The dual variational method in nonlocal semilinear Tricomi problems

The dual variational method in nonlocal semilinear Tricomi problems Daniela Lupo and Kevin R. Payne∗

Abstract We describe the results obtained in [15] and [16] concerning the use of the dual variational approach in order to prove the existence and multiplicity of solutions for a nonlocal variational Tricomi problem.

1

Introduction and setting of the problem

In this work, we provide a survey of the results obtained in [15] and [16] on the use of variational methods to establish the existence and multiplicity of nontrivial solutions, in a suitably generalized sense, to the following nonlocal semilinear Tricomi problem  T u ≡ −yuxx − uyy = Rf (u) in Ω (N ST ) u=0 on AC ∪ σ, where p ∈ R with 0 < p < 1, T ≡ −y∂x2 − ∂y2 is the Tricomi operator on R2 , R is the reflection operator on L2 (Ω) induced by composition with the map Φ : R2 → R2 defined by Φ(x, y) = (−x, y), Ω is a bounded region in R2 that is symmetric with respect to the y-axis and has a piecewise smooth boundary ∂Ω of the classical Tricomi form. That is, ∂Ω consists of a smooth arc σ in the elliptic region y > 0, with endpoints on the x-axis at A = (−x0 , 0) and B = (x0 , 0), and two characteristic arcs AC and BC for the Tricomi operator in the hyperbolic region y < 0 issuing from A and B and meeting at the point C on the y-axis. One knows that 2 2 AC : (x + x0 ) − (−y)3/2 = 0 and BC : (x − x0 ) + (−y)3/2 = 0, 3 3 Finally, f (u) indicates a nonlinear term associated to an f ∈ C 0 (R) with an asymptotically linear or sublinear growth at infinity and such that f (0) = 0; we note that this last hypothesis implies that u ≡ 0 is always a solution of (N ST ). ∗ Dipartimento di Matematica, Politecnico di Milano Piazza Leonardo da Vinci, 32; 20133 Milano, Italy; e-mail: [email protected]; [email protected]

1

First of all, let us specify in which sense we are looking for generalized solutions. We denote by CΓ∞ (Ω) the set of all smooth functions on Ω such that u ≡ 0 on Γ, by WΓ1 the closure of CΓ∞ (Ω) with respect to the W 1,2 (Ω) norm, and by WΓ−1 the dual of WΓ1 which can be shown to be the norm closure of L2 (Ω) with respect to the norm |(w, ϕ)L2 | kwkWΓ1 = sup . (1.1) 06=ϕ∈WΓ1 kϕkWΓ1 In this way, one obtains rigged triples of Hilbert spaces with inclusion chains such as WΓ1 ⊂ L2 (Ω) ⊂ WΓ−1 . 1 The space WAC∪σ is a Hilbert space in which one can find solutions u to T u = f 2 for all f ∈ L (Ω) in a strong sense as is clarified in section 2. 1 Definition 1.1. One says that u ∈ WAC∪σ is a generalized solution of (N ST ) if

T u = Rf (u) in L2 (Ω),

(1.2)

∞ and there exists a sequence {uj } ⊂ CAC∪σ (Ω) such that

1 lim kuj − ukWAC∪σ = 0 and

j→∞

lim kT uj − R (f (u)) kW −1

j→∞

= 0.

(1.3)

BC∪σ

The form of the problem (N ST ) arises from various considerations. In a general sense, we are interested in the use of variational methods for boundary value problems involving mixed (elliptic-hyperbolic) type partial differential equations. Our motivation is twofold. On the one hand, there are interesting physical problems such as transonic potential flow past profiles which are modeled by nonlinear mixed type boundary value problems which admit variational characterizations (cf. section 4 of [4]). On the other hand, global variational treatments of mixed type problems would improve the basic understanding as to why solutions to such problems should exist at all. Even for linear problems, the techniques employed often involve pasting together solutions found independently in elliptic and hyperbolic regions; with notable exceptions such as works based upon the positive symmetric systems technique of Friedrichs (cf. [11], [18]). Variational tools would provide another approach which is independent of type, with some added ability to interpret the results. The results presented here can be thought of a first step towards understanding the obstructions to variational formulations for mixed type problems and proposing a possible method for their resolution and represent the first variational treatment of a nonlinear Tricomi problem. More precisely, we consider a mixed type problem whose linear part is the most well understood, the seminal linear Tricomi problem [24] 2



Tu = f u=0

in Ω on AC ∪ σ ,

(LT )

and proceed to add to it the mildest kind of nonlinear structure, namely a semilinear term. The problem (LT ) has long been connected to the problem of transonic nozzle flow through the pioneering work of Frankl’ [10], in which, through + the T hodograph transformation, the elliptic behavior of the operator in Ω = Ω {(x, y) | y > 0} T describes the subsonic part of the flow, while the hyperbolic behavior in Ω− = Ω {(x, y) | y < 0} describes the supersonic flow. The term Tricomi problem refers to the placement of the boundary data on only the portion AC ∪ σ of the boundary; such a boundary condition is chosen because the presence of a hyperbolic region will overdetermine the problem for classical solutions if one attempts to place data on the entire boundary (cf. [2], for a maximum principle argument). More precisely, if one considers any portion of the boundary Γ ⊃ AC ∪ σ then the problem of finding classical solutions u ∈ C 2 (Ω) ∩ C 0 (Ω) to (LT) is overdetermined, that is n T u = 0 in Ω ⇒ u ≡ 0 on Ω. u = 0 on Γ On the other hand, under some restrictions on σ, there is a wealth of results on the linear Tricomi problem (LT ), including the existence of unique strong solutions in Hilbert spaces well adapted to this boundary condition. However, it should be noted that despite some 70 years of study, basic information remains unknown; for example, to our knowledge, the only established spectral result is the existence of one positive eigenvalue for the Tricomi operator supplemented with the Tricomi boundary condition (cf. [12] ). The main difficulty in using variational methods for semilinear Tricomi problems is a manifest asymmetry in the operator T that results from placing the boundary conditions on only a portion of the boundary. In cleanest terms, T does 1 into its dual, but rather into the dual of the adjoint problem not map WAC∪σ ∗ (LT ) , in which vanishing data is placed on BC ∪ σ. Our approach involves symmetrizing the linear operator T by first assuming that Ω is symmetric and then by composing T with the reflection R, which induces an isometric isomorphism be−1 1 tween the adjoint boundary spaces. In this way, RT does map WAC∪σ into WAC∪σ , and hence (N ST ) will admit a variational structure. One could consider a direct variational approach to the symmetrized problem, but this remains problematic since crucial information on the linear Tricomi operator remains unavailable for the study of the direct functional. The use of dual variational methods allows us to take advantage of the compactness of the inverse of the linear operator. In fact, the linear operator RT does possess a priori estimates with the loss of one derivative and hence it admits an inverse (RT )−1 which is compact on L2 (Ω). Furthermore, given the present knowledge of the linear operator, the necessity of obtaining the continuity of the Nemistkii operator associated to the nonlinearity constrains one to consider nonlinearities with at most an asymptotically linear 3

growth. In order to treat superlinear growth cases, one would need an appropriate Lp theory for the linear Tricomi operator, which is not present in the literature. A few additional remarks on the kind of boundary conditions one could consider can be found in [15]. Finally, while we cannot say that the presence of the nonlocal effect in (N ST ) results directly from physical reasoning for example, there are reasons to believe that the problem is sound. Not only does it possess a variational structure, but it is possible that the corresponding problem without the reflection possesses only the trivial solution. This is, in fact, the case for sublinear increasing nonlinearities that are C 1 as follows from the uniqueness theorems of [21] (our nonlinearities need not to be Lipschitz). These considerations are analogous with the problem of finding nontrivial solutions for a nonlocal semilinear O.D.E. problem whose linear part is the simplest second order ordinary differential operator, which we consider on a bounded interval with homogeneous Cauchy conditions at one endpoint. This non self-adjoint problem can be symmetrized by composition with a reflection operator and the same methods apply in the same way. The analog suggests the robustness of the phenomena described and allows for additional comparative remarks on the nonlocal (with R) versus nonlocal (without R) forms of the problem in terms of uniqueness of the trivial solution.

2

The linear results

In this section, we recall the main results for the linear problem (LT ) and its adjoint problem (LT )∗ . The main tools for treating such problems are the classical (a, b, c)integral method of Friedrichs and the theory of spaces of positive and negative norms in the sense of Leray and Lax, as developed by Berezanskii [3] and Didenko [8]. Denoting by TAC and TBC the unique continuous extensions of T relative to ∞ ∞ the dense subspaces CAC∪σ (Ω) and CBC∪σ (Ω) respectively, one gets the continuity estimate 1 1 kTAC ukW −1 ≤ C1 kukWAC∪σ , u ∈ WAC∪σ , (2.1) BC∪σ

with an analogous result for TBC . In order to obtain solvability results for the problem (LT ) and its adjoint problem (LT )∗ , in which the formal transpose T 0 of T is again T and the boundary conditions are placed on BC ∪ σ, one exploits the fact that a priori estimates with the loss of one derivative are often possible to establish for such a mixed type differential operator. We encode this principle into the following definition. Definition 2.1. A Tricomi domain Ω will be called admissible if there exist positive constants C2 and C3 such that kukL2 ≤ C2 kTAC ukW −1

BC∪σ

4

,

1 u ∈ WAC∪σ

(2.2)

and kvkL2 ≤ C3 kTBC vkW −1

,

AC∪σ

1 v ∈ WBC∪σ .

(2.3)

−1 −1 Such L2 − WAC∪σ and L2 − WBC∪σ a priori estimates were first obtained by Didenko [8]. Moreover, it is shown that it is possible to choose admissible domains which are symmetric with respect to the y-axis and that the admissibility holds for any piecewise C 2 graph σ that obeys an explicit bound on ||σ||C 1 (cf. [15]). Standard functional analysis then gives the following solvability result.

Proposition 2.2. (cf. [8], [15]) Let Ω be an admissible Tricomi domain. Then for 1 every f ∈ L2 (Ω) there exists a unique strong solution u ∈ WAC∪σ to the problem ∞ (LT ) in the following sense: there exists a sequence {uj } ⊂ CAC∪σ (Ω) such that 1 lim kuj − ukWAC∪σ = 0 and

j→∞

lim kT uj − f kW −1

j→∞

= 0.

BC∪σ

An analogous statement holds for the adjoint problem (LT )∗ . For an admissible Tricomi domain, one may then define 1 W = {u ∈ WAC∪σ : TAC u ∈ L2 (Ω)}

(2.4)

which is a dense subspace of L2 (Ω) as it contains C0∞ (Ω). One easily verifies that TAC |W admits a continuous left inverse −1 1 TAC : L2 (Ω) → W ⊂ WAC∪σ ,→ L2 (Ω), −1 TAC

(2.5)

2

such that is a compact operator on L (Ω). It is this compactness property on L2 (Ω) which suggests the use of a dual variational method for the nonlinear problem. However, for a generic Tricomi domain, an application of the divergence theorem yields the fundamental identity R (T u, v)L2 = Ω (yux vx + uy vy ) dxdy (2.6) ∞ ∞ = (u, T v)L2 , u ∈ CAC∪σ (Ω), v ∈ CBC∪σ (Ω). This identity clearly demonstrates a manifest asymmetry that results from the imposition of the boundary conditions on only a portion of the boundary. The closest statements to TAC being symmetric that one can make here are hTAC u, viBC = hu, TBC viAC ,

1 1 u ∈ WAC∪σ , v ∈ WBC∪σ ,

(2.7)

or 2 2 u ∈ WAC∪σ , v ∈ WBC∪σ ,

(TAC u, v)L2 = (u, TBC v)L2 ,

(2.8)

2 2 ∞ ∞ where WAC∪σ and WBC∪σ are W 2,2 (Ω) norm closures of CAC∪σ (Ω) and CBC∪σ (Ω) respectively (cf. [9] for (2.7) and [3] for (2.8)).

5

In order to circumnavigate this asymmetry, we will consider from now on symmetric admissible Tricomi domains. On such symmetric domains, we will introduce a reflection operator in the obvious way which effectively symmetrizes the Tricomi operator TAC for use in the variational method. We consider the linear map Φ : R2 → R2 with Φ(x, y) = (−x, y), and the induced operator R : L2 (Ω) → L2 (Ω) with Ru = u ◦ Φ.

(2.9)

Clearly, R is a norm preserving, self-adjoint automorphism on L2 (Ω) such that R2 = Id. Moreover, by restriction, R establishes an isometric isomorphism between 1 1 WAC∪σ and WBC∪σ ; thus one can easily verify the following proposition. Proposition 2.3. (cf. [15]) Let Ω be a symmetric admissible Tricomi domain and let R be the reflection operator defined by (2.9). Then the operator gotten by composing TAC with R 1 RTAC : W ⊂ WAC∪σ → L2 (Ω),

(2.10)

1 D(TAC ) = W = {w ∈ WAC∪σ : TAC w ∈ L2 (Ω)}.

(2.11)

where satisfies the following properties. (a) RTAC is a closed, densely defined operator which admits a continuous left inverse 1 (RTAC )−1 : L2 (Ω) → W ⊂ WAC∪σ . (b) (RTAC )−1 : L2 (Ω) → L2 (Ω) is a compact operator (c) RTAC is symmetric in the sense that (RTAC u, v)L2 = (u, RTAC v)L2 ,

u, v ∈ W = D(RTAC ).

(2.12)

Since K = (RTAC )−1 : L2 (Ω) → L2 (Ω) is a compact linear operator which is injective, non-surjective, and self-adjoint, one knows that the spectrum of K consists of {0} ∪ {µj }, where µj is a sequence of eigenvalues, of finite multiplicity, whose only possible accumulation point is zero. We denote by µ+ j the positive eigenvalues of K, written in non-increasing order, counting their multiplicities, and similarly we denote by µ− j the negative eigenvalues written in non-decreasing order, counting their multiplicity. The operator K is not positive definite on L2 (Ω) and hence may admit both positive and negative eigenvalues. In fact, there are infinite number of each sign, as one can show using the variational characterization of eigenvalues (cf. [7] and [17]). Proposition 2.4. (cf. [16]) If Ω is a symmetric admissible Tricomi domain, then the inverse of the reflected Tricomi operator K = (RTAC )−1 has an infinite number of positive eigenvalues and an infinite number of negative eigenvalues.

6

We conclude this section with a few additional observations which follow from 2 straightforward considerations. We will denote by e± j , j ∈ N, the L eigenvectors of (RTAC )−1 with L2 norm equal to one that are associated to the eigenvalues µ± j , 1 where, in fact, e± ∈ W ⊂ W . A AC∪σ j Corollary 2.5. If Ω is a symmetric admissible Tricomi domain, then the reflected 1 Tricomi operator RTAC : WA ⊂ WAC∪σ → L2 (Ω) admits infinitely many positive ± ± −1 and negative eigenvalues λj = (µj ) with associated eigenfunctions {e± j }j∈N ± where λj → ±∞ as j → +∞. 1 1,2 Remark 2.6. The eigenfunctions e± (Ω) must lie in Lq (Ω) for j ∈ WAC∪σ ⊂ W all q ∈ [1, +∞) by the Sobolev imbedding theorem. In addition, since L2 (Ω) is separable and K is self-adjoint, {e± j }j∈N forms a complete orthonormal basis of 2 L (Ω). We will denote by σ((RTAC )−1 ) the spectrum of (RTAC )−1 .

Remark 2.7. In all that follows, Ω will be a symmetric admissible Tricomi domain so that all of the results of section 2 will apply.

3

A sketch of the dual variational formulation

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. Then 1 ∃u0 ∈ WAC∪σ such that u0 6= 0 and TAC u0 = Rf (u0 ) in L2 (Ω)

(3.1)

is equivalent to 1 ∃u0 ∈ WAC∪σ such that u0 6= 0 and RTAC u0 = f (u0 ) in L2 (Ω).

(3.2)

Hence if we are able to find a v0 ∈ L2 (Ω) such that v0 6= 0 and g(v0 ) = Kv0 ,

(3.3)

where K = (RTAC )−1 , then u0 = Kv0 is a solution of our problem. Indeed f (g(v0 )) = f (Kv0 ) ⇔ v0 = f (u0 ) ⇔ RTAC u0 = K −1 u0 = f (u0 ). Therefore our goal will be to set up a vartiational formulation of the dual problem. To this end, we define J : L2 (Ω) → R as Z Z 1 J(v) = G(v) dxdy − vKv dxdy, 2 Ω Ω Rv where G(v) = 0 g(t)dt denotes the primitive of g such that G(0) = 0. Then it is easy to check that if g satisfies the growth condition (cf. [1]): |g(t)| ≤ K1 + K2 |t|, then 7

(3.4)

• J ∈ C 1 (L2 (Ω), R) R R • J 0 (v)[w] = Ω g(v)w dxdy − Ω wKv dxdy and hence critical points of J will be weak solutions of the dual problem. Let us remark explicitly that the key point in proving the second statement is the symmetry of K.

4

The asymptotically linear case

In this section, we will consider two different situations. First we suppose that the nonlinearity is asymptotically linear at infinity, but sublinear at zero, while in the second case we suppose that the nonlinearity is asymptotically linear (with different slope) both at zero and at infinity. In both of these cases we will be able to show the existence of a nontrivial solution of (N ST ), which in the first case will be the preimage of a mountain pass critical point of the dual functional, while the second one will be the preimage of a linking critical point. Let us start with the first case. Let Ω be a symmetric admissible Tricomi domain and suppose that the nonlinear term in (N ST ) satisfies the following set of hypotheses. (f1) f ∈ C 0 (R, R) and is strictly increasing Furthermore, suppose that the nonlinearity f is written as f (s) = s/a + f∞ (s), where a is a real number such that (f2) a ∈ / σ((RTAC )−1 ) and 0 < a < µ+ 1 , and sf∞ (s) < 0 for s 6= 0, where the perturbation f∞ ∈ C 0 (R, R) satisfies (f3) |f∞ | ≤ c1 + c2 |s|p , with c1 ≥ 0, c2 > 0 and 0 < p < 1, (f4) |f∞ (s)| ≥ c3 |s|p , with c3 > 0. Our main result is the following theorem. Theorem 4.1. Let Ω be a symmetric admissible Tricomi domain and assume that f satisfies (f 1), (f 2), (f 3) and (f 4). Then (N ST ) admits at least one nontrivial generalized solution. To begin, we note that by using the hypotheses (f i), i = 1, . . . , 4 one can check that the continuous inverse g = f −1 satisfies: (g1) g ∈ C 0 (R, R) and is strictly increasing, g(t)t > 0 for t 6= 0; (g2) g(t) = at + g∞ (t);

8

and the perturbation g∞ ∈ C 0 (R, R) satisfies (g3) |g∞ (t)| ≤ C1 + C2 |t|p , with C1 ≥ 0, C2 > 0, (g4) |g∞ (t)| ≥ C3 |t|p , with C3 > 0. Now, let us remark that (g2) and (g3) imply that g satisfies |g(t)| ≤ C1 + C2 |t| t ∈ R with C1 ≥ 0, C2 > 0,

(4.1)

and hence J ∈ C 1 (L2 , R). + 2 Denote by Vj = span{e+ 1 , . . . , ej } and note that, by Remark 2.8, every v ∈ L ⊥ can be written as v = z + w where z ∈ Vj and w ∈ Vj ; furthermore one has Z 2 − wKw dxdy ≥ −µ+ (4.2) j+1 kwkL2 , Ω

−

Z

2 zKz dxdy ≥ −µ+ 1 kzkL2 ,

(4.3)

2 zKz dxdy ≤ −µ+ j kzkL2 .

(4.4)

Ω

and −

Z Ω

Lemma 4.2. Let Ω be a symmetric admissible Tricomi domain and assume that g satisfies (g1), (g2) and (g3). Then J satisfies the Palais-Smale condition on L2 (Ω). Proof. (cf. [16], Lemma 4.3). One can show that: • a∈ / σ(K) implies that any Palais-Smale sequence is bounded; • the continuity of the Nemistkii operator f : L2 → L2 implies that the PS sequence must converge. The following lemma provides a geometrical structure suitable for constructing a mountain pass critical point for the dual functional J. Lemma 4.3. Let Ω be a symmetric admissible Tricomi domain and assume that g satisfies (g1), (g2), (g3) and (g4). Then there exists a ρ > 0 and an α > 0 such that J(v) ≥ α for every v ∈ L2 with kvk = ρ. Furthermore, J(te+ 1 ) → −∞ for t → +∞. Proof. (cf [16], Lemma 4.2) By (f 2), there exists a k such that µk+1 < a < µk . Then the decomposition L2 = Vk ⊕Vk⊥ and the inequality (4.2) imply the existence of a positive constant C > 0 such that Z Z Z a 1 p+1 2 J(v) ≥ C |v| dxdy + v dxdy − vKv dxdy 2 Ω 2 Ω Ω 9

≥ C

Z

p+1

|z + w| Z a − µ+ 1 z 2 dxdy. 2 Ω Ω

+

a − µ+ k+1 dxdy + 2

Z

w2 dxdy +

(4.5)

Ω

We want to show that there exist ρ > 0 and α > 0 such that, J(v) ≥ α > 0, for every v ∈ L2 such that kvk = ρ. We note that in (4.5) the last term is negative, and hence we want to show that, near zero, the positive “subquadratic” term (given by an Lp+1 -norm to the power p + 1 with p < 1) dominates the negative “quadratic” term (given by the square of an L2 norm); an argument by contradiction can be given. On the other hand, by (g2) and (g3), one has for t > 0, a 2 t2 p+1 |e+ | dxdy + t − 1 2 2 Ω + a − µ 1 2 t , ≤ C5 + C6 tp+1 + 2

p+1 J(te+ 1 ) ≤ C5 + C4 t

Z

Z

+ e+ 1 Ke1 dxdy

Ω

+ and hence J(te+ 1 ) → −∞ for t → +∞ since p + 1 < 2 and a − µ1 < 0.

Proof of Theorem 4.1. If f satisfies (f 1), (f 2), (f 3) and (f 4) then g = f −1 satisfies (g1), (g2), (g3) and (g4), thus the functional J will admit, by Lemmas 4.2 and 4.3, a mountain pass critical point v0 such that J(v0 ) > 0 and hence v0 6= 0 in L2 . Such critical points of J are weak solutions to the dual equation and 1 u0 = Kv0 ∈ WAC∪σ is a nontrivial generalized solution to (N ST ). We consider now the case in which the nonlinearity f is asymptotically linear both at zero and at infinity, with two different slopes. More precisely, we suppose that the nonlinear term in (N ST ) satisfies the following set of hypotheses: (f 1) and it is possible to write f (s) = s/a + f∞ (s) = s/b + f0 (s), where + (f2)0 the numbers a and b such that a ∈ / σ((RTAC )−1 ) and µ+ j+1 < b < µj ,

and the perturbations f∞ , f0 ∈ C 0 (R, R) satisfy (f3)0 f∞ is bounded and lims→0 f0 (s)/s = 0, (f4)0 −s2 /b ≤ sf0 (s) ≤ 0, s ∈ R. We obtain the following Theorem. Theorem 4.4. Let Ω be a symmetric admissible Tricomi domain and assume that f satisfies (f 1), (f 2)0 , (f 3)0 , and (f 4)0 . Then, if a > µ+ j , the problem (N ST ) admits at least one nontrivial solution. 10

The solution is found as the preimage of a linking critical point (cf. [16]) of the dual action functional J. To show that there is a nontrivial linking critical point, one needs J to satisfy an appropriate geometrical situation (cf. Lemma 4.5) and the Palais Smale compactness condition. We remark that the totality of conditions imposed above will require that a ≥ b, and that the compactness condition is given in Lemma 4.3. However, the linking structure here (Lemma 4.5) does exploit a > µ+ j > b. To begin with, one can check that the hypotheses (f 1), (f i)0 , i = 2, 3, 4 imply that the inverse g = f −1 satisfies (g1)0 g ∈ C 0 (R, R), g is invertible g(t) = at + g∞ (t) = bt + g0 (t), where a and b are as in (f 2)0 and the perturbations g0 and g∞ belong to C 0 (R, R) and satisfy (g2)0 g∞ is bounded and limt→0 g0 (t)/t = 0, (g3)0 tg0 (t) ≥ 0, t ∈ R. In this situation, the dual action functional J can be decomposed, depending on the situation, into Z

Z Z 1 b 2 v dxdy − vKv dxdy J(v) = G0 (v) dxdy + 2 Ω 2 Ω ZΩ Z Z a 1 = G∞ (v) dxdy + v 2 dxdy − vKv dxdy, 2 Ω 2 Ω Ω where G0 and G∞ satisfy, for every t ∈ R, |G∞ (t)| ≤ M |t| and 0 ≤ G0 ≤

1 (a − b)2 t2 + M |t|. 2

(4.6)

Denoting by ∂Bρ (Vj ) = {v ∈ Vj | kvk = ρ}

(4.7)

and by + + ⊥ ∂QR (Vj⊥ ⊕Re+ j ) = {w +αej | α > 0 and kw +αej k = R}∪{w ∈ Vj | kwk ≤ R},

the following lemma provides a geometrical structure suitable for constructing a linking for the dual functional J. Lemma 4.5. Let Ω be a symmetric admissible Tricomi domain and assume that g satisfies (g1)0 , (g2)0 , and (g3)0 . If a > µ+ j then there exist ρ > 0 and R > ρ such that the following inequality holds sup J(v) < ∂Bρ (Vj )

inf

∂QR (Vj⊥ ⊕Re+ ) j

11

J(v).

(4.8)

Proof. (cf. [16], Lemma 4.5.) The inequality (4.4) applied to arbitrary v ∈ Vj yields Z Z Z b 1 2 J(v) = v dxdy + G0 (v) dxdy − vKv dxdy (4.9) 2 Ω 2 Ω Ω Z 1 2 ≤ (b − µ+ G0 (v) dxdy.. j )kvk + 2 Ω Since it is possible to show that lim

kvk→0 v∈Vj

R

Ω

G0 (v) dxdy = 0, kvk2

we obtain the claim that there exists a ρ > 0 such that sup J(v) < 0.

(4.10)

∂Bρ (Vj )

On the other hand, to estimate the inf from below, we begin by observing that the inequality (4.2) and the lower bound in (4.6) yield, for arbitrary w ∈ Vj⊥ Z Z b 1 1 + 2 2 J(w) ≥ G0 (w) dxdy + w2 dxdy − µ+ j+1 kwk ≥ (b − µj+1 )kwk , (4.11) 2 2 2 Ω Ω ⊥ which is non negative for kwk ≤ R. Furthermore, for arbitrary w + αe+ j ∈ Vj ⊕ Re+ j , the inequalities (4.2), (4.3) together with the upper bound on G∞ from 2 2 2 (4.6) yields, setting R2 = kw + αe+ j k = kwk + α , taking into account that + + a > µj ≥ µj+1 ,

1 2 1/2 (a − µ+ R. (4.12) j )R − M |Ω| 2 It is clear that it is possible to choose R sufficiently large to make the right hand side of (4.12) strictly positive, which together with (4.10) and (4.11) completes the lemma. J(w + αe+ j )≥

Proof of Theorem 4.4. If f satisfies (f 1), (f 2)0 , (f 3)0 and (f 4)0 then g = f −1 satisfies (g1), (g2)0 , and (g3)0 , thus the functional J will admit, by Lemma 4.5, a linking geometrical structure. Furthermore, the (P S) condition holds, and hence there will exist a nontrivial critical point v0 6= 0. In fact, the Linking Theorem provides two distinct critical levels (cf. [14]) and thus a nontrivial one; hence 1 u0 = Kv0 ∈ WAC∪σ is a nontrivial generalized solution to (N ST ).

5

The sublinear case

We suppose that the nonlinearity f satisfies (f 1) and 12

(f2)00 |f (s)| ≤ c1 + c2 |s|p , s ∈ R, for some p ∈ (0, 1), with c1 ≥ 0, c2 > 0, (f3)00 |f (s)| ≥ c3 |s|p − c4 , s ∈ R for some p ∈ (0, 1), with c3 ≥ 0, c4 > 0. Remark 5.1. The hypothesis (f 2)00 implies that f induces a continuous Nemitski operator from Lp+1 (Ω) into L(p+1)/p (Ω) for p ∈ (0, 1). In this situation the dual action functional will be J : L(p+1)/p (Ω) → R defined by Z Z 1 J(v) = G(v) dxdy − vKv dxdy, (5.1) 2 Ω Ω which is C 1 and whose critical points are weak solutions to the dual equation (3.3). In this case, the operator K is given by K = j ◦ (RTAC )−1 ◦ i : L(p+1)/p (Ω) → Lp+1 (Ω), where i and j are the inclusion maps j

i

L(p+1)/p (Ω) ,→ L2 (Ω) ,→ Lp+1 (Ω) which are well defined and continuous since p+1 < 2 < (p+1)/p and Ω is bounded. The hypotheses on the nonlinearity f imply that the inverse function g satisfies: (g1), (g2)00 |g(t)| ≤ C1 + C2 |t|1/p , t ∈ R with C1 ≥ 0, C2 > 0, (g3)00 |g(t)| ≥ C3 |t|1/p − C4 , t ∈ R with C3 > 0, C4 ≥ 0. Utilizing (g1), (g2)00 and (g3)00 , it is easy to see that J is still weakly lower semicontinuous and coercive; thus, by a classical argument (cf. [23]), J is bounded from below and attains its minimum. The existence of a nontrivial generalized solution of (N ST ) can be shown since it is possible to construct a function v ∗ such that J(v ∗ ) < 0 and hence the minimum must be nontrivial. Theorem 5.2. Let Ω be a symmetric admissible Tricomi domain and assume that f satisfies (f 1), (f 2)00 and (f 3)00 . Then (N ST ) admits a generalized nontrivial 1 . solution WAC∪σ On the other hand, if one also utilizes the eigenvalue properties of section 2 together with a nonlinearity f that satisfies (in addition to (f 1), (f 2)00 , (f 3)00 ) the hypothesis (f4)00 f is odd, f ∈ C 1 (R \ {0}) and lims→0 f 0 (s) = +∞, a much stronger result is true.

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Theorem 5.3. Let Ω be a symmetric admissible Tricomi domain and assume that f satisfies (f 1), (f 2)00 , (f 3)00 and (f 4)00 . Then (N ST ) admits infinitely many 1 generalized solutions in WAC∪σ . Hypothesis (f 4)00 implies that the inverse g will also satisfy (g4)00 g is odd, g ∈ C 1 (R) and g 0 (0) = 0. In this case, one has • J ∈ C 2 (L(p+1)/p (Ω)(Ω), R) R • J 00 (0)[w1 ][w2 ] = − Ω w1 Kw2 dxdy i.e. J(v)

= J(0) + J 0 (0)[v] + J 00 (0)[v][v] + o(kvk2(p+1)/p) ) Z = − vKv dxdy + o(kvk2(p+1)/p) )

(5.2)

Ω

Example 5.4. The function f (s) = sign(s)|s|p with p ∈ (0, 1) satisfies (f 1), (f 2)00 , (f 3)00 and (f 4)00 . It is clear that the growth condition will imply that the functional J will still satisfy Lemma 4.2 on L(p+1)/p (Ω). Proof of Theorem 5.3. J is an even functional, and thus we may apply Theorem 8 of [6] to get our result. More precisely, denote by S(L(p+1)/p (Ω)) the set of A ⊂ L(p+1)/p (Ω) \ {0} which are symmetric with respect to the origin, and let Γm = {A | A closed, A ∈ S(L(p+1)/p (Ω)), γ(A) ≥ m}, where γ(A) denotes the Krasnoselski genus (cf. [23] or [6] for definition and properties), and finally denote by cm = inf

A∈Γm

sup J(v). A

¿From −∞ < cm < 0, it follows that cm is a critical level, and hence it suffices to produce, for every m ∈ N, a set A ∈ Γm such that J(v) < 0 for every v ∈ A. + To this aim, let us consider Vm = span{e+ 1 , . . . , em }, where, as in section 1, + 2 −1 ej are L eigenvectors of (RTAC ) associated to the positive eigenvalues µ+ 1 ≥ (p+1)/p + . These eigenvectors belong to L (Ω) by Remark 2.6. Consider µ+ ≥ . . . ≥ µ m 2 the (m − 1)-dimensional norm spheres Am, = {v ∈ Vm | kvk(p+1)/p = }; it is known that γ(Am, ) = m. Then, by (5.2) one gets, for every v ∈ Am, , 2 2 J(v) ≤ −Cµ+ m kvk(p+1)/p + o(kvk(p+1)/p ),

(5.3)

where the constant C > 0 comes from the fact that Vm ⊂ L(p+1)/p (Ω) is finite dimensional (all norms being equivalent on finite dimensional spaces). By choosing 14

 small enough, we get the desired result for A = Am, . Note that if cm = · · · = cm+k for some k ≥ 1, we still get infinitely many distinct critical points. In fact, in such a case, denoting by Kcm the set of critical points at level cm < 0, one has that 0 ∈ / Kcm and furthermore, since J is even, Kcm ∈ S(L(p+1)/p (Ω)). Hence, properties of the Krasnoselski genus give γ(Kcm ) ≥ 2, which provides infinitely many distinct critical points at level cm . The infinitely many distinct critical points vj ∈ L(p+1)/p (Ω) which result give 1 rise to infinitely many non trivial generalized solutions uj ∈ WAC∪σ , where uj is found as in the proof of Proposition 3.4 of [15].

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An ODE analog

In this section we consider an analogous nonlocal semilinear problem for a second order ordinary differential operator with Cauchy conditions whose manifest asymmetry can be resolved by composition with a reflection operator which results in a variational structure amenable to the very same dual variational methods we have used for (N ST ). Moreover, in this simplified setting some additional remarks concerning the role of the reflection operator with respect to existence of nontrivial solutions follow easily. We consider the question of finding nontrivial solutions u = u(x) to the problem  −u00 = Rf (u) on I = (−a, a) (N SO) u(−a) = u0 (−a) = 0 where R is the reflection operator about 0 in I and f will be a continuous strictly increasing nonlinearity with f (0) = 0. Just as in the case of (N ST ), u ≡ 0 will always be a solution of (N SO). Moreover, the operator T = −d2 /dx2 will fail to be symmetric on natural subspaces of L2 (I) associated to the boundary conditions; ∞ 2 (I) = {u ∈ C ∞ (I) : defined as the W 2,2 (I)-norm closure of C−a for example, W−a 0 u(−a) = u (−a) = 0}. One can easily prove (cf. [16]) the obvious analogs of Theorems 4.1, 4.2, 5.2, and 5.3 for the problem (N SO) by following the established lines used for the problem (N ST ). The only differences being that: 1) the solutions have some 2 added a priori regularity (they lie in W−a ) that results from the ellipticity of 2 2 T = −d /dx , 2) the spectrum of the linear part RT−a consists only of positive eigenvalues, and 3) the corresponding eigenfunctions also have higher regularity. On the other hand, if one considers the problem (N SO) without the reflection R, that is  −u00 = f (u) on I = (−a, a) (SO) u(−a) = u0 (−a) = 0, one can ask if there are still nontrivial solutions. Here it is important to note that the problems (N SO) and (SO) are not equivalent, since one cannot say that Rf = fˆ for some function fˆ; that is, Rf (u(x)) = f (u(−x)) = fˆ(u(x))? For

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f ∈ Lip(R), one certainly has the uniqueness of the trivial solution u ≡ 0 for (SO), but, for example, f (s) = sign(s)|s|p with p ∈ (0, 1) fails to be Lipschitz near the origin, and hence (SO) may admit the Peano phenomenon of an infinite number of solutions. However, the problem (SO) can be transformed into an initial value problem for a Hamiltonian system for which there is an isolated equilibrium at the origin in the phase plane, and hence one obtains uniqueness of the trivial solution for the unreflected problem (SO) (cf. [16], Prop. 5.1.). Therefore, one can say that the presence of the nonlocal effect in (N SO), as represented by R, not only yields a variational structure but allows for nontrivial solutions as well. For the problem (N ST ), similar uniqueness considerations may well hold, although the lack of regularity in the nonlinearity f does not allow one to apply known results such as those of [13] and [21] to conclude that the unreflected problem has the unique trivial solution.

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[11] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 338-418. [12] N. N. Gadai, Existence of a spectrum for Tricomi’s operator, Differential Equations 17 (1981), 20-25. [13] D. K. Gvazava, On uniqueness of solution of the Tricomi problem for a class of nonlinear equations, Soviet Math. Dokl. 11 (1970), 65-69. [14] D. Lupo and A. M. Micheletti, Multiple solutions for Hamiltonian systems via limit relative category, J. Comp. Appl. Math. 52 (1994), 325-335. [15] D. Lupo and K. R. Payne, A dual variational approach to a class of nonlocal semilinear Tricomi problems, NoDEA Nonlinear Differential Equations Appl., to appear. [16] D. Lupo, A. M. Micheletti and K. R. Payne, Existence of eigenvalues for reflected Tricomi operators and applications to multiplicity of sultions for sublinear and asymptotically linear nonlocal Tricomi problems, Advances in Diff. Equations, to appear. [17] A. Manes and A. M. Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7(1973), 285-301. [18] C. S. Morawetz, Non-existence of transonic flow past a profile I, II, Comm. Pure Appl. Math. 9 (1956), 45-68, Comm. Pure Appl. Math. 10 (1957), 107-131. [19] C. S. Morawetz, The Dirichlet problem for the Tricomi equation, Comm. Pure Appl. Math. 23 (1970), 587-601. [20] K. R. Payne, Interior regularity for the Dirichlet problem for the Tricomi equation, J. Math. Anal. Appl. 199 (1996), 271-292. [21] J. M. Rassias, On three new uniqueness theorems of the Tricomi problem for nonlinear mixed type equations, in “Mixed Type Equations”, TeubnerTexte Math., Vol. 90, Leipzig, 1986, 269-279. [22] C. Rebelo, Periodic solutions of nonautonomous planar systems via the Poincar´e-Birkhoff theorem, Ph.D. Dissertation, Faculade de Ciˆencias da Universidade Cl´ assica de Lisboa, Lisboa, 1996. [23] M. Struwe, “Variational Methods”, Springer Verlag, Berlin, 1990. [24] F. G. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine, di tipo misto, Atti Acad. Naz. Lincei Mem. Cl. Fis. Mat. Nat. (5) 14 (1923), 134-247.

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