Quasi-linear equations in Bbb R^N perturbation results

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS BENEDETTA PELLACCI1 Abstract. This paper deals with existence results for the quasi–linear elliptic problem

(

−div((I + εA(x, u))∇u) + u u ∈ H 1 (RN )

+εH(x, u, ∇u) = |u|p−1 u, in ∩W 2,q (RN ), q > N

RN ,

where 2 < p < (N + 2)/(N − 2), N > 2 and the operator −div((I + εA(x, u))∇u) +εH(x, u, ∇u) will be considered as a perturbation of the Laplacian. We use a perturbation method recently developed in [1], [2], [3] and we get results both in the variational and in the non–variational framework.

1. introduction In this paper we study a class of nonlinear elliptic equations in RN of the form ( −div((I + εA(x, u))∇u) + u + εH(x, u, ∇u) = |u|p−1 u, in RN , (Pε ) u(x) ∈ H 1 (RN ) ∩ W 2,q (RN ) q > N, with 2 < p < (N + 2)/(N − 2), N > 2 and ε is a small parameter so that (Pε ) is a perturbation of the problem ( −∆u + u = |u|p−1 u, in RN , (P0 ) u(x) ∈ H 1 (RN ) ∩ W 2,q (RN ) q > N. A(x, s) = (aij (x, s)), aij (x, s) : RN × R → R is a matrix of class C 2 , with respect of the variables (x, s). Moreover, we assume that for every compact set C ⊆ R there exists a positive constant β = β(C) such that for every s in C, for almost every x in RN and for every i = 1, . . . , N |A(x, s)| ≤ β,

(A0 ) (A1 ) (A2 )

|A0s (x, s)| ≤ β, |A00s (x, s)| ≤ β,

|∂i A(x, s)| ≤ β,

|∂i A0s (x, s)| = |∂s ∂i A(x, s)| ≤ β.

where ∂i A(x, s) is the matrix whose components are the partial derivative of the matrix A(x, s) with respect to the variable xi . Regarding the function 1

Partially supported by Cofin. National Project “Variational methods and nonlinear differential equations”. 1

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BENEDETTA PELLACCI

H(x, s, ξ) : RN × R × RN → R we assume that H is of class C 1 with respect to the variables (s, ξ) for almost every x in RN . In addition, we suppose that there exist a positive constant C0 and functions mi , di : RN → R+ , i = 1, . . . , 3 such that ( mi ∈ Lq (RN ) ∩ L2 (RN ), q > N, (1.1) di (x) ∈ L∞ (RN ) + Lq (RN ) ∩ L2 (RN ), i = 1, 2, 3. satisfying for every s ∈ R, for every ξ ∈ RN and a.e. in RN (H0 )

|H(x, s, ξ)| ≤ C0 [|s|r + m1 (x) + d1 (x)|ξ|t ],

r, t ≥ 1,

(H1 )

|Hs0 (x, s, ξ)| ≤ C0 [|s|r + m2 (x) + d2 (x)|ξ|t ],

r, t ≥ 1,

(H2 )

|Hξ0 (x, s, ξ)| ≤ C0 [|s|r + m3 (x) + d3 (x)|ξ|t ],

r, t ≥ 1,

where Hξ (x, s, ξ) is the vector in RN whose components are the partial derivatives of H(x, s, ξ) with respect to ξi . Notice that the growth conditions on the function H(x, s, ξ) with respect to ξ generate a lack of regularity in the problem. Indeed, consider, for instance, the case in which H = |ξ|2 , then H(x, u, ∇u) ∈ L1 (RN ) even if u ∈ H 1 (RN )∩C 1 (RN ). So that, the weak formulation of (Pε ) can be written only for test functions v in H 1 (RN ) ∩ L∞ (RN ). Moreover, the quadratic dependence of H(x, s, ξ) is in some sense critical. Indeed, even if we consider the simple example H(x, s, ξ) = s(1+|ξ|2 )1+µ , with µ > 0 in [17] it is proved that there can be no solution of problem (Pε ). Neverthless, here we will prove existence results for problem (Pε ) for any superlinear growth of H with respect to ξ, moreover we remark that our theorems apply also for the simple example H(x, s, ξ) = d(|x|)g(s)|ξ|t , where d(|x|) is a monotonous C 1 function that belongs to L2 (RN ) ∩ Lq (RN ) + L∞ (RN ), g ∈ C 1 (R) and it does not change sign for every s ∈ R+ , t is any number greater than one. We will set (Pε ) on H 1 (RN )∩W 2,q (RN ) in order to reformulate it as a fixed point problem. We will apply a perturbation method that has been recently developed in [1], [2], [3] and adapted to the Hilbertian nonvariational framework in [14]. Notice that our problem is not set on an Hilbert space so that in Section 2 we will first adapt the result of [14] in order to deal with operators defined in Banach spaces (Theorem 2.1). By following this approach we will apply the Implicit Function Theorem in order to make a nonlinear finite dimensional reduction. Denote with B(x, s) = I + εA(x, s) and K(x, s, ξ) = εH(x, s, ξ). Quasi– linear problem of this kind have been widely studied in bounded domains in the non–perturbative setting (see [10], [4], [5], [12] and the references therein). In unbounded domains, Problem (Pε ) is studied, in the absence of the term |u|p−1 u, in [13] where the authors assume that B(x, s) is uniformly bounded and elliptic, and the function K(x, s, ξ) has at most a quadratic growth with respect to ξ.

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

3

The nonvariational case with the presence of the non–linearity |u|p−1 u has been treated neither in bounded nor in unbounded domains. It is clear that on unbounded domains Problem (Pε ) becomes more difficult because of the lack of compactness. As a general feature of our approach, we remark that no problem of compactness arises as we work on a finite dimensional manifold near solutions of problem (P0 ). In the non-variational framework our existence results are proved in Theorems 3.5, 3.7, 3.8, 3.9. The variational case on RN is studied in [11] for uniformly bounded and elliptic matrices B(x, s). Our existence results are proved in Theorems 4.1, 4.2, 4.4 (see also Remark 4.3). In Section 5 we will consider perturbation with super– linear growth with respect to the gradient in the divergence operator (see Theorems 5.4, 5.6, Remark 5.5 and Corollary 5.7). Finally, we point out that we do not need to prove any regularity result for the solutions of (Pε ) as we will work in H 1 (RN ) ∩ W 2,q (RN ), with q > N so that any solution we find will be regular. The paper is organized as follows. In section 2 we will adapt the result of [14] in order to deal with operators defined in Banach spaces and following the method developed in [1], [2] we will apply the Implicit Function Theorem in order to reduce the problem to a finite dimensional one. In section 3 we apply the results of section 2 in order to find existence results in the non–variational case (see Theorems 3.5, 3.7, 3.8, 3.9). In section 4 we study the variational framework (see Theorems 4.1, 4.4). Finally in section 5 we will consider perturbation with super–linear growth on the gradient in the divergence operator. Notation In the sequel we will use the following notation: • Lp (RN ), 1 ≤ p ≤ +∞ denotes the standard Lebesgue space over RN and k · kp its norm. • W m,r (RN ) is the usual Sobolev space endowed with the norm: kukm,r = kukr + · · · + k∇r ukr ,

• • • • •

m, r ≥ 1.

The norm in H 1 (RN ) = W 1,2 (RN ) will be denoted with k·k1,2 = k·k. We will denote with (·|·) the inner product of H 1 (RN ). H −1 (RN ) is the dual space of H 1 (RN ) and, for every f ∈ H −1 (RN ), u ∈ H 1 (RN ), we denote with hf, ui the duality pairing between f and u. Br ⊆ RN is the open ball centered in the origin of radius r. L(E, F ) is the set of all the linear and continuous operator between two Banach spaces E and F . {ei } is a vector of the orthonormal basis of RN .

4

BENEDETTA PELLACCI

2. Abstract Setting Let X be a Banach space and let E be an Hilbert space with X ⊆ E. We will deal with existence of solutions of the following functional equation (2.1)

Qε (u) = S(u) + G(ε, u),

where S : X → X satisfies the following hypotheses (S0 ) S ∈ C 1 (X, X), (S1 ) there exists φ : Rd → X, continuous such that S(φ(θ)) = 0, for every θ in Rd , (S2 ) S 0 (φ(θ)) is a compact perturbation of a linear homeomorphism and it is such that dimKerS 0 (φ(θ)) = d. Note that as S 0 (φ(θ)) is a compact perturbation of a linear homeomorphism we have that imS 0 (φ(θ)) is closed and the operator is Fredholm of index 0. From now on we will use the notation Kθ = kerS 0 (φ(θ)) and Rθ = imS 0 (φ(θ)). As φ is continuous and S is C 1 , there exists orthonormal bases (ei (θ))i=1,...,d and (ki (θ))i=1,...,d of Rθ⊥ and Kθ respectively, where Rθ⊥ and Kθ are defined with respect of the inner scalar product of E. (ei (θ))i=1,...,d and (ki (θ))i=1,...,d depend continuously on θ. The operator G : R × X → X satisfies the following hypotheses (G0 ) G(0, u) = 0 for all u ∈ X, (G1 ) G is of class C 1 with respect to u in X, (G2 ) G : R × X → X and G0 : R × X → L(X, X) are continuous. Under the previous assumptions we can use the implicit function Theorem in order to find a smooth map w = w(ε, θ) ∈ Kθ⊥ such that Qε (φ(θ) + w) ∈ Rθ⊥ . More precisely we will prove the following theorem. Theorem 2.1. Assume (S0−2 ) and (G0−2 ), then for all r > 0, there exists εr and a smooth map w : [−εr , εr ] × Br → X such that w = w(ε, θ) ∈ Kθ⊥ and Qε (φ(θ) + w) ∈ Rθ⊥ , Proof. Let H : R × Rd × X × Rd → X × Rd be defined by H1 (ε, θ, w, a)

=

S(φ(θ) + w) + G(ε, φ(θ) + w) −

d P

ai ei (θ),

i=1

H2 ((ε, θ, w, a)

=

((w|k1 (θ)), . . . , (w|kd (θ))).

We want to prove that near the point (0, θ, 0, 0) we can write w = w(ε, θ), and a = a(ε, θ) such that H(ε, θ, w(ε, θ), a(ε, θ)) = 0. Solving the equation H1 = 0 implies that Qε ∈ Rθ⊥ , while solving the equation H2 = 0 means that w ∈ Kθ⊥ . By (S1 ) and (G0 ) we have that H(0, θ, 0, 0) = (0, 0), and thanks to conditions (S0 ) and (G1 ) we get that H is of class C 1 . To apply is invertible. the implicit function Theorem we have to show that ∂H(0,θ,0,0) ∂(w,a)

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

Note that

Then

∂H1 (0,θ,0,0) [v] ∂w ∂H1 (0,θ,0,0) [b] ∂a

∂H(0,θ,0,0) ∂(w,a) [v, b]

5

= S 0 (φ(θ))v, d P = − bi ei (θ). i=1

= (w, c) iff

S 0 (φ(θ))v −

d X

bi ei (θ) = w,

i=1

((v|k1 (θ)), . . . , (v|kd (θ))) = c. Let w = w1 + w2 where w1 ∈ Rθ and w2 ∈ Rθ⊥ . Take b as the unique vector such that bi = (w, ei (θ)) and consider v1 ∈ Kθ⊥ such that S 0 (φ(θ))v1 = w1 . Finally take v2 as the unique vector such that (v2 , ki (θ)) = ci . This proves that ∂H(0,θ,0,0) is bijective. ∂(w,a) By the implicit function Theorem there exists w = w(ε, θ) such that the equation H(ε, θ, w, a) = (0, 0) is satisfied in a neighbourhood of (0, θ0 ) for all θ0 ∈ Rd . Since Br is compact it follows that there exists εr > 0 and w : [−εr , εr ] × Br → X such that Qε (φ(θ) + w(ε, θ)) ∈ Rθ⊥ for all (ε, θ) ∈ [−εr , εr ] × Br . t u Let ε ∈ [−εr , εr ] and define Zεr = {φ(θ) + w(ε, θ) : θ ∈ Br }. Remark 2.2. We want to point out that Theorem 2.1 holds even if there is not an Hilbert space E with X ⊆ E. Indeed as S 0 (φ(θ)) is a Fredholm operator of index zero, Rθ has a topological supplement. So it is possible to use projection operators instead of the scalar product of E. On the other hand the presence of an Hilbert space will be fundamental in order to analize Qε on Zε . From now on it is possible to follow the same argument of [14] in order to analize Qε on Zε . We will need some informations on the dependence of G(ε, s) with respect of ε, more precisely we will suppose that (G3 ) There exists a positive number α > 0 and a continuous function Ψ : Rd → Rd such that (2.2)

lim ε−α (G(ε, φ(θ))|ei (θ)) = (Ψ(θ))i ,

ε→0

uniformly in θ in Br .

Let Θε : Rd → Rd be defined by (Θ(θ))i = (Qε (φ(θ) + w(ε, θ))|ei (θ)) . By the definition of w we get that the zeroes of Θε in Br correspond to the zeroes of Qε in Zεr . Moreover it is possible to prove that Θε (θ) can

6

BENEDETTA PELLACCI

be approximated by the map Ψ(θ) for ε sufficiently small (see for more details [14]). Therefore it holds the following existence result. Theorem 2.3. suppose that S and G satisfy conditions (S0−2 ), (G0−3 ). If there exists a bounded open set Ω ⊆ Rd such that deg(Ψ, Ω) 6= 0, then there exists ε1 such that for all ε, 0 < |ε| < ε1 , the equation (2.1) has at least a solution in ZεΩ . For the proof see [14]. Consider now the case in which the functional equation (2.1) has a variational structure, i.e. the operator Qε is the derivative of a functional fε = f0 + gε , where f00 = S and ∂s gε = G(ε, s). Then there exists also a map Γ(θ) such that Ψi (θ) = ∂i Γ(θ). The counter part of Theorem 2.3 in the variational framework is the following result proved in [1] Theorem 2.4. Let (S0−2 ), (G0−3 ) hold and assume that Γ has a proper local minimum (or maximum) at some point θ0 . Then there exists ε1 such that for all ε, 0 < |ε| < ε1 , the equation (2.1) has at least a solution which corresponds to a critical point of the functional fε . For the proof see [1]. 3. Existence Results: The non–variational case In this section we want to apply the results of the previous section to problem (Pε ). Following the notation of the previous section we consider E = H 1 (RN ) X = W 2,q (RN ) ∩ H 1 (RN ). Consider the operators S, G defined by
(3.1) S(u) = u − K |u|p−1 u (3.2)

G(ε, u) = εK [−div(A(x, u)∇u) + H(x, u, ∇u)] ,

where K is defined by K := (−∆ + 1)−1 . Note that S : X → X; indeed if u ∈ X, then, as 1 < p <
(N + 2)/(N − 2), |u|p belongs to L1 (RN ) ∩ L∞ (RN ), so that K |u|p−1 u is in X. Moreover for every u ∈ X we have that A(x, u)∇u belongs to W 1,q (RN ) ∩ L2 (RN ) for every q ∈ (N, ∞), so that K (divA(x, u)∇u) belongs to X; finally, under assumption (H0 ), we have that H(x, u, ∇u) belongs to Lq (RN ) with q > N , which implies K(H(x, u, ∇u)) belongs to X. Note that S(u) = f00 (u) where the functional f0 is defined by 1 1 kukp+1 f0 (u) = kuk2 − p+1 , 2 p+1 so that every u that is in KerS is a critical point of f0 . It is well known (see [8], [15]) that Problem (P0 ) has a unique positive radial solution which we call z0 (x). As (P0 ) is traslation invariant every zθ (x) = zeta(x + θ) is also a

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

7

solution. Moreover z0 (x) is strictly decreasing and decays exponentially to zero as |x| → +∞. Thus, f0 has a d-dimensional manifold of critical points Zθ : = {zθ = z0 (x + θ), θ ∈ Rd }, where z0 (x) is the positive radial solution (P0 ) So in this case φ : Rd → X is given by φ(θ) = zθ . Since it is well known (see [18], [19]) that the only solutions of the linearized equation of problem (P0 ) are the partial derivatives of zθ , then Rθ⊥ = Tzθ Z and ei (θ) = ∂i zθ . In order to apply Theorem 2.1 we have to prove that G satisfies hypotheses (G0−2 ). Lemma 3.1. Suppose that assumptions (A0−1 ), (H0 ) hold, then the operator G defined in (3.2) is continuous between R × X and X. Proof. It is sufficient to show the continuity of G with respect of u. Take un , u ∈ X such that un → u in X; as K is an homeomorphism between H −1 (RN ) and H 1 (RN ) and it is continuous between Lq (RN ) and W 2,q (RN ), it is enough to prove that A(x, un )∇un → A(x, u)∇u,

strongly in W 1,q (RN ) ∩ L2 (RN ),

H(x, un , ∇un ) → H(x, u, ∇u),

strongly in L2 (RN ) ∩ Lq (RN ).

From Sobolev imbedding Theorems we get that un → u and ∇un → ∇u strongly in Ls (RN ) for every s ∈ [2, ∞]. Then A(x, un )∇un → A(x, u)∇u, almost everywhere in RN , moreover un and ∇un live in compact sets and we can use condition (A0 ) to deduce that |A(x, un )∇un | ≤ β|∇un |, then by Lebesgue dominated convergence Theorem we have (3.3)

A(x, un )∇un → A(x, u)∇u,

strongly in Lr (RN ) with r ∈ [2, ∞).

Moreover condition (A1 ) implies that |∂i (A(x, un )∇un ) | ≤ c0 β (|∇un |(1 + k∇un k∞ ) + |∂i ∇un |) , again Lebesgue dominated convergence Theorem and (3.3) yield A(x, un )∇un → A(x, u)∇u,

strongly in W 1,q (RN ) ∩ L2 (RN ).

Now let us deal with H(x, un , ∇un ), as before we know that H(x, un , ∇un ) converges to H(x, u, ∇u) almost everywhere. From hypothesis (H0 ) we derive t−1 |H(x, un , ∇un )| ≤ C0 (kun kr−1 ∞ |un | + m1 (x) + d1 (x)k∇un k∞ |∇un |),

we conclude observing that the sequences on the right hand side strongly converge in L2 (RN ) ∩ Lq (RN ). t u Lemma 3.2. Suppose that assumptions (A0−2 ), (H0−2 ) hold, then G defined in (3.2) is of class C 1 with respect to u in X, and G0 : R × X → L(X, X) is continuous.

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BENEDETTA PELLACCI

Proof. It is easy to see that it is possible to compute the Gateaux derivative of G(ε, u) with respect to the variable u and it is given by (3.4)

∂G (ε, u)[v] := εK {−div(A(x, u)∇v + A0s (x, u)∇uv)} ∂u n o

+εK Hs0 (x, u, ∇u)v + Hξ0 (x, u, ∇u)∇v .

In order to show that ∂G/∂u is continuous in X, take a sequence vn which strongly converges to v in X and note that it is sufficient to prove that for every fixed u in X ( A(x, u)∇vn → A(x, u)∇v, (a) A0s (x, u)∇uvn → A0s (x, u)∇uv, strongly in W 1,q (RN ) ∩ L2 (RN ), and that ( Hs0 (x, u, ∇u)vn → Hs0 (x, u, ∇u)v, (b) 0 Hξ (x, u, ∇u)∇vn , → Hξ0 (x, u, ∇u)∇v, strongly in L2 (RN ) ∩ Lq (RN ). By condition (H1 ) we have (3.5)

|Hs0 (x, u, ∇u)vn | ≤ C0 (kukr∞ |vn | + m2 (x)kvn k∞ )
+ C0 d2 (x)k∇ukt∞ |vn | ,

as the last term is strongly convergent in L2 (RN ) ∩ Lq (RN ) we get the first assertion of (b). The term involving Hξ0 (x, u, ∇u)∇vn can be handled in the same way thanks to condition (H2 ). Now let us prove (a). Note that (A0−1 ) imply |A(x, u)∇vn + A0s (x, u)∇uvn | ≤ β (|∇vn | + k∇uk∞ |vn |) , and as the sequences on the right hand side strongly converge in L2 (RN ) ∩ L∞ (RN ) we get the convergences in (a) in L2 (RN ) ∩ Lq (RN ). Moreover we have |∂i (A(x, u)∇vn ) | ≤ c0 β (|∇vn | + k∇uk∞ |∇vn | + |∂i ∇vn |) ,

|∂i A0s (x, u)∇uvn | ≤ c0 β k∇uk∞ |vn | + k∇uk2∞ |vn | + c0 β (k∇uk∞ |∇vn | + k∇vn k∞ |∂i ∇u|) ,

so that applying Lebesgue dominated convergence Theorem we deduce (a). In order to conclude it remains to show that the map G0 : R × X → L(X, X) is continuous. Consider un → u in X and take a sequence vn in X such that kvn kX = 1; we have to prove that ( (Hs0 (x, un , ∇un ) − Hs0 (x, u, ∇u))vn → 0, (c) 0 0 (Hξ (x, un , ∇un ) − Hξ (x, u, ∇u))∇vn → 0,

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

9

strongly in L2 (RN ) ∩ Lq (RN ). Regarding the terms involving A(x, s) and A0s (x, s), we have to prove that ( (A(x, un ) − A(x, u))∇vn → 0, (d) 0 0 (As (x, un )∇un − As (x, u)∇u)vn → 0, strongly in L2 (RN ) ∩ W 1,q (RN ). By condition (H2 ) we get
|Hξ0 (x, un , ∇un )∇vn | ≤ C0 k∇vn k∞ (kun kr−1 ∞ |un |


+ C0 d3 (x)k∇un kt−1 ∞ |∇un | + m3 (x) ,

so that |Hξ0 (x, un , ∇un )∇vn | ≤ C0 (k∇vn k∞ (c1 |un | + c2 d3 (x)|∇un |) + m3 (x)) , and we can use Lebesgue dominated convergence Theorem to get the second assertion of (c), while the first one can be obtained using (3.5). Now let us deal with (d). Notice that the mean value theorem and condition (A0−1 ) imply (3.6)

kA(x, un ) − A(x, u)k∞ ≤ βkun − uk∞ ,

(3.7)

kA0s (x, un )∇un − A0s (x, u)∇uk∞ ≤ β (k∇(un − u)k∞ ) + β (k∇uk∞ kun − uk∞ ) .

Then (3.6) and (3.7) yield (3.8)

(3.9)

(A(x, un ) − A(x, u))∇vn → 0,

in L2 (RN ) ∩ Lq (RN ),

(A0s (x, un )∇un − A0s (x, u)∇u)∇vn → 0,

in L2 (RN ) ∩ Lq (RN ).

Moreover from (A0−1 ) and (3.6), (3.7) we deduce k∇x ((A(x, un ) − A(x, u))∇vn )kq ≤ βk∇vn kq kun − uk∞ (1 + k∇uk∞ ) +β(k∇(un − u)k∞ k∇vn kq ) +β(kun − uk∞ k∇2 vn kq ), proving, together with (3.8), the first assertion of (d). Computing the gradient of (A0s (x, un )∇un − A0s (x, u))∇u)vn gives ∇x ((A0s (x, un )∇un − A0s (x, u)∇u)vn ) = (A0s (x, un )∇un − A0s (x, u)∇u)∇vn + (A00s (x, un )∇un · ∇un − A00s (x, u)∇u · ∇u)vn + (A0s (x, un )∇2 un − A0s (x, u)∇2 u)vn + (∇x A0s (x, un )∇un − ∇x A0s (x, u)∇u)vn .

10

BENEDETTA PELLACCI

By applying hypotheses (A1−2 ) we obtain the following estimates on the gradient of (A0s (x, un )∇un − A0s (x, u))∇u)vn k∇x ((A0s (x, un )∇un − A0s (x, u)∇u)vn )kq ≤ c1 kvn k∞ k∇(un − u)kq +c2 kvn k∞ {k∇x (A0s (x, un ) − A0s (x, u))∇ukq } +c3 kvn k∞ {kA00s (x, un )∇un − A00s (x, u)∇ukq } +c4 kvn k∞ {kA0s (x, un )∇2 un − A0s (x, u)∇2 ukq } +c5 k∇vn k∞ {k(A0s (x, un ) − A0s (x, u))∇ukq }, and so we also get the second assertion in (d), proving the Lemma. t u Lemma 3.1 and 3.2 allow us to apply Theorem 2.1. Moreover the operator G satisfies condition (G3 ) with α = 1, and by (2.2) we can evaluate the components of the map Ψ(θ), obtaining (3.10) Z 1 (Ψ(θ))i = ( G(ε, zθ )|∂i zθ ) = A(x, zθ )∇zθ ∇∂i zθ + H(x, zθ , ∇zθ )∂i zθ dx . ε RN Moreover it is well known (see [8]) that z0 (x) is a radial function so that we can define γ(r) = z0 (x) (r = |x|). Performing a simple change of variable we have Z Ψ(θ)i = A(x − θ, z0 (x))∇z0 (x)∇∂i z0 (x) dx (3.11) +

RZN

H(x − θ, z0 (x), ∇z0 (x))∂i z0 (x) dx .

RN From now on we will suppose that the function H(x, s, ξ) is of class C 1 with respect to the variable x. Using (3.11) and computing the partial derivative yield Z ∂θj Ψ(θ)i = − ∂xj A(x − θ, z0 (x))∇z0 (x)∇∂i z0 (x) dx (3.12)

ZR − ∂xj H(x − θ, z0 (x), ∇z0 (x))∂i z0 (x) dx . N

RN Theorem 2.3 states that if we find a bounded open set Ω ⊆ Rd such that deg(Ψ, Ω) 6= 0, then there exists ε1 such that for all 0 < ε < ε1 , the equation

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

11

2.1 has at least a solution in ZεΩ . We define with Ψ1 (θ) and Ψ2 (θ) as follows Z 1 (3.13) Ψ (θ)i = A(x − θ, z0 (x))∇z0 (x)∇∂i z0 (x), (3.14)

Ψ2 (θ)i =

RZN

H(x − θ, z0 (x), ∇z0 (x))∂i z0 (x) dx .

RN Applying the results of the previous section and by Lemma 3.1, 3.2 we get the following existence result Theorem 3.3. Assume conditions (A0−2 ), (H0−2 ) are satisfied. Suppose that there exists a point θ in RN such that i) ψ(θ)i = 0, ∀
i = 1, . . . , N , ii) ∂j ψ(θ)i |θ=θ is invertible, then there exists a positive ε0 such that problem (Pε ) has a solution for every ε with 0 < |ε| < ε0 . Proof. By i) and ii) we get that there exists a bounded open set Ω ⊆ RN such that deg(Ψ, Ω) is well defined and deg(Ψ, Ω) 6= 0. Then Theorem 2.3 yields the conclusion. t u Now we will study some model examples in which it is possible to find a point θ saisfying i) and ii) of Theorem 3.3. From now on we will suppose that A(x, s) : RN × R → R is a scalar function, in addition H(x, s, ξ) and A(x, s) satisfy the following conditions (3.15)

H(x, s, ξ) = H(|x|, s, |ξ|) = H(r, s, η),

(3.16)

A(x, s) = A(|x|, s) = A(r, s).

We will prove that θ0 = 0 is an isolated zero of Ψ and deg(Ψ(θ), BR0 , 0) 6= 0. Remark 3.4. We remark that if either H(x, s, ξ) = 12 A0s (x, s)|ξ|2 or A(x, s) ≡ 0 and H does not depend on ξ problem (Pε ) is variational and we will treat this case in the following section. Theorem 3.5. Suppose that conditions (H0−2 ), (3.15) hold and assume that the map H(x, s, ξ) is derivable with respect to r and it satisfies the following condition (3.17)

Hr0 (r, s, |ξ|) > 0, or Hr0 (r, s, |ξ|) < 0,

∀s > 0.

12

BENEDETTA PELLACCI

then there exists a positive number ε0 such that for every 0 < |ε| < ε0 there is a solution of the problem ( −∆u + u + εH(x, u, ∇u) = |u|p−1 u, in RN , (PεH ) u ∈ H 1 (RN ) ∩ W 2,q (RN ). Remark 3.6. Note that if H(x, s, ξ) = d(x)|ξ|t with d(x) > 0 and d(x) = d(x) + d(x) where d ∈ L2 (RN ) ∩ Lq (RN ), d ∈ L∞ (RN ), then condition (3.17) implies that d 6≡ 0. Proof. First notice that in this case Ψ(θ) ≡ Ψ2 (θ). Conditions (H0−2 ) allow us to apply Theorems 2.1, 2.3. It will be sufficient to prove that θ0 = 0 is a zero of Ψ(θ)i such that ∂j Ψ(θ)i is invertible. From (3.15) it results Z Ψ(0)i = H(r, z0 (x), ∇z0 (x))∂i z0 (x) dx =

RZN

H(r, z0 (x), ∇z0 (x))γ 0 (r)

RN

xi dx = 0, r

because the previous integrals are odd in the variable xi . By (3.12) we have Z xi xj ∂j (Ψ(0)i ) = − Hr0 (r, z0 (x), ∇z0 (x))γ 0 (r) 2 . r RN For i = 6 j the last integral is zero since it is odd in some variable, while for i = j we have Z (xi )2 ∂i (Ψ(0)i ) = − Hr0 (r, z0 (x), ∇z0 (x))γ 0 (r) 2 r RN Z (xi )2 = Hr0 (r, z0 (x), ∇z0 (x))|γ 0 (r)| 2 . r RN Using condition (3.17) we get the conclusion applying Theorem 3.3. t u We now consider problem (PεA ). Theorem 3.7. Suppose that the matrix A(x, s) satisfies (A0−2 ), (3.16). Moreover assume that for every s in R+ and almost everywhere in RN one of the following conditions is satisfied (3.18)

A0r (|x|, s) ≥ 0,

∂s A0r (|x|, s) ≥ 0,

A00r (|x|, s) ≥ 0,

(3.19)

A0r (|x|, s) ≤ 0,

∂s A0r (|x|, s) ≤ 0,

A00r (|x|, s) ≤ 0.

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

13

Then there exists a positive number ε0 such that for every 0 < |ε| < ε0 there is a solution of the problem ( −∆u + u − εdiv(A(x, u)∇u) = |u|p−1 u, in RN , (PεA ) u ∈ H 1 (RN ) ∩ W 2,q (RN ). Proof. Notice that in this case Ψ(θ)i ≡ Ψ1 (θ)i , moreover from (3.13) and integrating by part we obtain Z X (∂k z0 (x))2 ∂i A(x − θ, z0 (x)) dx. Ψ(θ)i = − 2 k RN Then θ0 = 0 is a zero of Ψ(θ)i . Computing the partial derivatives of Ψ(θ)i yields Z xi xj xi xj i (γ 0 (r))2 h 00 ∂j (Ψ(0)i ) = Ar (x, γ) 2 + ∂s A0r (x, γ)γ 0 (r) dx 2 r r N R    Z δij xi xj (γ 0 (r))2 + A0r (x, γ) − 3 dx. 2 r r RN Then we deduce that ∂j (Ψ(0)i ) = 0 for every i 6= j, for i = j we have   Z (γ 0 (r))2 (xi )2 (xi )2 0 00 0 ∂i (Ψ(0)i ) = Ar (r, γ) + ∂s Ar (r, γ)γ (r) dx 2 r2 r RN    Z (γ 0 (r))2 1 (xi )2 0 + Ar (r, γ) − 3 dx. 2 r r RN Condition (3.18) or (3.19) and Theorem 3.3 yield the conclusion. t u Finally let us consider the case in which both H(x, s, ξ) and A(x, s) are present. Theorem 3.8. Assume that conditions (A0−2 ), (H0−2 ), (3.15)–(3.19) hold. Then there exists a positive number ε0 such that problem (Pε ) has at least a solution for every ε with 0 < |ε| < ε0 . Proof. The proof immediately follows from Theorems 3.5, 3.7. t u Suppose now that H(x, s, ξ) = h(|x|, s)|ξ|2 . This case is particularly interesting as there is an interaction between the functions H(x, s, ξ) and A(x, s) and we can prove the following.

14

BENEDETTA PELLACCI

Theorem 3.9. Assume that conditions (A0−2 ), (H0−2 ), (3.15), (3.16) hold. If A(x, s) and H(x, s, ξ) satisfy, for every s in R+ and almost everywhere in RN , one of the following conditions
(3.20) A00r (|x|, s) ≥ 0, A0r (|x|, s) ≥ 0, ∂s A0r (|x|, s) − h0r (|x|, s) ≤ 0, or (3.21)


A00r (|x|, s) ≤ 0, A0r (|x|, s) ≤ 0, ∂s A0r (|x|, s) − h0r (|x|, s) ≥ 0,

then there exists a positive number ε0 such that problem (Pε ) has a solution for every ε with 0 < |ε| < ε0 . Proof. As before condition (3.16) and (3.15) imply that θ = 0 is a zero of the function Ψ. Moreover we have Z xi xj (γ 0 (r))2 00 ∂j (Ψ(0)i ) = Ar (x, γ) 2 dx 2 r RN    Z δij xi xj (γ 0 (r))2 0 + Ar (x, γ) − 3 dx 2 r r RN Z  xi xj (γ 0 (r))2  + ∂s A0r (x, γ) − h0r (r, s) γ 0 (r) dx. 2 r N R Condition (3.20) or (3.21) and Theorem 3.3 imply the conclusion. t u Remark 3.10. In the non perturbative case problem (Pε ) has been studied without the presence of the non-linearity |u|p−1 u on unbounded domains in [13] where the authors consider an uniformly bounded matrix A(x, s) and the Hamiltonian satisfies the following growth condition |H(x, s, ξ)| ≤ %(x) + b(|s|)[k(x)|ξ| + |ξ|2 ] where %(x) ∈ L2 (RN ) ∩ L∞ (RN ) , k(x) ∈ Lp (RN ) ∩ L∞ (RN ) and b : R+ → R+ is an increasing function. We have showed that, in the perturbative case and if H and A satisfy conditions (3.15)–(3.19), we can admit any superlinear growth on the function H and we can consider at the same time bounded and unbounded maps A(x, s). 4. Existence Results : the variational case Now let us consider the case in which 1 (4.1) H(x, s, ξ) = A0 (x, s)|ξ|2 . 2 Notice first that H satisfies condition (H0−2 ) with r = 0, mi = 0 di = 1 for i = 1, . . . , 3 and t = 2. In this setting we can apply Theorem 2.4 in order to

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

15

find a solution of (Pε ). Indeed it will be sufficient to find a local minimum (or maximum) of the function Z 1 (4.2) Γ(θ) = A(x, zθ (x))∇zθ (x)∇zθ (x). 2 RN Theorem 4.1. Assume conditions (A0−2 ), (4.1). If there exists a point θ such that Γ(θ) 6= 0, then there exists a positive number ε0 such that problem (Pε ) has a solution for every ε with 0 < |ε| < ε0 . Proof. From the exponential decay of the function z0 (x) we derive lim Γ(θ) = 0.

|θ|→∞

Then if there exists a θ such that Γ(θ) 6= 0 the map Γ has a relative minimum or maximum. t u The following existence result is a simple application of Theorem 4.1. Theorem 4.2. Suppose that the matrix A(x, s) is definite for every s in R+ and for almost every x in RN . Then there exists ε0 > 0 such that for every ε, 0 < |ε| < ε0 there exists a solution of Problem (Pε ). Remark 4.3. Problem (Pε ) has been studied in the non–perturbative case and in the variational framework on buonded domains in [4], [5], [12]. While on unbounded domains it has been studied in [11] where the authors consider uniformly bounded matrices B(x, s) imposing that either B does not depend on x or it is such that there exists γ ∈ (0, p − 2) satisfying (CG1 ) 0 ≤ sBs0 (x, s)ξ · ξ ≤ γB(x, s)ξ · ξ, (CG2 ) (CG3 )

a.e. in RN , ∀ s ∈ R, ∀ ξ ∈ RN ,

B(x, s) → δij , Bs0 (x, s)s → 0, if|x| → ∞

a.e. in RN , ∀ s ∈ R,

B(x, s)ξ · ξ ≤ |ξ|2 .

We point out that condition (CG1 ) is assumed also in [4], [12] and it is fundamental in order to prove that any Palais–Smale sequence is bounded and compact. Conditions (CG2 ), (CG3 ) are fundamental in order to recover the compactness of the Palais–Smale sequences on unbounded domains. We point out that here, thanks to the perturbative formulation of the problem, we do not require any of the assumptions (CG1−3 ), more precisely we consider B(x, s) = I + εA(x, s) and we do not impose any sign condition on A as in (CG1 ), moreover we treat as well unbounded matrices which obviously do not satisfy (CG2−3 ). In the following result we deal with a model case in which it is possible to obtain the existence of a solution without applying Theorem 4.1.

16

BENEDETTA PELLACCI

Theorem 4.4. Assume conditions (A0−2 ). Moreover we suppose that the matrix A(x, s) = A(r, s), with r = |x| and that it satisfies one of the following conditions ( A0r (r, s) ≥ 0, ∀ s ∈ R+ ∀ r ∈ R+ , (4.3) A00r (r, s) ≥ 0, ∀ s ∈ R+ ∀ r ∈ R+ , ( A0r (r, s) ≤ 0, A00r (r, s) ≤ 0,

(4.4)

∀ s ∈ R+ ∀ r ∈ R+ , ∀ s ∈ R+ ∀ r ∈ R+ ,

Then there exists a positive number ε0 such that for every 0 < ε < ε0 there is a solution of the problem (Pε ). Proof. The proof follows applying Theorem 3.9 in the particular case in which h(r, s) = A0s (r, s). u t Remark 4.5. Consider a particular variational case i.e. A(x, s) = 0 and H(x, s, ξ) = g(x, s), if we denote with G(x, t) the primitive with respect of Rt t of the function g(x, t), G(x, t) = 0 g(x, s)ds, the map Γ will be defined as follows Z Γ(θ) =

G(x, zθ (x)) dx.

RN As before lim Γ(θ) = 0 so that we can apply Theorem 4.1, in order to find |θ|→∞

a solution, for sufficiently small ε, of the problem ( −∆u + u = |u|p−1 u − εg(x, u), in RN (P ) u ∈ H 1 (RN ) ∩ W 2,q (RN ). For example if g(x, s) has constant sign for almost every x in RN and for every s in R+ , hypotheses of Theorem 4.1 are satisfied and we get a solution for sufficiently small ε. We remark that Problem (P ) is studied in [9] assuming that G(x, s) ∈ C 2 (RN × R, R), G(x, 0) = g(x, 0) = 0 and the second derivative with respect to the variable s satisfies a natural growth condition. 5. Super–Linear Growth in the Divergence Operator In this section we briefly consider the case of perturbation with general divergence operator as −div(a(x, u, ∇u)); we obtain a model example if a(x, s, ξ) = A(x, s)|ξ|m−2 ξ with m > 1. On the function a(x, s, ξ) : RN × R × RN → RN we assume that a is of class C 2 with respect of the variables (x, s, ξ) and for every compact set C ⊆ R we suppose that there exist β = β(C), m ≥ 1 and σ ≥ 0 such that for almost every x in RN and for every (s, ξ) in R × RN the following conditions are satisfied (a0 )

|a(x, s, ξ)| ≤ β|ξ|m ,

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

(a1 )

( |∂xi a(x, s, ξ)| ≤ β|ξ|m , |∂ξi a(x, s, ξ)| ≤

17

|a0s (x, s, ξ)| ≤ β|ξ|m , β |ξ|m ,

where ai (x, s, ξ) are the components of the map a(x, s, ξ) and we denote with ∂xi a(x, s, ξ) (∂ξi a(x, s, ξ)) the vector whose components are the partial derivatives of ai (x, s, ξ) with respect to xi (to ξi ), and with a0s (x, s, ξ) the vector whose components are the partial derivatives of ai (x, s, ξ) with respect to s.  0 m  |∂xj ∂ξj a(x, s, ξ)| ≤ β|ξ|m , |∂xi as (x, s, ξ)| ≤ β|ξ| , 00 m (a2 ) |as (x, s, ξ) ≤ β|ξ| , |∂ξi a0s (x, s, ξ) ≤ β|ξ|σ ,   |∂ξj ∂ξi a(x, s, ξ)| ≤ β|ξ|σ .

We will be interested in the following problem ( −∆u + u + ε{−div(a(x, u, ∇u)) + H(x, u, ∇u)} = |u|p−1 u, (Pε ) u ∈ H 1 (RN ) ∩ W 2,q (RN ), q > N,

where H(x, s, ξ) satisfies hypotheses (H0−2 ). Now the perturbation G(u) is of the form G = G1 + G2 where Gi are defined by G1 (u) = K(−div(a(x, u, ∇u))), G2 (u) = K(H(x, u, ∇u)). By Lemma 3.1, 3.2 G2 is of class C 1 . Let us show that also G1 is of class C 1. Lemma 5.1. Under assumptions (a0−2 ) G1 is of class C 1 . Proof. Let un → u strongly in X. Then by (a0 ), up to a subsequence a(x, un , ∇un ) → a(x, u, ∇u),

a.e. in RN ,

|a(x, un , ∇un )| ≤ β|∇un |m ≤ c0 h(x), where h(x) belongs to L2 (RN )∩Lq (RN ) as ∇un → ∇u strongly in L2 (RN )∩ Lq (RN ). From now on we denote with c possibly different positive constants. (a1 ) yields |∂xi [a(x, un , ∇un )]| ≤ βc(|∇un | + |∂xi ∇un |). Then a(x, un , ∇un ) converges to a(x, u, ∇u) in W 2,q (RN ) ∩ L2 (RN ) and this implies that G1 is continuous. When we compute the derivative of G1 with respect of u we get ∂G1 (u)[v] = K(−div(a0s (x, u, ∇u)v + ∇ξ a(x, u, ∇u)∇v)), ∂u where ∇ξ a(x, s, ξ) is the matrix whose components are the map ∂ξi aj (x, s, ξ). 1 (u) It is easy to show that ∂G∂u (u)[v] is continuous with respect to v for every fixed u in X. Consider now {un } and {vn } in X such that kvn kX = 1 and un → u strongly in X. We have by (a1 ) |a0s (x, un , ∇un )vn + ∇ξ a(x, un , ∇un )∇vn | ≤ βckvn kX |∇un |,

18

BENEDETTA PELLACCI

which implies that the sequences a0s (x, un , ∇un ) and ∇ξ a(x, un , ∇un ) strongly converge in L2 (RN )∩Lq (RN ). Moreover by hypotheses (a1−2 ) we deduce the following estimates |∂xi [a0s (x, un , ∇un )vn ]| ≤ βkvn k∞ [|∇un |σ (1 + |∇2 un |)] + 2βkvn k∞ k∇un km−1 ∞ |∇un |, |∂xi [∇ξ a(x, un , ∇un )∇vn ]| ≤ βk∇vn k∞ c[|∇un | + |∂xi ∇un |] + k∂xi ∇vn kq |∇ξ a(x, un , ∇un )|. Notice that ∇ξ a(x, un , ∇un ) strongly converges in L∞ (RN ), indeed by (a2 ) k∇ξ a(x, un , ∇un ) − ∇ξ a(x, u, ∇u)k∞ ≤ βk∇un kσ∞ k∇un − ∇uk∞ + βk∇un kσ kun − uk∞ . Therefore ∇ξ a(x, un , ∇un )∇2 vn converges strongly in Lq (RN ), and this completes the proof. t u Remark 5.2. Consider the particular interesting case in which a is of the form a(x, s, ξ) = A(x, s)|ξ|m−2 ξ where A(x, s) satisfies hypotheses (A0−2 ). Then if m ≥ 3 we obtain a perturbation of class C 1 . As before we can apply Theorem 2.1. Moreover the operator G satisfies condition (G3 ) with α = 1, and by (2.2) we can evaluate the components of the map Ψ(θ) = Ψ1 + Ψ2 . Ψ2 is given in (3.14), while Ψ1 is defined by Z (5.1) (Ψ1 (θ))i = a(x, zθ (x), ∇zθ (x))∇∂i zθ (x).

RN We obtain the following existence result. Theorem 5.3. Assume conditions (a0−2 ), (H0−2 ) are satisfied. Suppose that there exists a point θ in RN such that i) ψ(θ)i = 0, ∀
i = 1, . . . , N , ii) ∂j ψ(θ)i |θ=θ is invertible, then there exists a positive ε0 such that problem (Pε ) has a solution for every ε with 0 < ε < ε0 . From now on we will consider the following model case (5.2)

a(x, s, ξ) = A(r, s)|ξ|m−2 ξ,

where A(r, s) is a map depending radially with respect of the variable x. Theorem 5.4. Assume conditions (a0−2 ), (3.18), (3.19), (5.2). Then there exists a positive ε0 such that Problem (Pεa ) ( −∆u + u − εdiv(a(x, u, ∇u)) = |u|p−1 u, in RN , (Pεa ) u ∈ H 1 (RN ) ∩ W 2,q (RN ) q > N,

QUASI–LINEAR EQUATIONS ON RN : PERTURBATION RESULTS

19

has a solution for every ε, 0 < |ε| < ε0 . Proof. Integration by part gives Z 1 1 Ψ (θ)i = |γ 0 (r)|m ∇i A(|x − θ|, z0 (x)) dx. m RN From now on it is possible to follow the argument used in the proof of Theorem 3.7. t u Remark 5.5. We remark that it is possible to obtain the same existence results of Section 3 and 4 under hypotheses (3.15), (3.17). Clearly in this case we will have interaction between H and the divergence operator if H has a growth of order m, and it is possible to prove the same existence result as Theorem 3.9 under conditions (3.20)–(3.21). Consider the case in which (5.3)

a(x, s, ξ) = h(x)|ξ|m−2 ξ.

Then Problem (Pεa ) is variational and we can prove analogous versions of Theorems 4.1, 4.2. More precisely, the following result holds. Theorem 5.6. Assume conditions (a0 )-(a2 ) and (5.3). Define the map Z 1 Γ(θ) = h(x − θ)|∇z0 (x)|m dx. m RN Suppose that there exists a point θ such that Γ(θ) = 6 0. Then there exists a positive number ε0 such that Problem (Pεa ) has a solution for every ε with 0 < |ε| < ε0 . Proof. As before, we have that Γ(θ) → 0 when |θ| → ∞. If there exists a point θ such that Γ(θ) 6= 0, then the map Γ has a relative minimum or maximum. Theorem 2.4 yields the conclusion. t u As a corollary we get the following existence result. Corollary 5.7. Assume conditions (a0 )-(a2 ) and (5.3). Moreover, suppose that for almost every x in RN the following condition holds h(x) > 0,

or

h(x) < 0.

Then there exists ε0 > 0 such that for every ε, 0 < |ε| < ε0 there exists a solution of Problem (Pεa ). Proof. The conclusion follows applying Theorem 4.1. t u

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BENEDETTA PELLACCI

References [1] Ambrosetti, A., Badiale, M., Homoclinics : Poincar´e-Melnikov Type Results via a Variational Approach. Ann. Ist. H.P. Analyse non Lin´eare 15 (2) (1998), 233-252. [2] Ambrosetti, A., Badiale, M.,Variational Perturbative Methods and Bifurcations of Bound State From The Essential Spectrum. Proc. Roy. Soc. Edin. Sect-A 128 (6) (1998), 1131-1161. [3] Ambrosetti, A., Badiale, M., Cingolani, S., Semiclassical States of Nonlinear Schr¨ odinger Equation. Arch. Rat. Mech. Anal. 140 (3) (1997), 285-300. [4] Arcoya, D. Boccardo, L., Critical Points for Multiple Integral of the Calculus of Variations. Archive Rat. Mech. Anal. 134 (3) (1996), 249-274. [5] Arcoya, D., Boccardo, L., Some Remarks on Critical Point Theory for Nondifferentiable Functionals. NoDEA 6 (1999), 79-100. [6] Badiale, M., Duci, A., Concentrated Solutions for a non Variational Semilinear Elliptic Equations. To appear on Houston Journ. of Math. [7] Badiale, M. Garc´ia Azorero, J., Peral I., Perturbation Results for an Anisotropic Schr¨ odinger Equation via a Variational Method. NoDEA 7 (2000), 201-230. [8] Beresticki, H., Lions, P.L., Nonlinear Scalar Fields Equation I. Existence of a Ground State. Arch. Rat. Mech. Anal. 82 (1983), 313-346. [9] Berti, M., Bolle, Homoclinics and Chaotic Behaviour for Perturbed Second Order Systems. Annali di Mat. Pura e Appl. CLXXVI n.IV (1999), 323-378. [10] Boccardo, L., Murat, F., Puel, J.P., L∞ (RN ) Estimates for Some Nonlinear Partial Differential Equations and an Application to an Existence Result. SIAM Journ. Math. Anal. 23 (1992), 326-333. [11] Conti, M. Gazzola, F., Positive Entire Solutions of Quasilinear Elliptic Problems via Nonsmooth Critical Point Theory. Topol. Meth. in Nonlinear Anal. 8 (2) (1996), 275-294. [12] Canino, A. Degiovanni, M., Non Smooth Critical Point Theory and Quasilinear Elliptic Equations. Topological Methods in differential equations and Inclusions (Montreal, 1994), Nato, Asi Series Kluwer, Dordrecht. [13] Donato, P., Giachetti, D., Quasilinear Elliptic Equations with Quadratic Growth in Unbounded Domains. Nonlinear Anal. TMA 10 (8) (1986), 791-804. [14] Henrard, M., Homoclinic and Multibump Solutions for Perturbed Second Order Systems Using Topological Degree. Preprint n.66 December 1997 Institut de Math´ematique Pure et Appliqu´ee, Univesit´e Catholique de Louvain. [15] Kwong, M. K., Uniqueness of Positive Radial Solution of −∆u − u + up = 0 in RN . Arch. Rat. Mech. Anal. 105 (1989), 495-505. [16] Lions, P.L., R´esolutions de Probl`emes Elliptiques Quasilin´eaires. Arch. Rat. Mech. Anal. 74 (1980), 335-353. [17] Nagumo, M., On Principally Linear Elliptic Equations of the Second Order. Osaka Math. Journ. 6 (2) (1954), 207-229. [18] Oh, Y.G., Existence of Semiclassical Bound State of Nonlinear Schr¨ odinger equations. Comm. Math. Phys. 209 (1993), 223-243. [19] Weinstein, A. Modulation Stability of Ground State of Nonlinear Schr¨ odinger Equations. SIAM Journ. Math. Anal. 16 (3) (1985), 472-491. (Benedetta Pellacci) S.I.S.S.A. v. Beirut 2-4, 34013, Trieste E-mail address, B. Pellacci: [email protected]