HETEROCLINIC SOLUTIONS FOR NON-AUTONOMOUS BOUNDARY VALUE PROBLEMS WITH SINGULAR Φ-LAPLACIAN OPERATORS

We prove the solvability of the following boundary value problem on the real line { Φ(u′(t))′ = f(t, u(t), u′(t)) on R, u(−∞) = −1, u(+∞) = 1, with a singular Φ-Laplacian operator. We assume f to be a continuous function that satisfies suitable symmetry conditions. Moreover some growth conditions in a neighborhood of zero are imposed.


1.
Introduction.The study of the existence of travelling wave solutions for reactiondiffusion equations has motivated in the recent years many papers concerning the existence of heteroclinic solutions for second order equations (see for instance [1,10,11,13]).
In the recent paper [4] Bianconi and Papalini study the non-autonomous problem Φ(u (t)) = f (t, u(t), u (t)), a.e. on R, u(−∞) = 0, u(+∞) = 1, where the usual linear second order operator u is replaced by the nonlinear one Φ(u (t)) .Here Φ : R → R is an increasing homeomorphism with Φ(0) = 0.The paradigm for this operator is the classical one-dimensional p-Laplacian The p-Laplacian operator arises in non-Newtonian fluid theory (as well as in the diffusion of flows in porus media or in nonlinear elasticity) and has became a very popular subject in the last decades (see [8,12,15,14] and references therein).Some existence results for the p-Laplacian in the presence of lower and upper solutions were extended for arbitrary increasing homeomorphisms Φ with different kinds of boundary conditions in [5,6].
Recently some papers have appeared where the authors consider Φ-Laplacian type equations with homeomorphisms Φ : (−a, a) → (−b, b) for 0 < a, b ≤ +∞ (see [2,3,7,9]).When b < +∞ the Φ-Laplacian is said to be bounded or non-surjective and the classical model is the mean curvature operator Φ(s) = s √ 1+s 2 for s ∈ R. On the other hand if a < +∞ then the Φ-Laplacian is said to be singular, in the terminology of Bereanu and Mawhin [3], and in this case the model is the relativistic operator Φ(s) = s √ 1−s 2 for s ∈ (−1, 1).In this paper we contribute to the literature studying the following boundary value problem on the real line where Φ is singular.
In [3] Bereanu and Mawhin have proven the striking result that for a singular Φ-Laplacian the Dirichlet problem is always solvable for every continuous function f and every T > 0 without additional assumptions (see also [7]).This "universal" solvability is related with the fact that all solutions of this problem have their derivatives a priori bounded.In this paper we exploit this fact in order to perform an approximation procedure to deal with our infinite interval problem.
We shall approximate problem (1)-( 2) by problems defined on compact intervals.The following result shall be very useful for us.
For each n ∈ N the modified problem has by Theorem 2.1 a solution u n : [0, n] → R with u n ∞ < a. Moreover it is easy to show that −1 ≤ u n (t) ≤ 1 and therefore u n is also a solution of (3).On the other hand (f 2) implies that u n is concave and then 0 ≤ u n (t) ≤ 1.
On the other hand, from the uniform continuity of function Φ −1 on compact sets it follows that the sequence {u n } is an equicontinuous family, and as consequence it is verified that Φ(u (t)) = f (t, u(t), u (t)) ≤ 0. So we deduce that u is nonincreasing.If u (t 0 ) < 0 at some point t 0 ≥ 0 then u (t) ≤ u (t 0 ) < 0 for all t ≥ t 0 and consequently lim  Since u is concave and bounded there exists lim t→+∞ u(t) = l ∈ (0, 1].Suppose that l < 1.From (f2) and the facts that 0 ≤ u(t) ≤ l < 1 and lim t→+∞ u (t) = 0 it follows that there exist a suitable compact set K ⊂ (−1, 1) × R, t K > 0 for which 0 < u (t K ) < 1 and a continuous function But in this case Φ(u (t)) → −∞ and then u (t) → −a < 0, which is a contradiction.Thus l = 1 and the proof is over.1−s 2 for all s ∈ (−1, 1) models mechanical oscillations subject to relativistic effects and f (t, x, y) = t 3 (x 2 −1)(y 2 n +1).Clearly conditions of Theorem 3.1 are fulfilled and so its solvability is guaranteed.

Remark 1 .
With some technical minor modifications the result of Theorem (3.1) also holds for L 1 -Carathéodory nonlinearities instead of continuous ones.Example 1.Let n ∈ N be given.Consider the problem    u (t)