applying Theorem 1. to the dual functional J. Lemma 3.. Let be a symmetric admissible Tricomi domain and assume that g satises (g1) (g3). If a j + ; there exist 1 0 and R 1 1 such that the following inequality holds: sup J (v) inf J (v): 1 (V R1 (V ;e + j ) Proof. We give a briefsketch ofthe proofwhich is contained in [7]. We want to show that it is possible to choose a 1 small enough to make the sup strictly negative and to choose R 1 large enough to make the inf nonnegative. Then, choosing R 1 even

7 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) larger, ifneed be so that R 1 1, will complete the lemma. To estimate the sup from above, we begin by noting that inequality (3.5) applied to arbitrary v V yields J (v)= b v dx dy + G 0 (v) dx dy 1 vkv dx dy 1 (b + j ) v + G 0 (v)dxdy: Ifwe show that lim v 0 v V G 0(v) dx dy v =0; (3.10) then we will obtain the claim: there exists a 1 0 such that J (v) 1 4 (b + j ) v 0 for any v V; v = 1 : (3.11) In fact, it is shown in Lemma 4:5 of[7], that G 0(v)dxdy v =0; W 1; lim v W 1; 0 v V j and, since all norms are equivalent on V with dim V +, the desired limit property on G 0 follows. On the other hand, to estimate the inf from below, we begin by observing that inequality (3.7) and the lower bound on G 0 in (3.1) yields, for arbitrary w V J (w) G 0 (w)dxdy + b w dx dy 1 + j+1 w 1 (b + j+1 ) w 0: (3.1) Furthermore, for arbitrary w + e j + V Re j +, the inequality (3.7) together with the lower bound on G in (3.1) yields J (w + e j + )=a w + a + G (w + e j + )dxdy 1 wkw dx dy + j 1 (a + j+1 ) w + (a + j ) M 1= ( w + e j + ): (3.13) Now, by recalling that w is orthogonal to e j +, we can set R = w + e j + = w + and from (3.13) taking into account that a j + + j+1,weget J (w + e + j ) 1 (a + j )R M 1= R: (3.14)

8 598 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) It is clear that it is possible to choose R suciently large to make the right-hand side of(3.14) strictly positive, and hence to choose, ifneed be, a bigger R 1 so that R 1 1 and hence the inequality (3.9) follows. Remark 3.3. Inequality (3.9) oflemma 3. gives the rst inequality in the hypotheses oftheorem 1. with the choices H = L (); Z= V; W = V and e 1 = e j +. Moreover, inequalities (3.1) and (3.13) yield the second inequality in the hypotheses of Theorem 1. and hence Lemma 3.1 completes the proofoftheorem A. Now we will show that iftwo eigenvalues fall between a and b, then one can construct a second linking geometry that is suitable for the application of Theorem 1.. Lemma 3.4. Let be a symmetric admissible Tricomi domain and assume that g satises (g1) (g3). If for some j ; j+1 + b + j + j 1 a then there exist 0; R and a value b ( j+1 + ;+ j ) such that for every b (b ; j + ) the following inequality holds: sup J (v) inf J (v): (V V R1 (V V j;e + j 1 ) Proof. Since V V j = j 1 k=1 V k, one can choose a, independently of b, such that for every v V V j with v = one has J (v) 1 4 (+ j + j 1 ) 0: (3.16) Indeed, for every v V V j one has J (v)= b v dx dy + G 0 (v)dxdy 1 1 (b + j 1 ) v + G 0 (v)dxdy 1 (+ j j 1 + ) v + G 0 (v)dx dy; vkv dx dy and hence, by using the same reasoning that leads to the inequality (3:11), one can x a independently of b so that (3:16) holds. On the other hand, let e j 1 + V j 1 be such that e j 1 + = 1. Then, for every w V ;v V j and for every R, keeping in mind that j j + j 1 and that w; v and e j 1 + are mutually orthogonal, by choosing R = w+v+e j 1 + = w + v +, one gets J (w + v + e j 1 + )=a w + a v + a + G (w + v + e j 1 + )dxdy 1 wkw dx dy + j v 1 + j 1

9 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) (a + j+1 ) w + 1 (a + j ) v + (a + j 1 ) and hence M 1= w + e + j + e + j 1 J (w + v + e + j 1 ) 1 (a + j 1 )R M 1= R: (3.17) Hence, one can choose an R large enough to ensure that the right-hand side of(3.17) strictly positive. Therefore, with R max{r; R 1 ; }, one has J (w + v + e j 1 + ) 0 (3.18) for every (w; v) V V j and for every R such that w + v + e j 1 + = R. On the other hand, for every w V and every v V j, taking into account (3.3) and j j, one has J (w + v)= G (w + v)dxdy + a (w + v) dx dy (w + v)k(w + v)dxdy (b a) ( w + v )+ a ( w + v ) + j+1 w + j v 1 (b + j )[ w + v ]: Therefore, for every w + v V V j with 0 w + v R, one gets J (w + v) 1 (b + j )[ w + v ] 1 (b + j )R : (3.19) From (3.19), since b j + 0 when b j +, it is possible to x a value b ( j+1 + ;+ j ) such that for every b (b ; j + ) and for every w + v V V j, with w + v R, one has J (w + v) 1 6 (+ j + j 1 ) : (3.0) From (3.16), (3.18) and (3.0) one gets sup J (v) (V V j) 4 (+ j j 1 + ) 1 6 (+ j j 1 + ) inf J R1 (V V j;e + j 1 ) and hence the result. Remark 3.5. Inequality (3.15) gives the third inequality in the hypotheses oftheorem 1. with the choices H = L (); Z= V; W = V ;Z 1 = V j ;e 1 V j and e = e + j 1. Moreover, the fourth inequality in the hypotheses of Theorem 1. follows from the estimates of J from below that lead to inequalities (3.17) (3.19).

10 600 D.Lupo, K.R.Payne / Nonlinear Analysis 46 (001) Proof of Theorem B. Suppose that, for some j ; j+1 + b + j + j 1 a and that f satises (f1) (f3). Then g = f 1 satises (g1) (g3); hence, the functional J satises, by Lemmas 3.1, 3. and 3.4 and Remarks 3.3 and 3.5, the hypotheses of Theorem 1.. Therefore, there will exist two distinct nontrivial critical points v 1 and v of J. Hence u i = Kv i WAC 1 for i =1; are distinct nontrivial generalized solutions to (NST). References [1] A. Ambrosetti, G. Prodi, A Primer ofnonlinear Analysis, Cambridge University Press, Cambridge, [] V.P. Didenko, On the generalized solvability ofthe Tricomi problem, Ukrainian Math. J. 5 (1973) [3] D. Lupo, Existence and multiplicity ofsolutions for superquadratic noncooperative variational elliptic systems, Topol. Methods Nonlinear Analysis 1 (1998) [4] D. Lupo, A.M. Micheletti, Multiple solutions for Hamiltonian systems via limit relative category, J. Comput. Appl. Math. 5 (1994) [5] D. Lupo, A.M. Micheletti, Two applications ofa three critical point result, J. Dierential Equations 13 (1996) 38. [6] D. Lupo, A.M. Micheletti, K.R. Payne, Existence ofeigenvalues for reected Tricomi operators and applications to multiplicity ofsolutions for sublinear and asymptotically linear nonlocal Tricomi problems, Adv. Dierential Equations 4 (1999) [7] D. Lupo, K.R. Payne, A dual variational approach to a class ofnonlocal semilinear Tricomi problems, NoDEA Nonlinear Dierential Equations Appl. 6 (1999) [8] A. Marino, A.M. Micheletti, A. Pistoia, A nonsymmetric asymptotically linear elliptic problem, Topol. Methods Nonlinear Analysis 4 (1994)

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems NoDEA Nonlinear differ. equ. appl. 6 (1999) 247 266 1021-9722/99/030247-20 $ 1.50+0.20/0 c Birkhäuser Verlag, Basel, 1999 Nonlinear Differential Equations and Applications NoDEA A Dual Variational Approach