Solvability for some boundary value problems with φ-Laplacian operators

We study the existence of solution for the one-dimensional φ-laplacian equation (φ(u′))′ = λf(t, u, u′) with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate λ0 is given such that the problem possesses a solution for any |λ| < λ0.

The study of the φ-laplacian equation is a classical topic that has attracted the attention of many researchers because of its interest in applications. Usually, a φlaplacian operator is said singular when the domain of φ is finite (that is, a < +∞), 728 J.ÁNGEL CID AND PEDRO J. TORRES on the contrary the operator is said regular. On the other hand we say that φ is bounded if its range is finite (that is, b < +∞) and unbounded in other case. There are three paradigmatic models in this context: • a = b = +∞ (Regular unbounded): the p-laplacian operator • a < +∞, b = +∞ (Singular unbounded): the relativistic operator • a = +∞, b < +∞ (Regular bounded): the one-dimensional mean curvature operator Among them, the p-laplacian operator has deserved a lot of attention and the number of related references is huge (see for instance ([5, 6, 7, 8, 9, 10] and references therein). For the relativistic operator, it has been proved in the recent paper [2] that the Dirichlet problem is always solvable. This is a striking result closely related with the "a priori" bound of the derivatives of the solutions. For the curvature operator, this is no longer true, but other results about existence and multiplicity of solutions can be obtained by variational [4] or topological approaches (see the thesis [3] for a more complete bibliography). The purpose of this note is to contribute to the literature by proving the existence of solution for small λ, giving an explicit estimate. This complements in part the results in [4]. Moreover we extend some previous results of Bereanu and Mawhin [1,2]. The proof is elementary and relies on Schauder's fixed point theorem after a suitable reduction of the problem to a first order integro-differential equation.
For convenience, for each 0 < r < b let M r be defined as where 2. The Dirichlet boundary value problem. Let us consider the boundary value problem under the conditions given in the introduction. Let us define the space Of course, b 2 must be understood as +∞ when b = +∞. The following result is a slight modification of [2, Lemma 1], but we include the proof for the sake of completeness.
Besides, |Q φ [y]| ≤ y ∞ and the function Q φ : Proof. By the properties of φ, it is clear that By Bolzano's theorem, there exists α verifying (6) with |α| ≤ y ∞ . Moreover, this constant is unique by the increasing character of φ −1 . To check the continuity assume that {y n } ⊂ H is a sequence converging to some y ∈ H. Then Q φ [y n ] → c (taking a subsequence if it is necessary) and by the dominated convergence theorem we have Therefore c = Q φ [y] and the proof is complete.
By means of a suitable change of variables we relate the problem (5) with the non-local first order equation Lemma 2.2. If y is a solution of problem (7) with y ∞ < b 2 then is a solution of problem (5).
The proof of the lemma is direct and thus we omit it. Now, we are in a position to prove the main result of this section: the solvability of problem (5) for small λ. Proof. Let 0 < r 1 < b/2 be such that |λ| ≤ r1 For each y ∈ B r1 define the operator It is easy to show that T is completely continuous. Moreover by our assumptions and the choice of r 1 we have which implies that T (B r1 ) ⊂ B r1 . Thus Schauder's fixed point theorem yields a fixed point for T which is a solution of equation (7) and therefore by Lemma 2.2 it is also a solution for problem (5 (1), for small and/or large λ > 0, where φ(u) = u √ 1+u 2 is the mean curvature operator. The main advantage of our approach is the simplicity on the assumptions and the fact that the constant λ 0 is established explicitly. Regrettably our method doesn't avoid in general the existence of the trivial solution. Now we are going to apply Theorem 2.3 to study the solvability of the Dirichlet

Remark 2. In [4] the authors obtain the existence of a positive solution to problem
extending some previous results in [1,2]. We point out that problem (8) presents interesting different features depending on the bounded or unbounded behavior of φ.
Corollary 1. Assume that φ is unbounded (that is, b = +∞) and there exists h ∈ L 1 (0, 1) such that Then the Dirichlet problem (8) has at least one solution.
Proof. By condition (9) it is clear that Therefore and then Theorem 2.3 ensures us that problem (5) has a solution for each λ ∈ R, and in particular for λ = 1.

2.2.
Bounded φ-laplacian (b < +∞). In the case of bounded φ-laplacian the "universal" solvability of (8) is not longer true even for a constant nonlinearity f (t, u, v) ≡ M as we show in the following result.
Proposition 1. Assume that φ is bounded (that is, b < +∞), let M ∈ R and consider the Dirichlet problem Then the following claims hold: (i) If |M | ≥ 2b then the problem (10) has no solution.
(ii) If |M | < 2b and moreover φ is odd then the problem (10) has a solution.
In this case u given by (11) would not be well defined since the domain of φ −1 is the interval (−b, b) and thus a solution of (10) can not exist.
Both claims of Proposition 1 apply to the one-dimensional mean curvature operator φ(s) = s √ 1+s 2 since it is a bounded and odd homeomorphism. Corollary 2. The Dirichlet boundary value problem has a solution if and only if |M | < 2.
As consequence of Theorem 2.3 we obtain the following sufficient condition for the solvability of the Dirichlet problem which extends a previous result in [1].
Then the Dirichlet problem (8) has at least one solution.
Proof. Now, for each 0 < r < b we have that M r = h 1 < b 2 . Therefore λ 0 = sup 0<r< b 2 r M 2r > 1, and thus Theorem 2.3 implies the existence of a solution for problem (5) with λ = 1.
3. The mixed boundary value problem. If compared with the Dirichlet problem, the mixed boundary value problem (φ(u ′ )) ′ = λf (t, u, u ′ ) for a.a. t ∈ I, u(0) = 0 = u ′ (1), is less studied in the related literature. In this case, by means of the change of variables y = φ(u ′ ) we have that a solution u : I → R of (12) is equivalent to a solution y : I → (−b, b) of the following non-local first order terminal value problem y ′ (t) = λf t, t 0 φ −1 (y(s))ds, φ −1 (y(t)) for a.a. t ∈ I, y(1) = 0.
By using the same idea as in Theorem 2.3, we can prove the following result.