Existence of positive solutions of a superlinear boundary value problem with indefinite weight

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^{p}$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.


Introduction
In this paper we are interested in the study of positive solutions for the nonlinear two-point boundary value problem u + a(x)g(u) = 0 u(0) = u(L) = 0, (1.1) where a : [0, L] → R is a Lebesgue integrable function and g : R + → R is a continuous function, where R + := [0, +∞[ denotes the set of non-negative real numbers. We recall that a positive solution of (1.1) is an absolutely continuous function u : [0, L] → R + such that its derivative u (x) is absolutely continuous, u(x) satisfies (1.1) for a.e. x ∈ [0, L] and u(x) > 0 for every x ∈ ]0, L[. This issue has been considered by many authors.
As classical examples, we mention [1,2,3,5,8,11] (see also the references therein), where different techniques are used to face this type of problem. Our work benefits from a new approach based on the Leray-Schauder topological degree, so, to obtain a positive solution, our goal is to prove that the degree of a suitable operator is non-zero on an open domain of C([0, L]) not containing the trivial solution.
Our assumptions allow the weight function a(x) to change its sign a finite number of times and, concerning the nonlinearity, we suppose that g(s) can change its sign, even an infinite number of times, and that, roughly speaking, it has a superlinear growth at zero and at infinity. More in detail, with respect to the growth of g(s)/s at zero, we assume a very general condition which depends on the sign of g(s) in a right neighborhood of zero.
Our main result states that, under the conditions just presented, problem (1.1) has at least a positive solution. This theorem clearly covers the case g(s) = s p , with p > 1. Moreover, the results concerning the BVP (1.1) where is assumed that a(x)g(s) ≥ 0 for a.e. x ∈ [0, L] and for all s ≥ 0 (see [5,8,11]) or that g(s) > 0 for all s > 0, when a(x) is allowed to change sign (see [3,6,7]), do not contain our result and, in some cases, are easy consequences of it. Figure 1 and Figure 2 show examples of nonlinearities g(s) satisfying our assumptions and which are not covered by previous results. g(s) = min{20s 6/5 − 6s 3 + s 4 , 400 s arctan(s)}. On the left we have shown the graph of g(s). We underline that g(s) changes sign and g(s)/s → +∞ as s → +∞. On the right we have represented the image of the segment {0} × [0, 12] through the Poincaré map in the phase-plane (u, u ). It intersects the negative part of the u -axis in a point, hence there is a positive initial slope at x = 0 from which departs a solution which is positive on ]0, 1[ and vanishes at x = 1. The plan of the paper is as follows. In Section 2 we present some basic facts. More in detail we list the hypotheses and we introduce an equivalent fixed point problem that permits to face the problem with a topological approach. In fact, using the technical assumptions, we are able to compute the degree on suitable small and large balls, in the same spirit of [6].
In Section 3 we present our main result. The theorem we state is an immediate corollary of the results exhibited in the previous section. In particular, we prove that the topological degree is non-zero on an annular domain. Therefore a nontrivial fixed point exists, this corresponds to a positive solution (using a standard maximum principle). Straightforward corollaries are then obtained.
Section 4 shows an important existence result of radially symmetric solutions on annular domains.

Preliminaries
In this section we state the hypotheses on a(x) and on g(s), we recall some classical results and we prove two preliminary lemmas that are then employed in Section 3 for the main result.
Consider the nontrivial closed interval [0, L], pointing out that different choices of a nontrivial compact interval contained in R can be made. Let a : [0, L] → R be a L 1 -weight function. Clearly the case of a continuous function can be treated as well. We assume that (H1) there exist m ≥ 1 intervals I 1 , . . . , I m , closed and pairwise disjoint, such that We underline that assumption (H1) trivially includes the case where a(x) ≥ 0 for a.e. x ∈ [0, L], taking m = 1 and I 1 = [0, L]. As standard notation, we define a + (x) := max{a(x), 0}, a − (x) := max{−a(x), 0}.
Concerning the nonlinearity, we suppose that g : R + → R is a continuous function such that (H2) g(0) = 0 and g ≡ 0.
We set We stress that we do not suppose g(s) ≥ 0 on R + and, in particular, it is not required that g(s) > 0 for all s > 0 (cf. [5,6,7,8]). Consequently, the nonlinearity g(s) could be non-negative, non-positive or it could change sign, even an infinite number of times, on a compact neighborhood of zero. Now we show how the superlinearity of g is expressed at zero and at infinity. As first step we impose a condition on the growth of g(s)/s at 0, depending on the sign of g(s). Precisely we assume that • if g(s) changes sign an infinite number of times in every neighborhood of zero, it holds that where λ 0 > 0 is the first eigenvalue of the eigenvalue problem The functions a(x) and g(s) introduced in Figure 1 satisfy the first condition of hypothesis (H3), while the example shown in Figure 2 corresponds to the third case. As second step we define the superlinear behavior at infinity. We suppose that Now we describe the topological approach we adopt to face problem (1.1). Our first goal is to introduce a completely continuous operator and to define an equivalent fixed point problem.
Letg : R → R be the standard extension of g(s) defined as We deal with the boundary value problem From conditions (H2) and (H3) and by a classical maximum principle (cf. [6,9]), it follows that all possible solutions of (2.1) are non-negative. Moreover, if these solutions are nontrivial, then they are strictly positive on ]0, L[ and hence positive solutions of (1.1). The next step is to define the classical operator Φ : where G(x, s) is the Green function associated to the equation u + u = 0 with the two-point boundary condition. The operator Φ is completely continuous in C([0, L]), endowed with the sup-norm · ∞ , and such that u is a fixed point of Φ if and only if u is a solution of (2.1). Therefore we have transformed problem (1.1) into an equivalent fixed point problem.
We close this section by proving two technical lemmas that allow us to find a nontrivial fixed point of Φ, hence a positive solution of (1.1). The approach we use now is based on the Leray-Schauder topological degree and it is in the same spirit of [6].
Using this first lemma we are able to compute the degree of Id − Φ on small balls. Proof. We divide the proof in two steps.
Step 1. We prove that there exists r 0 > 0 such that every solution u(x) ≥ 0 of the two-point BVP The proof of this first step is given only when there exists δ > 0 such that g(s) ≥ 0, for all s ∈ [0, δ]. The two remaining cases can be treated in an analogous way.
Using condition (H3), we fix 0 < r 0 < δ such that We stress that ϕ(x) > 0, for all x ∈ ]0, L[. Using a Sturm comparison argument, we attain Step 2. Computation of the degree. Let us fix 0 ≤ ϑ ≤ 1. As remarked when we have introduced the operator Φ, the maximum principle ensures that every Proof. We divide the proof in two steps. Step , and such that the first positive eigenvalueλ of the eigenvalue problem ϕ + λ a + (x) ϕ = 0 ϕ| ∂I ε i = 0 The existence of ε is ensured by the continuity of the eigenvalue as function of the boundary condition (see [4,12]) and by hypothesis (H4). From the previous inequality it follows that there exists a constantR > 0 such that g(s) >λs, ∀ s ≥R.
By contradiction, suppose there is not a constant R i > 0 with the properties listed above. So, for each integer n > 0 there exists a solution u n ≥ 0 of (2.4) with max x∈Ii u n (x) =:R n > n.
We claim that there exists an integer N ≥R such that u n (x) >R for every x ∈ I ε i and n ≥ N . If it is not true, for every integer n ≥R there is an integer n ≥ n and xn ∈ I ε i such that un(xn) =R. We note that the solution un(x) is concave on each subinterval of I i where un(x) ≥R, since a(x)g(s) ≥ 0 for a.e. x ∈ I i and for all s ≥R. Then, without loss of generality, we can assume that there exists a maximum pointxn ∈ I i of un such that un(x) >R for all x between xn andxn (if necessary, we change the choice of xn). From the assumptions, it follows that n <Rn = un(xn) = un(xn) + xn xn u n (ξ) dξ ≤R + (τ i − σ i )|u n (xn)|. (2.6) Since h(x, s) is a L 1 -Carathéodory function, there exists γR ∈ L 1 ([0, L], R + ) such that |h(x, s)| ≤ γR(x), for a.e. x ∈ [0, L] and for all |s| ≤R. Then, we fix a constant C > 0 such that Using (2.6), we have that for every n ≥ (τ i − σ i )C +R there existsn ≥ n and xn ∈ I ε i such that un(xn) =R and |u n (xn)| > C. Let us fix n ≥ (τ i − σ i )C +R, n ≥ n and xn ∈ I ε i with the properties just listed. Suppose that u n (xn) > C and consider the interval [σ i , xn]. If u n (xn) < −C we proceed similarly dealing with the interval [xn, τ i ]. For every x ∈ [σ i , xn] u n (x) = u n (xn) + From this inequality we obtain that un(x) ≤R, for all x ∈ [σ i , xn], and therefore Then, we obtaiñ a contradiction. Hence the claim is proved. So, we can fix an integer N ≥R such that u n (x) >R for every x ∈ I ε i and for n ≥ N . We denote by ϕ the positive eigenfunction of the eigenvalue problem (2.5) with ϕ ∞ = 1. Then ϕ(x) > 0, for every x ∈ ]σ i + ε, τ i − ε[, and ϕ (σ i + ε) > 0 > ϕ (τ i − ε). We remark that u n (σ i + ε) > 0 and u n (τ i − ε) > 0, for every integer n, employing the maximum principle.
Using a Sturm comparison argument, for each n ≥ N , we obtain Step 2. Computation of the degree. We stress that the constant R i , i ∈ {1, . . . , m}, does not depend on the function h(x, s). Define and fix a radius R ≥ R * .
We the theorem is proved.
Let α ≥ 0. The maximum principle ensures that any nontrivial solution u ∈ C([0, L]) of u = Φ(u)+αv is a non-negative solution of u +a(x)g(u)+α1 A (x) = 0 with u(0) = u(L) = 0. Hence, u is a non-negative solution of (2.4) with By definition, we have that h(x, s) ≥ a(x)g(s), for a.e. x ∈ A and for all s ≥ 0, and h(x, s) = a(x)g(s), for a.e. x ∈ [0, L] \ A and for all s ≥ 0. By the convexity of the solution u on the intervals of [0, L] \ A where u(x) ≥R, we obtain that

From
Step 1 and the definition ofR we deduce that u ∞ < R * ≤ R. Then (2.7) is proved and the theorem follows.

The main result
In this section we apply the two technical lemmas just proved to obtain the existence of a positive solution to the two-point boundary value problem (1.1). More in detail, we use the additivity of the topological degree to provide the existence of a nontrivial fixed point of the operator Φ defined in (2.2).
A first immediate consequence of Lemma 2.1 and Lemma 2.2 is our main theorem. Proof. Let r 0 be as in Lemma 2.1 and R * be as in Lemma 2.2. We observe that 0 < r 0 < R * < +∞. From the additivity property and the two preliminary lemmas it follows that Then there exists a nontrivial fixed point of Φ and hence a corresponding positive solution of (1.1), as already remarked.
From Theorem 3.1 we easily achieve the following two results.   [5]) are available to define an equivalent fixed point problem and to adapt the scheme shown in this paper.

Radially symmetric solutions
We denote by · the Euclidean norm in R N (for N ≥ 2). Let be an open annular domain, with 0 < R 1 < R 2 . Let a : [R 1 , R 2 ] → R be a continuous function. In this section we consider the Dirichlet boundary value problem −∆ u = a( x ) g(u) in Ω u = 0 on ∂Ω (4.1) and we are interested in the existence of positive solutions of (4.1), namely classical solutions such that u(x) > 0 for all x ∈ Ω. Since we look for radially symmetric solutions of (4.1), our study can be reduced to the search of positive solutions of the two-point boundary value problem w (r) + N − 1 r w (r) + a(r)g(w(r)) = 0, w(R 1 ) = w(R 2 ) = 0. Consequently, the two-point boundary value problem (4.3) is of the same form of (1.1) considering r(t) 2(N −1) a(r(t)) as weight function. Clearly the following result holds.