ON THE SOLVABILITY OF SINGULAR BOUNDARY VALUE PROBLEMS ON THE REAL LINE IN THE CRITICAL GROWTH CASE

. Combining ﬁxed point techniques with the method of lower-upper solutions we prove the existence of at least one weak solution for the following boundary value problem where ν 1 ,ν 2 ∈ R , Φ : R → R is a strictly increasing homeomorphism extending the classical p -Laplacian, a is a nonnegative continuous function on R × R which can vanish on a set having zero Lebesgue measure and f is a Carath´eodory function on R × R 2 .


1.
Introduction. This paper is concerned with the existence of at least one weak solution for the following boundary value problem ( Φ(a(t, x(t)) x (t)) ) = f (t, x(t), x (t)) in R, where ν 1 , ν 2 ∈ R, the function a ∈ C(R × R, [0, +∞)) and may vanish on a set with zero Lebesgue measure and f : R 3 → R is a Carathéodory function, that is (i) for any (x, y) ∈ R 2 , t → f (t, x, y) is a measurable function on R, (ii) (x, y) → f (t, x, y) is a continuous function for a.e. t ∈ R.
It is worth emphasizing that the first author in [5] established the existence of at least one weak solution to the following problem Φ(a(t, x(t)) x (t)) = f (t, x(t), x (t)) in I = [0, ∞), under the same assumptions on Φ, a and f . One of the main differences between our main result (see Theorem 2.3) and Theorem 1 in [5] concerns the relative behavior of f (t, x, ·) and Φ(·) as y → 0, and of f (·, x, y) as t → ∞.
To be more precise, one of the key assumptions in [5,Theorem 1] is the existence of a constant θ > 1 with the following property: for every L > 0 one can find K L ∈ W 1,1 loc (I) such that ∞ c 1 a (t) K L (t) − 1 ρ(θ−1) dt < ∞ (for a suitable c > 0) (2) where ρ is as in (1) and a is a continuous function satisfying a (t) ≥ a(t, x) for every (t, x) ∈ I × R and 1/a ∈ L p loc (I) for some p > 1 (actually, a is defined essentially as in (7)). As it is clear from some concrete examples (see Section 5 in [5]), assumption (2) excludes the possibility that f has the critical rate of decay −1 as t → ∞, that is The main aim of the present paper is to obtain an existence result which covers the case (3) and which takes care of the fact that problem (BVP) is formulated on the whole of R.
Roughly speaking, in order to deal with the critical case (3), one needs to reformulate assumption (2) for θ = 1. As it will be clear from the proof of Proposition 2, the appropriate assumption in such a case is of exponential type, that is |t|≥T0 1 a (t) e − K L (|t|) ρ < +∞ (for a suitable T 0 > 0).
Another interesting phenomenon, which is peculiar of the critical case (3), is that the solvability of (BVP) is also influenced by the relative behavior of f and Φ with respect to x.
One of the main reason to face with problem (BVP) comes from the amount of applications of the Φ-Laplacian operators in e.g. non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity and theory of capillary surfaces; see for instance [9,19].
In view of this fact several papers have been devoted to Φ-Laplacian type equations where Φ : (−a, a) → (−b, b) is a strictly increasing homeomorphism for some 0 < a, b ≤ ∞. When a = ∞ and b < ∞, the map Φ is usually called non-surjective Φ-Laplacian, and the main prototype is the mean curvature operator Φ(s) = s √ 1 + s 2 for s ∈ R; we refer the interested reader to [1,3] and references therein. When a < ∞, the Φ-Laplacian is said to be singular, and in this case the main prototype is the relativistic operator Φ(s) = s √ 1 − s 2 for s ∈ (−1, 1); see [2,4,14,20]. In all the aforementioned papers the ODEs considered are nonsingular in the following sense: they take the form (for suitable Φ and f ) Φ(a(t, x(t)) x (t)) = f (t, x(t), x (t)) a.e. on I ⊆ R where the function a is assumed to be continuous and strictly positive. This is one of the main differences with our setting: indeed we allow the function a to vanish on a subset of R × R having zero Lebesgue measure. If this is the case, we shall say that the ODE (4) is singular. Concerning the existence of heteroclinic solutions for (BVP) (which is an interesting problem of relevance in physics/dynamic of populations) we recall the papers [16,17] in which the authors deal with the following BVP (see also [8] for the case a ≡ 1, ν 1 = 0 and ν 2 = 1) and the more recent papers [21,22] where the author considered the following BVP We stress the fact that also in these papers the authors assumed a > 0 (that is the ODE is non-singular in the previous sense).
The literature concerning BVP for singular ODEs of the form (4) (both on compact and non-compact intervals) seems to be rather incomplete. To the best of our knowledge, we mention the paper [15] in which the authors obtained the solvability of (4) assuming that a(t, x) = k(t) and I = [0, T ] for some T > 0; more recently, this paper has been generalized in [7] to the case where a depends on (t, x) (and again I = [0, T ]).
As regards the case I = (0, ∞) or I = R, we highlight the paper [6] in which the authors establish the existence of at least one weak solution of where the function k can vanish on a set with zero Lebesgue measure. As already said, this existence result has been generalized in [5] to BVPs of the form where a(t, x) is allowed to vanish on a set with zero Lebesgue measure and f (t, ·, ·) ≈ |t| γ as t → ∞, for suitable γ > −1.
The main aim of this paper is to give a further contribute in this direction: in fact we prove the solvability of (BVP) assuming that f has the critical rate of decay −1 as |t| → ∞, that is Together with this assumption, we require that f satisfies a suitable form of the so-called Nagumo-Wintner growth condition (see, precisely, assumption (H 2 )); we stress that such condition (in some stronger forms) has been profitably exploited in [16,17,21,22]. In our context, the Nagumo-Wintner growth condition allows us to obtain a priori estimates of the first derivative of any solution of (BVP) on any compact interval of R. These estimates play a fundamental role in the proof of Theorem 2.3: in fact, first we use a fixed point technique and the method of the lower/upper solutions (see assumption (H 1 )) to prove the existence of a solution u n to ( Φ(a(t, x(t)) x (t)) ) = f (t, x(t), x (t)) a.e. t ∈ I n = [−n, n] with n ∈ N sufficiently large; then by means of the a priori estimates provided by the Nagumo-Wintner growth condition we are able to prove that the sequence u n converges (in a suitable sense) to a solution of (BVP). We point out that, since we are assuming (5) we need to require a balance between the behavior of f (t, x, ·) with respect to Φ as y → 0 (see assumption (H 3 )).
Finally, we explicitly note that Theorem 2.3 comprehends also the case when a(t, x) = a(t), that is a is independent of x. As a consequence, the result in this paper complete the study started in [6,7]. To the best of our knowledge it remains open the problem when f (t, ·, ·) ≈ |t| γ as t → ∞, for suitable γ < −1.

2.
Assumptions and main theorem. We begin by giving a couple of preliminary definitions which we will use along the paper. Definition 2.1. We say that x ∈ C(R, R) is a solution to the problem (BVP) if (1) x ∈ W 1,1 loc (R) and t → Φ(a(t, x(t)) x (t)) ∈ W 1,1 loc (R), (2) ( Φ(a(t, x(t)) x (t)) ) = f (t, x(t), x (t)) for a.e. t ∈ R, We point out that if x ∈ W 1,p (R) is such that t → Φ(a(t, x(t))x (t)) ∈ W 1,1 (R), then being Φ a homeomorphism there exists a unique A x ∈ C(R, R) such that At this point we can state our main result.
) and a measurable function ψ : (0, +∞) → (0, +∞) such that and every y ∈ R with |y| ≥ H; (H 3 ) for every L > 0 there exists a nonnegative function η L ∈ L 1 (R) and a continuous function K L ∈ W 1,1 loc ([0, ∞)) null on [0, T 0 ] and strictly increasing on (ii) setting, for every t ∈ R, then, for a.e. t ∈ (−∞, −T 0 ] ∪ [T 0 , +∞), every x ∈ [α(t), β(t)] and every y ∈ R verifying the bound |y| ≤ N L (t)/a(t, x), we have (iii) setting, for almost every t ∈ R, then, for almost every t ∈ R, every x ∈ [α(t), β(t)] and every y ∈ R verifying the bound |y| ≤γ L (t) we have and every y ∈ R verifying the bound |y| ≤γ L (t), we have Then there exists at least one weak solution x ∈ W 1,p Remark 1. Let us point out that N L (t) is continuous, and using the fact that 1/a ∈ L p loc (R) we have that γ L ∈ L p loc (R). Moreover, since K L (t) is strictly increasing in [T 0 , +∞) and Φ is a strictly increasing homeomorphism, we deduce that N L (t) is strictly decreasing for any t ∈ (−∞, −T 0 ] and for any t ∈ [T 0 , +∞). In particular, gathering the definition of N L (t) and the monotonicity of Φ, we can and combining the above considerations together with (H 3 )-(i) and the fact that on account of (H 3 )-(iv) we can re-write (H 3 )-(ii) as follows: and any y ∈ R s.t. |y| ≤ N L (t)/a(t, x).
3. Solvability on compact sets. In this section we establish the existence of at least one weak solution to the following auxiliary boundary value problem on the compact interval I n = [−n, n] where n ∈ N is such that n > T 0 . Let J = [−T 0 , T 0 ], and define and and let us observe that since Φ is a strictly increasing homeomorphism on R satis- Let us introduce the truncating operators T : and D : Moreover, we introduce the truncating function f * : We then consider the following truncated problem The forthcoming result guarantees the existence of a solution to (16) and it is based on the following abstract result [7, Theorem 2.1].
be general operators satisfying the following properties: (H1) A is continuous with respect to the uniform topology of C(I n , R); moreover, there exist two functions for every x ∈ W 1,p (I n ) and every t ∈ I n ; (H2) F is continuous (with respect to the usual norms) and there exists a nonnegative function Θ ∈ L 1 (I n ) such that for every x ∈ W 1,p (I n ) and a.e. t ∈ I n .
Then, for every ν 1 , ν 2 ∈ R there exists a solution x ∈ W 1,p (I n ) of the problem More precisely, there exists a function x ∈ W 1,p (I n ) such that Theorem 3.2. Assume that the assumptions (H 1 )-(H 3 ) hold true. Then, there exists at least one weak solution u n ∈ W 1,p loc (I n ) to (16). Proof. Let us define the operators A : W 1,p (I n ) → C(I n , R) and F : Then, setting ν 1 := α(−n) and ν 2 := β(n), problem (16) becomes Put α n := min t∈In α(t) and β n := max t∈In β(t). Exploiting the definition of T (x) and using assumption (H 1 ) we get Now, since T is continuous as an operator on C(I n , R), the uniform continuity of a on I n × [α n , β n ] implies that A is continuous with respect to the uniform topology of C(I n , R).
for any t ∈ I n , and from (7) and (18) we can infer that for any t ∈ I n it holds Arguing as in Theorem 3.1 in [15] (and by exploiting the fact that D(α ) ≡ α and D(β ) ≡ β ), we can prove that F is continuous; furthermore, for every x ∈ W 1,p (I n ) and for a.e. t ∈ I n .
Since by assumption (H 3 )-(iii) we have Θ ∈ L 1 (I n ), we are in the position to apply Theorem 3.1 to deduce the existence of a solution to (16).
be any solution to (16). Then Proof. First we show that α(t) ≤ u n (t) for every t ∈ I n . Assume by contradiction that there existst ∈ I n such that α(t) > u n (t).
Similarly we can prove that u n (t) ≤ β(t) for every t ∈ I n .
In the statement of Proposition 1 below, we use the definition of A x in (6); we also recall that J = [−T 0 , T 0 ] (where T 0 is defined in Theorem 2.3, see assumption (H 1 )-(ii)), I n = [−n, n] (with n ∈ N satisfying n ≥ T 0 ) and L ≥ N is a constant such that (15) holds. Proposition 1. Under the assumptions (H 1 )-(H 3 ), let u n ∈ W 1,p (I n ) be any solution to (16). Then |A un (t)| < L for every t ∈ J.
Proof. Assume by contradiction that there existst ∈ J such that |A un (t)| ≥ L; then either A un (t) ≥ L or A un (t) ≤ −L. Assume that A un (t) ≥ L.
Therefore we can find −T 0 ≤ t 1 < t 2 ≤ T 0 such that From (25), by gathering together the definition of A un in (6), the definition of a 0 in (13) (see also (19)) and the definition of a in (7), we infer that for a.e. t ∈ (t 1 , t 2 ); on the other hand, by (14) and the fact that for a.e. t ∈ (t 1 , t 2 ).
(26) Now, using a change of variable, (26), assumption (H 2 ), Hölder inequality and (12) we obtain which is in contrast with (15). Similarly, one can prove that also the case A un (t) ≤ −L leads to a contradiction.
We point out that from Proposition (1), in particular it follows that if u n ∈ W 1,p (I n ) is a solution to (16) such that α(t) ≤ u n (t) ≤ β(t) for every t ∈ I n , then for a.e. t ∈ J. , n] such that A un (t 2 ) = 0, then A un (t) = 0 for any t ∈ [t 2 , n]; (iv) |A un (t)| ≤ N L (t) for every t ∈ I n ; (v) |u n (t)| ≤ γ L (t) for a.e. t ∈ I n .
Similarly, we can prove that (Φ(A un (t))) ≤ 0 every t ∈ [−n, −T 0 ]. From the monotonicity of Φ it follows the desired result. (ii) Assume by contradiction that there exists t 1 ∈ [−n, −T 0 ] such that A un (t 1 ) < 0. From (i) for any t ∈ [−n, t 1 ] we have A un (t) ≤ A un (t 1 ) < 0; as a consequence, by taking into account the definition of A x in (6), we deduce that Now, since u n solves (16), using (28), (19) and assumption (H 1 )-(ii), we have which gives a contradiction. Similarly, arguing by contradiction we can find t 2 ∈ [T 0 , n] such that A un (t 2 ) < 0. From (i) for any t ∈ [t 2 , n] we have as a consequence, we obtain u n (t) = A un (t) a(t, u n (t)) < 0 a.e. t ∈ [t 2 , n].
By proceeding exactly as above, from (29) we obtain which is a contradiction.
(iii) The proof of this statement easily follows by combining (i)-(ii).
(iv) Combining Proposition 1 and Remark 1 we have that |A un (t)| < L = N L (t) for any t ∈ J. In the light of (ii) it remains to prove that Lett Assume by contradiction thatt > −n. Firstly, let us note that A un (t) > 0 for all t ∈ [t, −T 0 ]. Indeed, from the definition oft it follows that A un (t) < N L (t) for any t ∈ (t, −T 0 ]. Moreover, by (iii) if there existst ∈ [t, −T 0 ] such that A un (t) = 0, then A un (t) = 0 for any t ∈ [−n,t], and recalling that N L (t) > 0 we have that (6) and (7) we get Since u n solves (16), exploiting (31), the definition of D and assumption (H 3 )-(ii) we get where we have also used assumption (H 3 )-(iv). Recalling that A un (t) > 0 for any t ∈ [t, −T 0 ], Φ is strictly increasing and Φ(0) = 0 we can infer that Integrating both sides of this last inequality on [t, −T 0 ], and taking into account Now, let us note that by Proposition 1 we get A un (−T 0 ) < L, thus Φ(A un (−T 0 )) < Φ(L) (as Φ is increasing); as a consequence, by combining this last inequality with (32), we have From this, we obtain and this contradicts the definition oft. Thus we conclude thatt = −n. In a very similar way one can show that |A un | ≤ N L on [T 0 , n], and the proof is complete. 4. Solvability on the real line. Now, using a limit argument, we give the proof of Theorem 2. 3. In what follows, we inherit all the notation introduced so far; in particular, we use the definition of A x in (6), the definitions of N L and γ L in (8)-(9) (with L ≥ N as in (15)), and I n = [−n, n].
Proof of Theorem 2.3. Let u n ∈ W 1,p (I n ) be a solution to (16) given by Theorem 3.2. Taking into account the definitions of T , D and f * , and using Proposition 2-(v), we can infer (Φ(a(t, u n (t)) u n (t) )) = f * (t, u n (t), D(u n )(t)) = f (t, u n (t), u n (t)).
Thus u n is a solution to (11). Let us define {x n } n∈N ⊂ W 1,p (R) as follows For every n > T 0 , let otherwise .
Thus, by applying the Dunford-Pettis Theorem we deduce the existence of two functions g, h in L 1 (R) such that, up to a sub-sequence, y n g and z n h in L 1 (R). Therefore, for every measurable subset A ⊂ R we have Now, from Lemma 3.3 and Proposition 1 we deduce that the sequences {u n (0)} n∈N and {A un (0)} n∈N are bounded, so we can assume that u n (0) → u 0 and A un (0) → y 0 for some u 0 , y 0 ∈ R. Hence Let us define Then we have that x ∈ C(R), x(0) = u 0 and x (t) = g(t) a.e. t ∈ R. Moreover, from the definition of x n (t) and the fact that α, β ∈ C(R, R), it follows that α(t) ≤ x(t) ≤ β(t) for every t ∈ R. Now, if t ∈ I n , then x n (t) = u n (t) and y n (t) = u n (t), thus Taking into account that a ∈ C(R × R, (0, +∞)), from (36) we get Recalling (34), we can apply the Dominated Convergence Theorem to prove that Since x n = y n g in L 1 (R), we can infer that From (38), (39) and x (t) = g(t) a.e. t ∈ R we have Let us observe that by (37) and (39) we have x n (t) → g(t) = x (t) a.e. t ∈ R as n → +∞. Moreover, taking into account that u n is a solution to (11) on I n and x n = u n a.e. t ∈ I n , it is possible to find a set F ⊂ R, independent on n with vanishing Lebesgue measure, such that, for every n > T 0 and every t ∈ I n \ F we have z n (t) = (Φ(a(t, u n (t))u n (t))) = f (t, u n (t), u n (t)) = f (t, x n (t), x n (t)).
Since x n → x and f is a Carathéodory function, we get lim n→+∞ z n (t) = f (t, x(t), x (t)) for every t ∈ R \ F.
On the other hand, applying the dominated convergence theorem we obtain As a consequence, since z n h in L 1 (R) we obtain (Φ(a(t, x(t))x (t))) = h(t) = f (t, x(t), x (t)) a.e. t ∈ R, and this proved that x is a solution to the (ODE). Finally, we observe that see (33) and (35) = |u n (0) − u 0 | + x n − x L 1 (R) for every n ∈ N; as a consequence, since x n → x in L 1 (R) and u n (0) → u 0 as n → ∞, we conclude that x n → x uniformly on R. In particular,

A class of examples.
Let ν 1 , ν 2 ∈ R be such that ν 1 < ν 2 . We consider the following BVP: where the functions a, Φ, f 1 and f 2 fulfill the assumptions listed below: (I) a : R × R → R is a non-negative continuous function on R × R which is also bounded on R × [ν 1 , ν 2 ]. Furthermore, we suppose that it is possible to find a real p > 1 and a real σ > 0 such that, setting we have 1/h ∈ L p loc (R) and (II) Φ : R → R is an odd, strictly increasing homeomorphism from R onto itself; moreover, there exists ρ > 0 such that is a Carathéodory function enjoying the following properties: for a.e. t ∈ R and every x ∈ [ν 1 , ν 2 ]; (III) 2 it is possible to find a real number T 0 > 0, two real constants c 1 , c 2 > 0 and a real number δ ≥ −1 such that for a.e. |t| ≥ T 0 and every x ∈ [ν 1 , ν 2 ]; (III) 3 t · f 1 (t, x) ≤ 0 for every |t| ≥ T 0 and every x ∈ [ν 1 , ν 2 ].
(IV) f 2 ∈ C(R, R) and it enjoys the following properties: (IV) 1 f 2 > 0 on R \ {0} and f 2 (0) = 0; (IV) 2 there exists a real number y > 0, two real constants c 1 , c 2 > 0 and a real number γ ≤ 1 such that for every y ∈ R with |y| < y ; (IV) 3 there exist a real H > 0 and a real constant c 3 > 0 such that, if y ∈ R and |y| ≥ H, the following estimate holds true: (IV) 4 b is homogeneous of degree d > 0 on R, that is, f 2 (sy) = s d f 2 (y) for every s > 0 and every y ∈ R.
Finally, introducing the constant (see also assumption (I)) we suppose that the following relations hold: Our aim is to prove that, in the present setting, all the hypotheses of Theorem 2.3 are satisfied; as a consequence, there exists a solution x ∈ W 1,p loc (R) of (40). We explicitly point out that in the particular case when (43) holds with δ = −1 we cover the critical case Remark 3. Before proceeding we highlight, for a future reference, a few consequences of the above assumptions (I)-to-(IV) we shall use in the sequel.
(a) For every υ ∈ (−∞, p] one has In fact, since a(t, x) ≥ h(t) ≥ 0 for any (t, x) ∈ R × [ν 1 , ν 2 ] and a is bounded on the same set, we obviously have that h is bounded on R; hence, the map t → |t| −σ /h p being integrable on {|t| ≥ 1}, for every υ ∈ (−∞, p] we have In fact, if y ∈ R is such that |y| ≤ ζ, from the fact that Φ is an odd increasing function on R (see assumption (II)) we infer that (c) By combining assumption (III) 3 with estimate (43) we easily infer that t · f 1 (t, x) < 0 for every t ∈ R \ [−T 0 , T 0 ] and every x ∈ [ν 1 , ν 2 ].
Indeed, if x ∈ [ν 1 , ν 2 ] and |t| ≥ T 0 > 0 we have (d) By combining the growth condition (45) in assumption (IV) 3 with the dhomogeneity of f 2 in assumption (IV) 4 it is readily seen that In fact, if H > 0 is as in assumption (IV) 3 and if y > H is arbitrarily fixed, by (45) and the d-homogeneity of f 2 one has s d f 2 (y) = f 2 (sy) ≤ c 3 s 1−1/q y 1−1/q , for every s ≥ 1, but this is possible only if d ≤ 1 − 1/q.
We now prove that all the hypotheses of Theorem 2.3 are satisfied. First of all, on account of assumptions (II), we have that Φ satisfies (1) (with a suitable ρ > 0); furthermore, since f 1 is a Carathéodory function on R × R and f 2 is continuous on R (as it follows from assumptions (III) and (IV)), the function is a Carathéodory function in R 3 .
Hypothesis (H 1 ). We now prove that hypotheses (H 1 ) in the statement of Theorem 2.3 is satisfied. To this end we observe that, since f 2 (0) = 0 (see (IV) 1 ), the (constant) functions are, respectively, a lower and a upper solution of the ODE Moreover, it is straightforward to check that • α ≤ β on R (since ν 1 < ν 2 ); • α is increasing on (−∞, −T 0 ] and β is increasing on [T 0 , ∞) (where T 0 > 0 is the same number appearing in assumption (III)); Finally, since we have (by (7) and (48)) from assumption (I) we infer that 1/a = 1/h ∈ L p loc (R) (for a suitable p > 1). Hypothesis (H 2 ). In this paragraph we prove that also hypothesis (H 2 ) is satisfied. In fact, if H > 0 is as in assumption (IV) 3 , by exploiting estimate (45) and taking into account assumption (III) 1 we obtain and T 0 > 0 as in assumption (III) 3 . We explicitly point out that, since the function λ is in L ∞ loc (R) (see (III) 1 ), we have µ ∈ L q ([−T 0 , T 0 ]). Hypothesis (H 3 ). In this last paragraph of the section we prove that, in our setting, hypothesis (H 3 ) in the statement of Theorem 2.3 is satisfied.
To begin with, if T 0 > 0 is as in assumption (III), we consider the function where γ : R → R is defined as follows: Since, by assumption (III), the map x → f 1 (t, x) is continuous on R for a.e. t ∈ R, the function K 0 is well-defined; moreover, as and λ is in L ∞ loc (R), it is readily seen that K 0 ∈ W 1,1 loc ([0, ∞)). Finally, K 0 is strictly increasing on [T 0 , ∞) (see ) and, by (43), we have If c 1 > 0 is as in (44), we set and we consider the function H L : R → R defined by Since, by (51), K 0 (t) → ∞ as t → ∞, we have that H L (t) → 0 as t → ±∞; as a consequence, it is possible to find a real t L > T 0 such that We then claim that the function K L defined by satisfies all the properties in hypothesis (B4). Indeed, on the one hand we have ( * ) K L ∈ W 1,1 loc ([0, ∞)), since the same is true of K 0 (see also (42)); ( * ) K L ≡ K 0 ≡ 0 on [0, T 0 ] and K L is strictly increasing on [T 0 , ∞), since the same is true of K 0 and t L > T 0 (see also  and (52)).
Furthermore, since K L ≥ c(L) K 0 on [0, ∞) (note that c(L) ≤ c 1 /(M a ) d , see the above (52)), by the very definition of N L we have By combining this last inequality with (53) we then get As a consequence, for almost every t ∈ R with |t| > t L , every x ∈ [ν 1 , ν 2 ] and every y ∈ R with |a(t, x)y| ≤ N L (t) ≤ y we obtain by (44) and (46) ≥ K L (|t|) Φ(a(t, x)y) by the very definition of K L .
On the other hand, since N L ≤ L on R, for almost every t ∈ R with |t| ∈ [T 0 , t L ], every x ∈ [ν 1 , ν 2 ] and every y ∈ R with |a(t, x)y| ≤ N L (t) we have by the very definition of K L .
We now turn to prove that K L satisfies assumption (H 3 )-(i). To this end we first notice that, since K L is continuous on [0, ∞) and 1/a = 1/h is in L p loc (R) (see assumption (I) and (50)), we obviously have On the other hand, by the very definition of K L and by using (43) one has where c > 0 is a suitable constant. From this, since the map is integrable on {|t| ≥ 1} (see Remark 3-(a)) and since, by (47), we have By combining (55) with (56) we finally obtain the validity of (H 3 )-(i).
To this end we observe that, on account of (54), for a.e. t ∈ R, every x ∈ [ν 1 , ν 2 ] and every y ∈ R such that |a (t)y| ≤ N L (t) we have the following estimate:  We now show that η L is in L 1 (R). On the one hand, since 0 < d ≤ p (see Remark 3-(c)) and λ ∈ L ∞ loc (R), we have (also remind that 1/a = 1/h ∈ L p loc (R)) On the other hand, by crucially exploiting estimate (43) (and taking into account the very definition of N L , see (8)) we obtain (as d ≤ p, see Remark 3) and since, by (47), we have it is readily seen that By combining (57) with (59) we then conclude that it is possible to find (at least) one solution x ∈ W 1,p loc (R) (where p > 1 is as in assumption (I)) of the BVP (40), further satisfying ν 1 ≤ x(t) ≤ ν 2 for every t ∈ R.
Remark 4. It is worth noting that, in the particular case when also the homeomorphism Φ in assumption (II) is homogeneous of a certain degree g ∈ (0, p], our growth assumption (IV) 3 can be replaced with the following one: (IV) 3 there exist a real H > 0 and a real constant c 3 > 0 such that, if y ∈ R and |y| ≥ H, the following estimate holds: f 2 (y) ≤ c 3 |Φ(y)| α for some α ≤ 1.