Electronic Journal of Qualitative Theory of Differential Equations

Problem of the type −�pu = f(u) + h(x) in (a, b) with u = 0 on {a, b} is solved under nonresonance conditions stated with respect to the first


Introduction
This paper is mainly concerned with the following quasilinear two-point boundary value problem  We denote by the set of couples of positive numbers (µ + , µ − ) such that the homogeneous problem on {a, b} has a nontrivial solution u.Here u + = max(u, 0), u − = u + − u.The set is called the Fučik spectrum of the p-Laplacian operator −∆ p on W 1,p 0 (a, b).Denote respectively by λ 1 and λ 2 the first and the second eigenvalue of −∆ p on W 1,p 0 (a, b).It is well known that is composed of two trivial lines λ 1 × R and R × λ 1 , and of a sequence of hyperbolic-like curves (cf [2], [6]).The first curve C 1 passes through λ 2 and is the set where π p = 2(p − 1) 1/p 1 0 ds Let us denote by F the primitive of f defined by F (s) = s a f (t) dt.In some previous works (see for instance [1], [3], [4], [10] ) many authors have proved the solvability of (P) when h ∈ L ∞ (a, b) under various nonresonance assumptions on either the nonlinearity f , or on the primitive F , or on both f and F .As far as nonresonance conditions are considered at the right of λ 1 , the Dolph-type condition: is sufficient to yield solvability of (P) when h ∈ L ∞ (a, b) ( See [1]).It was observed in a recent work in [3], that weaker conditions with respect to the first curve in the Fučik spectrum such as yield the same conclusion.Adapting an example given in [5], one can observe that assumption (3) cannot be relaxed to lim inf Our purpose in the present paper is to weaken nonresonance conditions (2) and (3) at the light of a recent contribution in [11] for p = 2. Indeed, in [11] the solvability of (P) when p = 2 occurs under assumptions such as and where µ 1 is the first eigenvalue of −∆, on H 1 0 (a, b) and (µ, ν) is such that It is worth noticing that the roles of s at infinity in (6) are interchangeable.Clearly, assumption such as (6 )improves (3) in the particular case of p = 2 and the question naturally arises to know whether similar assumption can be extended to the p-Laplacian.The aim of this work is to investigate such a problem and as a result of this investigation we have the following.
As a consequence of our main result we have the following Needless to mention that the limits at ∞ are interchangeable.Thus, our result improves [3] in what concerns the conditions with respect to the first curve in the Fučik spectrum.The proof of Theorem 1-1 is given in section 4. Basically, it uses time-mapping estimates to yield the needed a-priori bounds for a suitable parametrized problem related to (P) and combines topological degree argument to conclude .Our section 2 is devoted to the establishment of general properties for quasilinear differential equations useful for the proof of our main result.In section 3 we have given new estimate results for the time-mapping related to the p-Laplacian and accordingly improved some estimate results stated in [10].Those estimates play a central role in the proof of Theorem 1-1.

General properties
Here we give general results for a large class of parametrized quasilinear problem of the form For any solution u of (Q γ ) we set here and henceforth the following Writing the first equation in (Q γ ) in the planar system with f(x, s, γ) = f (x, s, γ) + c and H(x) = x a (h(t) − c) dt, we derive the following.
Lemma 2.1.A positive constant L exists such that any solution u of (Q γ ) satisfying max u > L, fulfills the following conditions: there exist uniquely determined real numbers ρ and ρ, with α 0 < ρ ≤ x * ≤ ρ < β 0 such that (ii) u is strictly increasing on [α 0 , ρ], strictly decreasing on [ρ, β 0 ] and If furthermore lim be obtained in the case that u is a solution of the planar system with min u < −L and lim uniformly (for γ ∈ [0, 1] and a.e.x ∈ [a, b]).In this case f and H in the planar system are written f (x, s, γ The proof of Lemma 2.1 in the particular case p = 2 is given in [11].We give here the general case for any p > 1.So, let us consider α 0 , β 0 , x * as set in the definitions 2.1, 2.2.Since y ′ (x) = − f (x, u(x), γ), from the sign condition on f, we have that y is strictly decreasing on (α 0 , β 0 ) and accordingly y(α 0 ) > y(x * ) > y(β 0 ) Moreover u ′ (x * ) = 0, and then (7) yields be the maximal interval containing x * and such that Clearly, for such an interval, part (i) of Lemma 2.1 holds.Since ϕ p is a bijection on R, one can write (7) on the form u ′ (x) = ϕ −1 p (y(x) − γ H(x)) And then using the monotoniciity of ϕ p and (9), we have for s ∈ [ρ, ρ] Accordingly, for x ′ and x ′′ in [ρ, ρ], we get Consequently we get and then Since max u ≥ K, we have α K ∈ [α 0 , x * ] and then by using part (i) of the lemma and condition (10) we have And then Next, we derive from the integration of u ′ on [α K−L , α K ] the following inequality To go further with the integral in the left hand-side of (11), let us set p is an even strictly increasing function on R + .Let us denote by Ψ * −1 p (s) its positive inverse function on R + .
EJQTDE, 2009 No. 57, p. 7 On the other hand, the function is differentiable with respect to x and so Combining ( 11)and ( 12), we get Now, using the positive inverse of Ψ * p , one has One can easily see that the right hand-side of inequality ( 13) is equivalent to k −1/p at infinity.So when K tends to +∞, k tends to +∞, and then lim Proof Let us consider only the case max u < A, the second case min u > −A of course can be proved similarly.Thus, suppose on the contrary that there exist a sequence (γ n ) ∈ [0, 1] denote (γ) for sake of simplicity of notation, and corresponding solutions u n of (Q γ ), with max u n < A and min u n tending to −∞.Then, from the sign condition and the L 1 -Carathéodory condition on f we have where for any set E, χ E denote its characteristic function.Choose two points x * n and x * n such that u n (x * n ) = max u n and u n (x * n ) = min u n .We can suppose without loss of generality that Then after the multiplication of the first equation in (Q γ ) with ũn and its integration over [x * n , x * n ], we have and then we have From the Hölder inequality we have Combining the above inequality and (14), we get So the sequence (ũ n ) is bounded and hence (u n ) is bounded.This is a contradiction to the fact that min u n tends to −∞.

Time-mapping estimates
Let's consider the initial value problem Where g : R → R is a continuous function satisfying sgn(s)g(s) > −c for c > 0 and G(s) → +∞.
The function τ g defined by p * 1/p and p * = p p−1 is the time-mapping associated to (I).Under the assumptions sgn(s)g(s) > −c for c > 0 and G(s) → +∞ when |s| → +∞, τ g (s) is well defined for |s| large enough.By adapting arguments developed in [12] for the case p = 2, one can easily derive that for s large enough (I) admits a periodic solution u s with ||u s || ∞ = s and τ g (s) is the value of the half period.Time-mapping enables to provide a-priori estimates for solutions of boundary value problems ( cf [7], [10], [11], [12]).Here, we give new results on the time-mapping estimates extending and even improving some results in [7], [10], [11], [12].Lemma 3.1.Assume that there exist positive real numbers k ± and k ± such that One can notice that under the assumption sgn(s)g(s) > −c for c > 0 and the fact that k ± k ± are greater than 0, G(s) → +∞ when |s| → +∞ so that τ g (s) is well defined for s large enough.Let's limit the proof of the lemma to the cases lim sup So for ǫ > 0, there is a real number s 0 < 0 such that for s < s 0 we have Recalling the expression of τ g (s) and taking into account inequality above, we get Setting z = ξ/s, one has For the case lim inf s→−∞ pG(s)/|s| p = k − , we have ∀ǫ > 0, there is a real number s 0 < 0 such that for s < s 0 So, for ǫ sufficiently small such that k − − ǫ > 0, we have And by a simple computation as previously done we get

Auxiliary functions related to the time-mapping
Let us consider the following parametrized problem for each γ ∈ [0, 1], and where g(s, γ) = g(s, γ) + c for s ≥ 0 and g(s, γ) = g(s, γ) − c for s ≤ 0 The planar system equivalent to the first equation in (P γ ) is written for x ∈ (a, b) and γ ∈ [0, 1].It is clear that the planar system (16), ( 17) is a particular case of the planar system (7), ( 8) and hence Lemma (2.1) is valid for any solution of ( 16), (17) as well.
For any solution u of ( 16), (17), let us consider the function T ǫ where ǫ = ±1 and defined by with γ ∈ [0, 1] and p > 1.One can easily see that Recalling part (i) of Lemma (2.1), we derive that: for ǫ = −1 EJQTDE, 2009 No. 57, p. 12 for ǫ = 1 So recalling again part (i) of Lemma (2.1), one can easily check that Accordingly we have Taking into account the expressions of T −1 (x) and T 1 (x) and recalling again (i) of Lemma 2.1, we get Next, by setting ξ = u(x), we get From Lemma (2.1), u(ρ) and u(ρ) are greater than max u − L and then writing max u = s in (20), one has for s > L.
In conclusion we have for |s| > L.
Thus T ǫ provides lower estimates for the length of the intervals [α 0 , β 0 ] and [α ′ 0 , β ′ 0 ].Let us now deal with upper estimates provide by T ǫ .Going back to (18) and to the expressions of T 1 (x) and T −1 (x) we derive that . So let us consider inequalities in (24)and (25) respectively on [α 0 , α max u−L ] and [β max u−L , β 0 ] and let us assume that the following is satisfied (we will show EJQTDE, 2009 No. 57, p. 14 farther in section 4 that such a condition is indeed satisfied under suitable condition ): Then, we derive after the change of variable ξ = u(x) in ( 24) and (25), that Arguing in a similar way, one can show that One can easily see that according to (21), ( 22) with K = 2|| H|| ∞ On the other hand, the following lemma shows that τ γ (s) is a good approximation of T γ (s) for s large enough.Proof Without loss of generality, we can suppose that it is T γ (s) which is bounded uniformly with respect to γ. Furthermore the proof will be given only for the case s → +∞, the case s → +∞ can be dealed similarly.So, let us consider for s > 0.
Since lim s→+∞ g(s, γ) = +∞ uniformly with respect to γ, for any A > 0 there exists a real number d > 0 such that and Choose s such that s > d + L with L > 0, Let us split the integral I as follows Dealing with the first term of this decomposition, we get by using the monotonicity of ϕ −1 p Tending s to infinity, we notice that the right-hand side integral tends to zero.So for s large enough we have To deal with the second term I 2 ,we write it as follows In order to estimate I 2 the following inequalities will be useful.

Claim 1.
(i) A positive constant D exists such that for any real numbers a, b Proof.For the case (i), on can refer to [9].In order to prove (ii), let us consider the function Obviously r is derivable and its derivative function r ′ (y The second inequality in (ii) follows similarly and thus claim (1) is proved.Now, let's go ahead with the proof of the Lemma (3.2).Using (i) of claim ( 1) No. 57, p. 17 for p ≥ 2. For 1 < p < 2, we have 1 p−1 > 1 and then we can apply the first inequality in (ii) of claim (1) with 1  p−1 playing the role of p, a = K, b = ( Gγ (s) − Gγ (ξ)) p−1 p .Thus, we have In conclusion: For p > 2, Since T γ is uniformly bounded with respect to γ, by choosing A large enough we have EJQTDE, 2009 No. 57, p. 18 Here again, for A large enough, we have 2c p I 2 < ǫ/2 and finally, we get 2c p (I 1 +I 2 ) < ǫ for s large enough, that is τ γ (s)−T γ (s) < ǫ for s large enough.
An analogous of Lemma (3.2) holds when K is a negative real number.In order to state it let us start define and K a negative real number.
Lemma 3.3.Assume that lim |s|→+∞ g(s, γ) = +∞ uniformly with respect to γ and that at least one of the functions Tγ (s) and τγ (s) is uniformly bounded with respect to γ.Moreover, suppose that the following condition is satisfied: Then respectively lim Proof.The proof is not too different of that of Lemma (3.2).We will sketch it below.Suppose that Tγ (s) is uniformly bounded with respect to γ and let's give the proof when s → +∞ (the other cases being similar).EJQTDE, 2009 No. 57, p. 19 Since Tγ (s) > τγ (s) for s > L, we shall just have to prove that for any ǫ > 0, Tγ (s) − τγ (s) < ǫ for s sufficiently large.
with d as in ( 31)and (32).And then So for s large enough, we have 2c p Ĩ1 < ǫ/2.
To estimate Ĩ2 in the case p ≥ 2, we proceed similarly as in the proof of Lemma (3.2) by using (i)of claim (1) to yield Hence 2c p Ĩ2 ≤ ǫ/2 for s large enough and p ≥ 2. In the case 1 < p < 2, following the same way as in Lemma (3.2), we apply the (ii) of claim(1) by writing the numerator of I 2 in the form and by setting a = −K, b = K + ( Gγ (s) − Gγ (ξ)) p−1 p ; and then we obtain where A is as in (31).Since τγ (s) is uniformly bounded with respect to γ, we have 2c p Ĩ2 < ǫ/2 for s large enough and then Tγ (s) − τγ (s) < ǫ for s sufficiently large.
4 Proof of the Theorem.
Let us consider the following parametrized problem Notice that the function defined by : (s, γ) → (1 − γ)θϕ p (s) + γf (s) is a particular case of the function f and accordingly under the assumptions of Lemma (2.1)(respectively Lemma (2.2)), the conclusions of Lemma (2.1)(respectively Lemma (2.2)) are also valid for (S γ ) as well.
Under the assumptions (h 1 ), (h 3 ) of the theorem the following lemmas hold.(ii) When (h 1 ) and the second part of (h 3 ) hold, there exists a sequence T n → −∞ such that if u solves (S γ ) for some γ ∈ [0, 1] and u changes sign, then min u = T n for every n and every γ ∈ [0, 1].
Proof.We prove only the first statement, the proof of the second one being similar.For such a Gγ we associate the function H defined by for all s ∈ [0, s n [ with s n > L = (b − a)ϕ −1 p (2|| H|| ∞ ) Choose S n as a tail sequence of the sequence s n and suppose that with such a S n , Lemma 4.1 is false.Then, one can find a subsequence of s n still denoted by s n , and solutions u n of (S γ ) for γ = γ n ∈ [0, 1] satisfying max u n = s n → +∞, and hence according of Lemma (2.2), min u n → −∞.Let's show below that such a sequence solutions leads to a contradiction.So, let's consider the real numbers (α 0 , ρ, ρ, β 0 ) and (α ′ 0 , ρ ′ , ρ′ , β ′ 0 ) corresponding to the sequence solutions u n as respectively in Lemma (2.1) and in its dual version.For the sake of simplicity we will keep the notations, α 0 , ρ, ρ, β 0 and α ′ 0 , ρ ′ , ρ′ , β ′ 0 , however those numbers depend on n .Recalling inequality (29), that is (β 0 − α 0 ) ≥ T γ (max u n ) for EJQTDE, 2009 No. 57, p. 22 max u n > L and using Lemma 3.2, we get ∀ǫ > 0, (β 0 − α 0 ) ≥ τ γ (S n ) − ǫ for S n large enough.Next, combining (39) and inequality above, we get This is a contradiction so Lemma 4.1 is proved.
The following lemma provides a-priori bounds for solutions of (S γ ) having a constant sign.Then, there are two constants K > 0, K ′ < 0 such that there is no nonnegative solution or respectively no non-positive solution u of (S γ ) for some γ ∈ [0, 1] such that max u ≥ K or respectively min u ≤ K ′ .
We denote by x * the first point of maximum of u and x * the last point of minimum of u.