ASYMPTOTIC PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH BOUNDED Φ-LAPLACIAN

The asymptotic problem for the general Laplacian are
investigated and some discrepancies between general Laplacian
and classical Laplacian are pointed out.


Introduction
In this paper we deal with the second order nonlinear differential equation (a(t)Φ(x ′ ) ′ + b(t)F (x) = 0, (t ≥ t 0 ), (1) where: (i) Φ : R → (−σ, σ), 0 < σ ≤ ∞, is an increasing odd homeomorphism, such that Φ(u)u > 0 for u = 0; (ii) F : R → R is a continuous increasing function such that F (u)u > 0 for u = 0; (iii) a, b : [t 0 , ∞) → (0, ∞) are continuous functions and Our aim is to study the existence of positive solutions x of (1) satisfying the asymptotic boundary conditions The prototype of ( 1) is the equation where the map Φ C : R → (−1, 1) is given by This equation arises in the study of the radially symmetric solutions of partial differential equation with the curvature operator div g(|x|) ∇u where x = (x 1 , . . . i , E = {x ∈ R n : |x| ≥ d}, d > 0 and g : R + → R + is a weight function.Denote r = |x| and du dr = u r the radial derivative of u.Since ∇u = x r u r , we have g(r) ∇u and, by a direct computation, we get that the function u is a radially symmetric solution of (6) if and only if the function y = y(r) = u(|x|) is a solution of which is a special case of (1).
Boundary value problems on a compact interval associated to the partial differential equations with the mean curvature operator have been investigated in [4,5,6]; see also the references therein.
In a recent paper [9], the authors studied all possible types of nonoscillatory solutions of (1) and their mutual coexistence under the assumption that there exists λ > 0 such that λa −1 (t) ∈ Im Φ for any t ≥ t 0 .
This classification depends on the asymptotic behavior of the vector (x, x [1] ), where x is a solution of (1) and x [1] denotes its quasiderivative x [1]  7) is satisfied for any λ > 0. So, condition (7) plays a role only when Im Φ is bounded and requires lim inf t→∞ a(t) > 0 .
For these reasons, particular attention will be devoted to the equation (1) with Φ bounded and its special case (4).Consequently, throughout this paper, we assume It is easy to show (see below) that, when (Hp) holds, any nonoscillatory solution x of (1) satisfies lim t→∞ x [1] (t) = 0.Moreover, the global positiveness and uniqueness of solutions of (1)-(2) will be also considered.This problem is motivated by searching for positive radially symmetric solutions in a fixed exterior domain in R N for (6).
We will show that the lack of the homogeneity property of Φ can produce several new phenomena, which are illustrated by some examples.With minor changes, our results can be applied also when σ = ∞ and so they complement the previous ones stated in [7,9] for a general Φ and in [8] for the classical Φ-Laplacian.Similarities EJQTDE Spec.Ed.I, 2009 No. 9 and discrepancies with these cases complete the paper jointly with a discussion on the meaning of the assumption (Hp).
Finally, we introduce the integral where Φ * denotes the inverse function to Φ and µ is a positive constant.This integral plays a crucial role for the asymptotic behavior of solutions, similarly as in case when Φ is unbounded ( [7,8,9]).

Necessary Conditions
Throughout this paper we shall consider only the solutions of (1) which exist on some ray [t x , ∞), where t x ≥ t 0 may depend on the particular solution.As usual, a solution x of (1) defined in some neighborhood of infinity is said to be nonoscillatory if x(t) = 0 for large t, and oscillatory otherwise.
If x is eventually positive [negative], then its quasiderivative x [1] is decreasing [increasing] for large t.The following holds.
Proof.Let x be a nonoscillatory solution of (1) and, without loss of generality, assume x(t) > 0 for t ≥ T ≥ t 0 .From (Hp), there exists {t k }, t k → ∞ such that lim k x [1] (t k ) = 0 and, because x [1] is eventually decreasing, the assertion follows.
Proof.Let x be a nonoscillatory solution of (1).In view of Lemma 2.1 we can suppose, without loss of generality, x(t) > 0, x ′ (t) > 0 for t ≥ T .
Claim i 2 ).Now assume lim t→∞ x(t) = ∞.Using the same argument, we obtain for x is unbounded.The second assertion follows from claim i 2 ).
From (13) the following result follows.
Remark 2.1.Proposition 2.1-i 2 ) remains to hold if the unboundedness of F is replaced by The following example shows that the condition ( 14) is optimal.
Example 2.1.The equation has the unbounded solution x(t) = t/2, i.e., the statement of Proposition 2.1-i 2 ) does not hold.In this case is not verified.However, Proposition 2.1-i 3 ) is applicable and any nonoscillatory solutions is unbounded.

Nonoscillatory Bounded Solutions
In this section we deal with solutions of (1) satisfying the asymptotic conditions (2) and with their global positiveness and uniqueness.
and there exists a positive constant µ such that Then, for each L > 0, L sufficiently small, (1) has nonoscillatory solutions, x, such that lim t→∞ x(t) = L.
Proof.In view of (15), there exists L > 0 such that F (L) < µ and sup Denote with C[t 1 , ∞) the Fréchet space of all continuous functions on [t 1 , ∞) endowed with the topology of uniform convergence on compact subintervals of [t 1 , ∞) and con- Define on Ω the operator T as follows Let us show that T (Ω) is relatively compact, i.e.T (Ω) consists of functions equibounded and equicontinuous on every compact interval of [t 1 , ∞).Because T (Ω) ⊂ Ω, the equiboundedness follows.Moreover, in view of the above estimates, for any u ∈ Ω we have which proves the equicontinuity of the elements of T (Ω).The continuity of T in Ω follows by using the Lebesgue dominated convergence theorem and taking into account (16).Thus, by the Tychonov fixed point theorem, there exists a fixed point x of T.
Clearly, x is a solution of (1) such that lim t→∞ x(t) = L and the solvability of the BVP (1)-( 2) follows from Proposition 2.1-i 4 ).Theorem 3.1 answers the existence problem of bounded eventually positive solutions of (1).For the equation ( 4) this result can be improved by obtaining sufficient conditions for their global positivity and uniqueness.To this end, the following Gronwall type lemma is needed.
for some nonnegative constant A, then If, in particular, A = 0, then w(t) = 0 identically on [T, ∞).EJQTDE Spec.Ed.I, 2009 No. 9 Theorem 3.2.Assume (Hp) and suppose that F is continuously differentiable in a neighborhood of zero such that If conditions (15) and are verified, then for any positive and sufficiently small L, there exists a unique solution x of (4) such that Proof.In view of (15), there exists λ > 0 such that for any t ≥ t 0 and so, from (18), we have J λ < ∞.In virtue of (17), choose L > 0 such that F is continuously differentiable on (0, L] and From (21) we get and so ( 16) is satisfied with t 1 = t 0 .Reasoning as in the proof of Theorem 3.1-i 1 ), there exists at least one solution x of (4) satisfying the boundary conditions (19).It remains to show the uniqueness of this solution.Let z, y be two solutions of ( 4) satisfying (19).Since z and y are increasing, we have 0 EJQTDE Spec.Ed.I, 2009 No. 9 A direct calculation gives and so, the mean value theorem implies Using again the mean value theorem, we get where Remark 3.1.As already claimed, Theorem 3.2 plays an important role in searching for positive radially symmetric solutions in a fixed exterior domain in R N for the partial differential equation (6).Moreover, Theorem 3.2 can be easily extended to an equation involving a more general Φ, by assuming that Φ is continuously differentiable in a neighborhood of zero.The details are left to the reader.
A closer examination of proofs of Theorems 3.1 and 3.2 shows that these results hold also when Im Φ is unbounded.So, in particular, they can be applied to the equation associated to the Sturm-Liouville operator The boundedness of nonoscillatory solutions of (24) is strongly related to the boundedness of nonoscillatory solutions of (4).We make this observation precise in Corollary 3.1.To show this fact, the following lemma concerning the map Φ C is needed.
i 2 ) =⇒ i 3 ).From Proposition 2.2 we have J C µ < ∞ for a sufficiently small constant µ > 0. So, in view of Lemma 3.2, the assertion follows.

Asymptotic Estimates
When Φ is the classical Φ-Laplacian, for two bounded nonoscillatory solutions x, y, such that lim t→∞ x(t) = ℓ x = 0, lim t→∞ y(t) = ℓ y = 0 we have that the limit is finite and different from zero.Roughly speaking, all bounded nonoscillatory solutions of ( 1) with the classical Φ-Laplacian have an equivalent growth at infinity.This fact can fail for a general Φ, with Im Φ bounded, as the following example illustrates.
Example 4.1.Consider the equation where Φ : R → (−1, 1) is a continuous odd function such that Let us show that (27) has two bounded solutions such that We have we get for λ ∈ (0, 1] Taking into account that in virtue of Theorem 3.1 and its proof, there exist two bounded nonoscillatory solutions x, y of (27) such that lim Hence there exists T ≥ t 0 such x ′ (t) > t −3 , y ′ (t) < t −4 for t > T .Then x ′ (t) > ty ′ (t) and (29) follows.
EJQTDE Spec.Ed.I, 2009 No. 9 A sufficient condition, in order that bounded nonoscillatory solutions have an equivalent growth at infinity, is given by the following.
Corollary 4.1.Assume (Hp), (10) and that for some α > 0. If x, y are two bounded nonoscillatory solutions of (1) such that then the limit lim t→∞ x(t) − c x y(t) − c y is finite and different from zero.Moreover, any bounded nonoscillatory solution x of (1) satisfies Proof.Without loss of generality, let x, x [1] , y and y [1] be positive for t ≥ T ≥ t 0 .Hence, the l'Hopital rule gives x [1] (t) y [1] (t) = F (c x ) F (c y ) .
In virtue of Theorem 3.1 we have lim x ′ (t) y ′ (t) α which implies that the limit lim t→∞ x ′ (t) y ′ (t) is finite and different from zero and the first assertion follows.
From the equality x [1] (t) taking into account Lemma 2.1, by using the l'Hopital rule, the assertion follows.

Unbounded Solutions
In this section we study the existence of solutions of (1) satisfying the asymptotic conditions (3).
If there exists a positive constant µ ∈ Im F such that then the asymptotic problem (1)-( 3) is solvable.
Proof.Let L be such that In virtue of (35), we can choose t 1 > 0 so large that sup EJQTDE Spec.Ed.I, 2009 No. 9 Now, as in the proof of Theorem 3.1, denote by C[t 1 , ∞) the Fréchet space of all continuous functions on [t 1 , ∞) endowed with the topology of uniform convergence on compact subintervals of [t 1 , ∞) and consider the set Ω ⊂ C[t 1 , ∞) given by Define in Ω the operator T as follows In view of (38), we have Obviously, T (u)(t) ≥ L and so T maps Ω into itself.Reasoning as in the proof of Theorem 3.1 and applying the Tychonov fixed point theorem, there exists a solution x of the integral equation Clearly, x is a solution of (1).Because in virtue of (35), we obtain lim t→∞ x ′ (t) = 0.Moreover, from (37) we have Because all the assumptions of Theorem 5.1 are satisfied for any k > 0 and µ > 0, this equation has unbounded nonoscillatory solutions x such that lim t→∞ x ′ (t) = 0.
EJQTDE Spec.Ed.I, 2009 No. 9 Remark 5.2.When (8) holds and Im Φ is bounded, the existence of nonoscillatory solutions x of (1) such that lim t→∞ x [1] (t) = 0 has been obtained in [9] as limit of a sequence {z n }, where z n are solutions of (1) such that lim t→∞ z [1] n (t) > 0. Hence, in virtue of Lemma 2.1, the argument used in [9, Theorem 1] cannot be adapted to the case here considered.
In the next theorem, we give an asymptotic estimate for unbounded solutions of (1).Theorem 5.2.Let (Hp) be satisfied.Assume that for some α > 0 the function Φ satisfies (31) and the function F satisfies (35) and If x is a solution of the BVP (1)-( 3), then Proceeding as in the proof of Corollary 4.1, we have x [1] (t) ∞ t b(s)F (ks) ds and by (39) The assertion follows by the l'Hopital rule.

Coexistence Result
From Theorems 3.1 and 5.1 we have the following coexistence result.
The following example illustrates Corollary 6.1.x ′ (t) = 0 Example 6.1 also shows that the convergence of the integral J µ can depend on the values of the parameter µ.In view of Lemma 3.2, for the map Φ * C this fact does not occur when (15) holds.Because (35) implies (15), Corollary 6.1 cannot be applied to equation (4).

Open Problems and Suggestions
(1) Asymptotic estimations for bounded solutions.Does (33) hold for any bounded nonoscillatory solution, x, by assuming, instead of (31), that Φ is asymptotically homogeneous near zero, i.e. lim u→0 Φ(λu) Φ(u) = λ α for λ ∈ (0, 1] and some α > 0 ?(41) Condition (41) means that Φ is a regularly varying function at zero.This notion, and the analogous one at infinity, are often used both in searching for radial solutions of elliptic problems and in asymptotic theory of ordinary differential equations, see, e.g., [6,12] and references therein.
(2) The growth of solutions.When (Hp) and lim sup
2.1 and the l'Hopital rule, we obtain (33).It follows from the proof of Corollary 4.1 that bounded solutions for equations with the map Φ C have the same growth as that ones with Sturm-Liouville operator.EJQTDE Spec.Ed.I, 2009 No. 9