USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 65–87. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

On the existence of infinitely many solutions to a damped sublinear boundary-value problem ^{∗}

Anna Capietto & Marielle Cherpion

Abstract

We prove the existence of infinitely many solutions (with prescribed nodal properties) to a damped sublinear boundary-value problem. The proofs are performed by means of an abstract continuation theorem and the time-map technique for strongly nonlinear operators.

1 Introduction

We study the existence and multiplicity of solutions to the boundary-value problem

\begin{matrix}(r^{(k−1)}u′)′ + r^{(k−1)}a(u′)f(r, u) = r^{(k−1)}h(r, u, u′), \\ u′(0) = 0 = u(R) \end{matrix}  (1.1)

(k > 1). As it is well-known, solutions to (1.1) are radially symmetric solutions to the following elliptic boundary-value problem on a ball B = B(0, R)

\begin{matrix}∇ · (∇u) + a(|∇u|)f(|x|, u) = h(|x|, u, |∇u|) in B, \\ u = 0 on ∂B. \end{matrix}  (1.2)

We deal with a so-called ”sublinear” problem. More precisely, we assume that a(ξ) = a_{0} + |ξ|^{q} (a_{0} > 0, 0 < q < 2) and that the following conditions are satisfied:

(H_{f}) The function f : [0, R] × [−ε_{0}, ε_{0}] → ℝ is continuous and such that

f(r, 0) ≡ 0

and

lim_{s→0} \frac{f(r, s)}{s} = +∞ uniformly in r ∈ [0, R].

∗Mathematics Subject Classifications: 34B15.

Key words: Damped sublinear problem, continuation theorem, time-map technique.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

A.C. Supported by GNAFA-C.N.R.-Italy, and by EEC grant CHRX-CT94-0555.

M.C. Supported by F.N.R.S.-Belgium.

(H_{F}) For F (r, s) := ∫_{0}^{s} f(r, x) dx, F is differentiable with respect to r ∈ [0, R] and there exists a continuous positive function α : [0, R] → (0, +∞) such that for all r ∈ [0, R], all s ∈ [−ε_{0}, ε_{0}],

|\frac{∂F}{∂r} (r, s)| ≤ α(r)F (r, s).

(H_{h}) The function h : [0, R] × [−ε_{0}, ε_{0}] × ℝ → ℝ is continuous and there exists H > 0 such that for all (r, s, ξ) ∈ [0, R] × [−ε_{0}, ε_{0}] × ℝ,

|h(r, s, ξ)| ≤ H|ξ|.

Moreover, there exists a continuous function C : ℝ → (0, +∞) such that

lim_{s→0} \frac{h(r, s, ξ)}{s} = C(ξ) uniformly in r ∈ [0, R].  (1.3)

We point out that problem (1.1) can be considered “singular” in a two-fold sense. Indeed, on one hand, under condition (H_{f}) the uniqueness of the solutions to initial value problems associated to (1.1) must be guaranteed by (H_{F}); on the other hand, a singularity in the r-variable arises because of the boundary condition in zero. For more comments on (H_{F}) we refer to [3, 4, 6, 16].

Our main result is the following (cf. Theorem 4.1).

Theorem A Assume (H_{f})-(H_{F})-(H_{h}) and let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0. Then there exists n_{0} ∈ ℕ such that for every n > n_{0} problem (1.1) has at least two solutions u_{n} and v_{n} with u_{n}(0) > 0 and v_{n}(0) < 0, both having exactly n zeros in [0, R). Moreover, we have

lim_{n→+∞} |u_{n}(r)| + |u′_{n}(r)| = 0 = lim_{n→+∞} |v_{n}(r)| + |v′_{n}(r)|, uniformly in r ∈ [0, R].

Multiplicity results for a boundary-value problem of the form (1.1) can be found e.g. in [1], [2], [4], [6], [8], [12]. However, apart from [6] where additional regularity conditions are imposed, in those papers the authors considered the case a ≡ 1 and/or h ≡ 0. In some of the above quoted papers, the differential operator under consideration is strongly nonlinear. We refer to [4] for a more comprehensive list of references.

We work in the framework of topological degree methods and use some of the ideas developed in [4] (see also [5], [9]). In this situation, two main tasks have to be accomplished. First, one has to study an autonomous problem

\begin{matrix}u′′ + a(u′)g(u) = 0, \\ u′(0) = 0 = u(R), \end{matrix}  (1.4)

where g : [−ε, ε] → ℝ, ε > 0, is a continuous function such that

lim_{s→0} \frac{g(s)}{s} = +∞.

Secondly, suitable estimates on the (possible) solutions to a family of parameterdependent problems (cf. (P_{λ})) have to be established.

In this paper (cf. Section 2) we overcome the first difficulty by studying problem (1.4) in the equivalent form

\begin{matrix}(ϕ(u′))′ + g(u) = 0, \\ u′(0) = 0 = u(R), \end{matrix}  (1.5)

where ϕ is an odd increasing homeomorphism defined through a. In this way, we can use the time-map technique for equations containing the ϕ-Laplacian (see [4], [7], [8], [10], [11], [12], [13]) and establish a multiplicity result for (1.4) (cf. Theorem 2.2 and Theorem 5.4 in [4]).

Then, in order to show that the (nodal) properties of the solutions to (1.4) can be ”continued” to problem (1.1), some estimates on the number of zeros of the (possible) solutions to the associated parameter-dependent boundary-value problem (cf. (P_{λ})) have to be established. To this end, we argue on the lines of [4]; however, some technical difficulties due to the presence in (1.1) of the functions a and h have to be overcome (see in particular the proofs of Lemma 3.1, Lemma 3.3 and Claim 2 in Theorem 4.1).

We end this introductory section by observing that a result analogous to Theorem A can be performed for a more general strongly nonlinear boundaryvalue problem

\begin{matrix}(r^{(k−1)}ψ(u′))′ + r^{(k−1)}a(u′)f(r, u) = r^{(k−1)}h(r, u, u′), \\ u′(0) = 0 = u(R), \end{matrix}  (1.6)

where ψ is an odd increasing homeomorphism satisfying suitable assumptions.

Furthermore, on the lines of [4], one could prove the existence of an additional double sequence of solutions to (1.1) (whose norm tends to infinity) provided that g has a ”superlinear” behaviour at infinity and assumption (H_{h}) is modified accordingly.

This paper is organized as follows. In Section 2 we study the autonomous problem (1.4). In Section 3 we introduce a parameter-dependent non-autonomous problem and develop some estimates on its solutions. In Section 4 we recall an abstract continuation theorem which is then applied for the proof of the main result.

In what follows, for any Banach space X, for any linear compact operator L : X → X and for any subset Ω ⊂ X we will denote by deg(I −L, Ω) the Leray-Schauder degree of I − L (if defined). The space C^{1}([0, R]) of the continuously differentiable real functions u on [0, R] will be equipped with the norm

||u||_{1} = max \{\sqrt{|u(t)|^{2} + |u′(t)|^{2}} : t ∈ [0, R]\} .

Finally, C_{#}^{1}([0, R]) denotes the space of functions u ∈ C^{1}([0, R]) satisfying the boundary condition u′(0) = 0 = u(R).

2 An autonomous problem

Let us consider the second order ODE

\begin{matrix}u′′ + a(u′)g(u) = 0, \\ u′(0) = 0 = u(R), \end{matrix}  (2.1)

where a : ℝ → ℝ is defined by a(ξ) := a_{0} + |ξ|^{q}, 0 < q < 2, a_{0} > 0. Set

ϕ(s) = ∫_{0}^{s} \frac{1}{a(x)} dx.  (2.2)

We assume that g : [−ε, ε] → ℝ is continuous (ε > 0) and such that

lim_{s→0} \frac{g(s)}{ϕ(s)} = +∞.  (2.3)

We shall also assume (without loss of generality) that g(s)s > 0 for all s ∈ [−ε, ε] \ {0} and we set G(s) = ∫_{0}^{s} g(ξ) dξ.

We observe that problem (2.1) can be written in the form

\begin{matrix}(ϕ(u′))′ + g(u) = 0, \\ u′(0) = 0 = u(R). \end{matrix}

It is not difficult to check that ϕ is an odd increasing homeomorphism. Then, as in [10], it is possible to study (2.1) with the time-map technique by means of the system

\begin{matrix}u′ = ϕ^{−1}(y), \\ y′ = −g(u). \end{matrix}  (2.4)

More precisely, for

L(ξ) = ∫_{0}^{ξ} \frac{x}{a(x)} dx,  (2.5)

we shall use the fact that if u is a solution of (2.4), then E(r, u(r), u′(r)) := G(u(r)) + L(u′(r)) is constant. Observe that our assumptions on g ensure that the orbits of (2.1) are closed curves on the phase-plane. Then, denoting by L^{−1} the inverse of the restriction to ℝ^{+} of the function L, we can introduce the function T_{1} : (0, ε) → (0, +∞) by

T_{1}(α) = ∫_{0}^{α} \frac{dx}{L^{−1}(G(α) − G(x))}.  (2.6)

It is straightforward to check that T_{1}(α) represents the time needed for a rotation along the orbit of ”energy” G(α) in the upper (resp. lower) half plane from the point (0, L^{−1}(G(α))) to the point (α, 0) (resp. from (α, 0) to (0, −L^{−1}(G(α)))). Analogously, for α_{1} < 0 s.t. G(α_{1}) = G(α), the function T_{2} : (0, ε) → (0, +∞) defined by

T_{2}(α) = ∫_{α_{1}}^{0} \frac{dx}{L^{−1}(G(α) − G(x))}  (2.7)

is the time needed for a rotation along the orbit of ”energy” G(α) from the point (α_{1}, 0) to the point (0, L^{−1}(G(α))) (resp. from (0, −L^{−1}(G(α))) to (α_{1}, 0)). For a classical reference on this topic, the reader can consult [15]. See also [7].

For the completion of the study of the autonomous case, we need the following result.

Proposition 2.1 Let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0 and ϕ given by (2.2). Assume g : [−ε, ε] → ℝ is a continuous function such that g(s)s > 0 for all s ∈ [−ε, ε] \ {0} and satisfying (2.3). Then the functions T_{1}(α) and T_{2}(α) defined by (2.6) and (2.7) are such that for i = 1, 2 we have

lim_{α→0} T_{i}(α) = 0.

Proof. Observe that L(s) = (Φ_{∗} ∘ ϕ)(s) with Φ_{∗}(s) = ∫_{0}^{s} ϕ^{−1}(x) dx. The proof follows the same arguments as in Lemma 2.1 in [11] and Theorem 3.2 in [10] where the assumptions on the function g, as well as the result on the asymptotic behaviour of the time-maps, are relative to a neighbourhood of infinity. For a more detailed proof, one can also see Theorem 2.2.8 in [7]. □

Once the time-maps are defined, we can introduce the ”generalized Fučik spectrum” as in [3], [4], [7] in order to get a characterization of the existence of solutions with a fixed number of zeros. Indeed, using Proposition 2.1, one gets the following multiplicity result for the autonomous problem (2.1).

Theorem 2.2 [4, Th. 5.4] Let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0 and ϕ given by (2.2). Assume g : [−ε, ε] → ℝ is a continuous function such that g(s)s > 0 for all s ∈ [−ε, ε] \ {0} and satisfying (2.3). Then there exists k_{0} ∈ ℕ such that for every k ≥ 2k_{0} problem (2.1) has at least two solutions u_{k} and v_{k} with u_{k}(0) > 0 and v_{k}(0) < 0, both having exactly k zeros in [0, R).

We end this section by giving two important properties of L, which will be crucial in the sequel.

Proposition 2.3 Let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0 and L given by (2.5). Then for all ξ ∈ ℝ, we have

\frac{ξ^{2}}{a(ξ)} ≤ 2L(ξ).

Proof. An easy computation gives (\frac{s^{2}}{a(s)})′ ≤ 2L′(s), for all s ≥ 0. Then by integration, we get

\frac{ξ^{2}}{a(ξ)} ≤ 2L(ξ) .

Proposition 2.4 Let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0 and L given by (2.5). Then for any c_{1} > 1, c ≥ c_{1}^{2} + 1 and ξ > 0 small enough, we have

c_{1}L^{−1}(ξ) ≤ L^{−1}(cξ).  (2.8)

Proof. Notice that since lim_{x→0} \frac{L(c_{1}x)}{L(x)} = c_{1}^{2}, then for any c ≥ c_{1}^{2} + 1 and x > 0 small enough we have L(c_{1}x) ≤ cL(x). As L^{−1} : ℝ^{+} → ℝ^{+} is continuous, we have for ξ > 0 small enough

L(c_{1}L^{−1}(ξ)) ≤ cL(L^{−1}(ξ)) = cξ

and since L^{−1} is increasing c_{1}L^{−1}(ξ) ≤ L^{−1}(cξ). □

Remark 2.5 In general, if one sets Φ_{∗}(s) = ∫_{0}^{s} ϕ^{−1}(x) dx, where ϕ is an odd increasing homeomorphism, an inequality like (2.8) can be proved separately for the functions Φ_{∗}^{−1} and ϕ^{−1}. This is done in [4] under the ”lower σ-condition”

lim inf_{s→0} \frac{ϕ(σs)}{ϕ(s)} > 1, ∀σ > 1.

In our situation, we observe that L(s) = (Φ_{∗} ∘ ϕ)(s), so we could have proved Proposition 2.4 by combining the inequalities for Φ_{∗}^{−1} and ϕ^{−1}. A direct proof of Proposition 2.4 is simpler thanks to the fact that we can explicitly use the function L and its properties.

3 Preliminary results

We consider the boundary-value problem

\begin{matrix}(r^{(k−1)}u′)′ + r^{(k−1)}a(u′)f(r, u) = r^{(k−1)}h(r, u, u′), \\ u′(0) = 0 = u(R), \end{matrix}  (3.1)

where k > 1, a : ℝ → ℝ is defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0 and for a fixed ε_{0} > 0, the functions f and h satisfy the following properties.

(H_{f}) The function f : [0, R] × [−ε_{0}, ε_{0}] → ℝ is continuous and such that

f(r, 0) = 0

and

lim_{s→0} \frac{f(r, s)}{s} = +∞ uniformly in r ∈ [0, R].

(H_{F}) For F (r, s) := ∫_{0}^{s} f(r, x) dx, F is differentiable with respect to r ∈ [0, R] and there exists a continuous positive function α : [0, R] → (0, +∞) such that for all r ∈ [0, R], all s ∈ [−ε_{0}, ε_{0}],

|\frac{∂F}{∂r} (r, s)| ≤ α(r)F (r, s).

(H_{h}) The function h : [0, R] × [−ε_{0}, ε_{0}] × ℝ → ℝ is continuous and there exists H > 0 such that for all (r, s, ξ) ∈ [0, R] × [−ε_{0}, ε_{0}] × ℝ,

|h(r, s, ξ)| ≤ H|ξ|.

Moreover, there exists a continuous function C : ℝ → (0, +∞) such that

lim_{s→0} \frac{h(r, s, ξ)}{s} = C(ξ) uniformly in r ∈ [0, R].  (3.2)

A typical example for the function h is h(r, s, ξ) = η(s)|ξ|^{β} with β > 1 for |ξ| < 1, 0 < β < 1 for |ξ| ≥ 1 and η(s) ∼ s for s → 0. □

Following a degree approach, problem (3.1) will be treated by means of the parameter-dependent family of problems (λ ∈ [0, 1])

\begin{matrix}(r^{λ(k−1)}u′)′ + r^{λ(k−1)}a(u′)f_{λ}(r, u) = λr^{λ(k−1)}h(r, u, u′), \\ u′(0) = 0 = u(R), \end{matrix}  (P_{λ})

where f_{λ} : [0, R] × [−ε_{0}, ε_{0}] → ℝ is defined by

f_{λ}(r, s) = λf(r, s) + (1 − λ)g(s),  (3.3)

and g : [−ε_{0}, ε_{0}] → ℝ is a continuous nondecreasing function such that

(H_{g})  lim_{s→0} \frac{g(s)}{s} = +∞.

We shall also assume (without loss of generality) that g(s)s > 0, for every s ∈ [−ε_{0}, ε_{0}] \ {0}. Note that our assumptions on g guarantee that condition (2.3) is satisfied.

Set F_{λ}(r, s) := ∫_{0}^{s} f_{λ}(r, x) dx. It is immediate to remark that in the situation described above we have

(H_{F_{λ}}) F_{λ}(r, s) is differentiable with respect to r ∈ [0, R] and there exists a continuous positive function α : [0, R] → (0, +∞) such that for all r ∈ [0, R], all s ∈ [−ε_{0}, ε_{0}],

|\frac{∂F_{λ}}{∂r} (r, s)| ≤ α(r)F_{λ}(r, s).

Moreover, using (H_{f}) and (H_{g}), we have

lim_{s→0} \frac{f_{λ}(r, s)}{s} = +∞ uniformly in λ ∈ [0, 1]

and, by (H_{F_{λ}}), for all r ∈ [0, R], all s ∈ [−ε_{0}, ε_{0}] \ {0} and all λ ∈ [0, 1],

F_{λ}(r, s) > 0.  (3.4)

In our main result we will prove the existence of infinitely many solutions of (P_{1}) using an abstract continuation theorem. To this end, we need the following lemma concerning the Cauchy problem

\begin{matrix}(r^{λ(k−1)}u′)′ + r^{λ(k−1)}a(u′)f_{λ}(r, u) = λr^{λ(k−1)}h(r, u, u′), \\ u(0) = d, u′(0) = 0. \end{matrix}  (3.5)

Lemma 3.1 For all ε ∈ (0, ε_{0}], if u is a (local) solution of problem (3.5) with d small enough, then u can be defined on [0, R] and ||u||_{1} ≤ ε.

Proof. Let ε > 0 be fixed and u be a solution of (3.5). Assume that there exists ρ ∈ (0, R] such that for all r ∈ [0, ρ],

|u(r)| ≤ ε  and  |u′(r)| ≤ ε.

Let

E_{λ}(r, s, ξ) := F_{λ}(r, s) + L(ξ)

where L is given in (2.5) and for all r ∈ [0, ρ], we consider the function

v_{λ}(r) := E_{λ}(r, u(r), u′(r)).  (3.6)

We have, using (H_{h}), (H_{F_{λ}}) and Proposition 2.3

v′_{λ}(r) = \frac{∂F_{λ}}{∂r} (r, u(r)) + \frac{u′(r)}{a(u′(r))} (λh(r, u(r), u′(r)) − \frac{λ(k − 1)}{r}u′(r))
≤ \frac{∂F_{λ}}{∂r} (r, u(r)) + λh(r, u(r), u′(r))\frac{u′(r)}{a(u′(r))}
≤ \frac{∂F_{λ}}{∂r} (r, u(r)) + H \frac{(u′(r))^{2}}{a(u′(r))}
≤ α(r)F_{λ}(r, u(r)) + 2HL(u′(r))
≤ α̃(r)v_{λ}(r),

where α̃ : [0, R] → (0, +∞) is a continuous function. Integrating on (0, r), we get

v_{λ}(r) ≤ v_{λ}(0)e^{∫_{0}^{r} α̃(s) ds} = F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}

and by definition of v_{λ}, we have

L(u′(r)) ≤ v_{λ}(r) ≤ F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}.  (3.7)

For the rest of the proof, we argue as in the proof of Lemma 2.3 in [4]; however, we give the details for the reader’s convenience.

Consider (a_{1}, a_{2}) ∈ (0, 1)^{2} such that

a_{1} + Ra_{2} ≤ \frac{1}{2} and a_{2} ≤ \frac{1}{2}  (3.8)

(observe that, for every R > 0, a similar choice of a_{1} and a_{2} is always possible). Since lim_{d→0} L^{−1}(F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}) = 0 uniformly in λ ∈ [0, 1], for every ε ≤ ε_{0} there exists d_{ε} > 0 such that d_{ε} ≤ a_{1}ε and for all 0 < |d| ≤ d_{ε}, all λ ∈ [0, 1],

L^{−1}(F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}) ≤ a_{2}ε.

Then, for 0 < |d| ≤ d_{ε}, we deduce from (3.7) that for all r ∈ [0, ρ],

|u′(r)| ≤ L^{−1}(F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}) ≤ a_{2}ε ≤ \frac{ε}{2}.  (3.9)

The above estimate implies that for all r ∈ [0, ρ],

\begin{aligned}|u(r)| & ≤ & d + ∫_{0}^{r} |u′(s)| ds ≤ d + RL^{−1}(F_{λ}(0, d)e^{∫_{0}^{r} α̃(s) ds}) \\ & ≤ & a_{1}ε + Ra_{2}ε = (a_{1} + Ra_{2})ε ≤ \frac{ε}{2}. \end{aligned}  (3.10)

Since (3.9) and (3.10) hold independently on ρ, we can extend u on [0, R] as a C^{1}-function. Finally, (3.9) and (3.10) imply that

||u||_{1} = max_{r∈[0,R]} \sqrt{|u(r)|^{2} + |u′(r)|^{2}} ≤ ε.

Remark 3.2 We deduce from Lemma 3.1 that if u is a solution of (3.5) with d small enough then u′ is bounded. Hence condition (3.2) in (H_{h}) implies that there exists δ̃ > 0 such that for all r ∈ [0, R],

0 < |u(r)| ≤ δ̃ ⟹ |h(r, u(r), u′(r))| ≤ C|u(r)|,

with C independent of u′.

Lemma 3.3 There exists δ̄ > 0 such that if u is a solution of (P_{λ}) with |u(0)| = d̄ small enough then for all r ∈ [0, R],

|u(r)|^{2} + |u′(r)|^{2} ≥ δ̄.

Proof. Step 1 - Let 0 < ε̃ < a_{0} be fixed and u(·, d) = u(·) be a solution of (3.5). Integrating the equation in (3.5) from 0 to r, we have

−u′(r) = r^{−λ(k−1)} ∫_{0}^{r} s^{λ(k−1)}(f_{λ}(s, u(s))a(u′(s)) − λh(s, u(s), u′(s))) ds.

Let δ̃ be given by Remark 3.2. We deduce from (H_{f}) that there exists 0 < δ < δ̃ such that for all r ∈ [0, R],

0 < |u(r)| ≤ δ ⟹ |f(r, u(r))| ≥ \frac{C}{ε̃} |u(r)| and f(r, u(r))u(r) > 0. (3.11)

Hence if d > 0 and r are small enough, for every s ∈ [0, r] we have 0 < u(s) ≤ δ and

f_{λ}(s, u(s))a(u′(s)) − λh(s, u(s), u′(s)) ≥ λ\frac{C}{ε̃} u(s)a(u′(s)) − λCu(s)
= λCu(s) (\frac{a(u′(s))}{ε̃} − 1) > 0 ,

which proves that u is decreasing in [0, r]. In the same way, if d < 0 is small enough then

f_{λ}(s, u(s))a(u′(s)) − λh(s, u(s), u′(s)) ≤ λCu(s) (\frac{a(u′(s))}{ε̃} − 1) < 0

and u is increasing in [0, r]. Arguing as in [5], for every θ ∈ (0, 1) we can consider the first point r_{θ}(d) such that

u(r_{θ}(d); d) = θd.

Moreover, we denote by r_{0}(d) the first zero of u(·; d). Step 2 - There exists δ̄ > 0 such that if u is a solution of (P_{λ}) with |u(0)| = d̄ small enough then for all r ∈ [0, R] we have |u(r)|^{2} + |u′(r)|^{2} ≥ δ̄.

Let u be a solution of (P_{λ}) with |u(0)| = d̄ sufficiently small. To prove this Step we will need the following two claims. Claim 1 : For every θ ∈ (0, 1) and for every λ ∈ (0, 1), we have

r_{θ}(d̄) ≥ \sqrt{\frac{2d̄(1 − θ)(1 + λ(k − 1))}{(f̂(d̄) + g(d̄))a(ε_{0}) + Hε_{0}}} =: β(d̄) > 0,

where f̂ is defined by

f̂(s) = \{ \begin{matrix}sup\{f(r, x), r ∈ [0, R], x ∈ [0, s]\} & if 0 < s ≤ ε_{0}, \\ inf\{f(r, x), r ∈ [0, R], x ∈ [s, 0]\} & if − ε_{0} ≤ s < 0 . \end{matrix}

An analogous result holds for r_{θ}(−d̄).

Assume u(0) = d̄ > 0. If d̄ is small enough, we have for every s ∈ [0, r_{0}(d̄)] that 0 < u(s) ≤ δ and we deduce from Lemma 3.1 that |u′(s)| ≤ ε_{0}. Moreover, using (H_{h}), we have for all s ∈ [0, r_{0}(d̄)],

f_{λ}(s, u(s))a(u′(s)) − λh(s, u(s), u′(s)) ≤ (f̂(d̄) + g(d̄))a(ε_{0}) + Hε_{0}.

Hence for r ∈ [0, r_{0}(d̄)] we obtain

−u′(r) = r^{−λ(k−1)} ∫_{0}^{r} s^{λ(k−1)}(f_{λ}(s, u(s))a(u′(s)) − λh(s, u(s), u′(s))) ds
≤ \frac{r}{λ(k − 1) + 1}((f̂(d̄) + g(d̄))a(ε_{0}) + Hε_{0})

and integrating from 0 to r_{θ}(d̄) we get

θd̄− d̄= u(r_{θ}(d̄)) − u(0) ≥ −\frac{((f̂(d̄) + g(d̄))a(ε_{0}) + Hε_{0})(r_{θ}(d̄))^{2}}{2(λ(k − 1) + 1)}

which implies

r_{θ}(d̄) ≥ \sqrt{\frac{2d̄(1 − θ)(1 + λ(k − 1))}{(f̂(d̄) + g(d̄))a(ε_{0}) + Hε_{0}}}

and the Claim is proved.

A similar computation holds if u(0) = −d̄. Claim 2 : There exists δ_{0} > 0 such that if u is a solution of (P_{λ}) with |u(0)| = d̄ sufficiently small, we have E_{λ}(r, u(r), u′(r)) ≥ δ_{0} for every r ∈ [0, R].

First, we observe that, by (H_{F_{λ}}), there is a constant γ such that for all r ∈ (0, R], all s ∈ [−ϵ_{0}, ϵ_{0}] and all λ ∈ [0, 1],

\frac{∂F_{λ}}{∂r} (r, s) + \frac{γ}{r} F_{λ}(r, s) ≥ 0.  (3.12)

Recall that E_{λ}(r, s, ξ) := F_{λ}(r, s) + L(ξ). Using (H_{h}) and Proposition 2.3,

\frac{d}{dr} E_{λ}(r, u(r), u′(r)) + \frac{γ}{r} E_{λ}(r, u(r), u′(r))

= \frac{∂F_{λ}}{∂r} (r, u(r)) + \frac{γ}{r} F_{λ}(r, u(r)) + \frac{u′(r)}{a(u′(r))}λh(r, u(r), u′(r)) + \frac{γ}{r} L(u′(r))

−λ\frac{(k − 1)}{r} \frac{(u′(r))^{2}}{a(u′(r))}

≥ \frac{u′(r)}{a(u′(r))}λh(r, u(r), u′(r)) + \frac{γ}{r} L(u′(r)) − \frac{2(k − 1)}{r}L(u′(r))
≥ −H \frac{(u′(r))^{2}}{a(u′(r))} + \frac{γ}{r} L(u′(r)) − \frac{2(k − 1)}{r}L(u′(r))
≥ L(u′(r)) (−2H + \frac{γ}{r} − \frac{2(k − 1)}{r}) ≥ 0

if γ ≥ 2HR + 2(k − 1). Multiplying by r^{γ} and integrating from r_{θ}(d̄) to r, we obtain

E_{λ}(r, u(r), u′(r))r^{γ} − E_{λ}(r_{θ}(d̄), u(r_{θ}(d̄)), u′(r_{θ}(d̄)))r_{θ}(d̄)^{γ} ≥ 0

and

E_{λ}(r, u(r), u′(r)) ≥ E_{λ}(r_{θ}(d̄), u(r_{θ}(d̄)), u′(r_{θ}(d̄)))r_{θ}(d̄)^{γ}R^{−γ}
= R^{−γ} (L(u′(r_{θ}(d̄))) + F_{λ}(r_{θ}(d̄), u(r_{θ}(d̄)))) r_{θ}(d̄)^{γ}
≥ R^{−γ}F^{0}(θd̄)r_{θ}(d̄)^{γ}
≥ R^{−γ}F^{0}(θd̄)(β(d̄))^{γ},

where F^{0}(θd̄) = min{F_{λ}(r, θd̄) : r ∈ [0, R], λ ∈ [0, 1]} > 0. We finish the proof of the Claim by setting δ_{0} = R^{−γ}F^{0}(θd̄)(β(d̄))^{γ}.

If the claim in Step 2 were not true then for every δ̄ > 0, there exists r̄ ∈ [0, R] such that |u(r̄)|^{2} + |u′(r̄)|^{2} < δ̄, which contradicts Claim 2. □

Now, for every d ∈ ℝ_{0} and for every λ ∈ [0, 1] we can define

n : S_{d,λ} → ℕ : u ↦ n(u),

where

S_{d,λ} = {u : u is a solution of (P_{λ}) and u(0) > d if d > 0, u(0) < d if d < 0}

and n(u) is the number of zeros of u in [0, R). Arguing as in the proof of Lemma 3.1 and of Lemma 2.5 in [4], it is possible to prove that if u is a solution of the equation in (P_{λ}) such that u(r_{∗}) = 0 = u′(r_{∗}) (for some r_{∗} ∈ (0, R]) then u ≡ 0. This fact guarantees that n is well defined. Moreover, for d small enough, arguing as in Lemma 3.1 in [12] we can prove that n is continuous.

4 Main result

In this section we will prove the existence of infinitely many solutions of problem

\begin{matrix}(r^{(k−1)}u′)′ + r^{(k−1)}a(u′)f(r, u) = r^{(k−1)}h(r, u, u′), \\ u′(0) = 0 = u(R) \end{matrix}  (3.1)

(k > 1), using an abstract theorem. Note that solutions to (P_{1}) are solutions to (3.1), while solutions to (P_{0}) are solutions to the autonomous problem

\begin{matrix}u′′ + a(u′)g(u) = 0, \\ u′(0) = 0 = u(R). \end{matrix}

Our main result is the following.

Theorem 4.1 Assume (H_{f})-(H_{F})-(H_{h}) and let a : ℝ → ℝ be defined by a(ξ) := a_{0} + |ξ|^{q} with 0 < q < 2, a_{0} > 0. Then there exists n_{0} ∈ ℕ such that for every n > n_{0} problem (3.1) has at least two solutions u_{n} and v_{n} with u_{n}(0) > 0 and v_{n}(0) < 0, both having exactly n zeros in [0, R). Moreover, we have

lim_{n→+∞} |u_{n}(r)| + |u′_{n}(r)| = 0 = lim_{n→+∞} |v_{n}(r)| + |v′_{n}(r)|, uniformly in r ∈ [0, R].

To prove this theorem, we will need an abstract continuation result. In order to state this theorem, let us consider a Banach space X and a completely continuous operator N : dom N ⊂ X × [0, 1] → X. Moreover, let A, B be two open sets such that A ⊂ Ā ⊂ B ⊂ B̄ and (B̄ \ A) ⊂ dom N.

Let Σ be the set of the solutions of the abstract equation u = N (u, λ), i.e.

Σ = {(u, λ) : u = N (u, λ)}.

For any subset D ⊂ X × [0, 1], we denote the section of D at λ ∈ [0, 1] by D_{λ} = {x ∈ X : (x, λ) ∈ D} and we also set N_{λ} = N (·, λ). We are now in position to state the following

Theorem 4.2 [4, Th. 2.1] Let k : Σ ∩ (B̄ \ A) → ℕ be a continuous function; suppose that there exists a positive integer n satisfying the following conditions

n ∉ k(∂(B̄ \ A))  (4.1)

and

k^{−1}(n) is bounded.  (4.2)

Then, for an open set U_{0}^{n} such that (k^{−1}(n))_{0} ⊂ U_{0}^{n} ⊂ \overline{U_{0}^{n}} ⊂ (B̄ \ A)_{0} and Σ_{0} ∩ U_{0}^{n} = (k^{−1}(n))_{0}, the Leray-Schauder degree deg(I − N_{0}, U_{0}^{n}) is defined. If

deg(I − N_{0}, U_{0}^{n}) ≠ 0,  (4.3)

then there is a continuum C_{n} ⊂ Σ with

{λ ∈ [0, 1] : ∃u ∈ X : (u, λ) ∈ C_{n}} = [0, 1]

and such that

(u, λ) ∈ C_{n} ⟹ (u, λ) ∈ (B \ Ā) and k(u, λ) = n.

In particular there is at least one solution ũ ∈ (B \ Ā)_{1} of the operator equation

u = N (u, 1)

with

k(ũ, 1) = n.

Proof of Theorem 4.1. First note that problem (P_{λ}) can be put into the form u = N (u, λ) where

N : dom N ⊂ C_{#}^{1}([0, R]) × [0, 1] → C_{#}^{1}([0, R])

is a completely continuous operator (see for example [14]). In order to give the appropriate definition for the sets A and B, we need some estimates on n. Let δ be given by (3.11) and d̄≤ δ sufficiently small. Claim 1 : There exists n^{∗} ∈ ℕ such that for any solution u of (P_{λ}) we have

|u(0)| = d̄ ⟹ n(u) < n^{∗}.

The proof follows the same lines as the proof of Lemma 3.1 in [12], using Lemma 3.1 and Lemma 3.3. Claim 2 : For every N > 0 there exists d_{N} > 0, d_{N} < d̄, such that for any solution u ∈ S_{d,λ} (for some d) we have

|u(0)| ≤ d_{N} ⟹ n(u) > N.

Let us consider u ∈ S_{d,λ}. We observe that for every N > 0 there exists M(N) > 2^{2(2k−1)} such that for all |s| ≤ ε_{0}

\frac{1}{\sqrt{M(N)}} ∫_{0}^{s} \frac{du}{L^{−1}(s^{2} − u^{2})} < \frac{1}{N} .  (4.4)

Let M̃(N) := M(N) + \frac{C}{a_{0}} . By (H_{f}), there is η := η_{M̃(N)} such that for all r ∈ [0, R], all 0 < |s| ≤ η and all λ ∈ [0, 1],

|f_{λ}(r, s)| > M̃(N)|s|.  (4.5)

Let ε_{N} ≤ min{η, d̄}. From Lemma 3.1, we can consider d_{N} > 0 small enough such that

|u(0)| ≤ d_{N} ⟹ ||u||_{1} ≤ ε_{N}.

Now, let (u, λ) ∈ Σ with |u(0)| ≤ d_{N}. The equation in (P_{λ}) can be written as

\begin{matrix}u′ = \frac{y}{r^{λ(k−1)}} , \\ y′ = −r^{λ(k−1)}(f_{λ}(r, u)a(u′) − λh(r, u, u′)). \end{matrix}  (4.6)

We shall be concerned with the zeros {r_{i}}_{i=1,...,I} of u in the interval [R/2, R]. More precisely, we first estimate the distance between two successive zeros r_{i} and r_{i+1} of u in the case when

u′(r_{i}) > 0, u′(r_{i+1}) < 0, and u(r) > 0, ∀r ∈ (r_{i}, r_{i+1}).

From (4.6) we infer that y′(r) < 0 for every r ∈ (r_{i}, r_{i+1}). Since y(r_{i}) > 0 and y(r_{i+1}) < 0, we deduce that there exists exactly one point r^{∗} ∈ (r_{i}, r_{i+1}) such that y(r^{∗}) = 0 and again from (4.6) it follows that

u′(r) > 0, ∀r ∈ (r_{i}, r^{∗}), u′(r) < 0, ∀r ∈ (r^{∗}, r_{i+1}) and u′(r^{∗}) = 0.

Let A = (R/2)^{λ(k−1)} and B = R^{λ(k−1)}. Using (H_{h}), (3.11) and (4.5), we observe that

f_{λ}(r, u) − \frac{λh(r, u, u′)}{a(u′)} ≥ (M̃(N) − \frac{C}{a_{0}} ) u = M(N)u.

Hence for r ∈ (r_{i}, r^{∗}) ⊂ [R/2, R], we get

\begin{matrix}u′ ≤ \frac{y}{A}, \\ y′ ≤ −AM(N)ua (\frac{y}{B} ) . \end{matrix}  (4.7)

Multiplying the first inequality in (4.7) by \frac{A}{B} M(N)u and the second one by \frac{y}{ABa( \frac{y}{B})} and adding up, we obtain

\frac{A}{B} M(N)uu′ + \frac{yy′}{a(\frac{y}{B})AB} ≤ \frac{M(N)uy}{B} − \frac{M(N)uy}{B} = 0.

This means that the function \frac{A}{B} M(N)\frac{u^{2}}{2} + \frac{B}{A} L(\frac{y}{B}) is non-increasing in (r_{i}, r^{∗}). Hence, setting u^{∗} := u(r^{∗}), we obtain

\frac{A}{B} M(N)\frac{u(r)^{2}}{2} + \frac{B}{A} L (\frac{y(r)}{B}) ≥ \frac{A}{B} M(N)\frac{(u^{∗})^{2}}{2}

which implies for all r ∈ (r_{i}, r^{∗}),

u′(r) ≥ \frac{B}{r^{λ(k−1)}} L^{−1} (\frac{M(N)A^{2}}{2B^{2}} ((u^{∗})^{2} − u^{2}(r)))
≥ L^{−1} (\frac{M(N)A^{2}}{2B^{2}} ((u^{∗})^{2} − u^{2}(r))) .

Finally, using Proposition 2.4 with c = \frac{M(N)A^{2}}{2B^{2}} , c_{1} = (1/2)^{2k−1}\sqrt{M(N)}, we have for all r ∈ (r_{i}, r^{∗}),

u′(r) ≥ (1/2)^{2k−1}\sqrt{M(N)}L^{−1}((u^{∗})^{2} − u^{2}(r)))

(notice that with the above choices c_{1} > 1 and c > c_{1}^{2} + 1). Integrating from r_{i} to r^{∗}, we get

∫_{r_{i}}^{r^{∗}} \frac{u′(r)}{(1/2)^{2k−1}\sqrt{M(N)}L^{−1}((u^{∗})^{2} − u(r)^{2})} dr ≥ r^{∗} − r_{i}.

If we set u(r) = u, then using (4.4) we obtain

r^{∗} − r_{i} ≤ ∫_{0}^{u^{∗}} \frac{du}{(1/2)^{2k−1}\sqrt{M(N)}L^{−1}((u^{∗})^{2} − u^{2})} < \frac{2^{2k−1}}{N} .

For the completion of the proof of the Claim, it is now sufficient to observe that a computation analogous to the one developed above can be performed if we consider the interval (r^{∗}, r_{i+1}) or an interval (r_{i}, r_{i+1}) where u is negative.

Now, let n_{0} = max(n^{∗}, 2k_{0}) (for the definition of k_{0}, see Theorem 2.2). Next, let us consider n > n_{0} and the number d_{n} arising from Claim 2. In order to prove the existence of the solutions with exactly n zeros by an application of Theorem 4.2, we introduce the sets

B = {(u, λ) ∈ dom N : u(0) < d̄}

(d̄ as in Lemma 3.3 and Claim 1) and

A_{n} = {(u, λ) ∈ dom N : u(0) < d_{n}}.

Moreover, the functional

k : Σ ∩ (B̄ \ A_{n}) → ℕ

will be defined by

k(u, λ) = n(u).

Let us now prove that conditions (4.1) and (4.2) are satisfied. We observe that

∂(B̄ \ A_{n}) = {(u, λ) : u(0) = d̄} ∪ {(u, λ) : u(0) = d_{n}}.

If (u, λ) ∈ Σ and u(0) = d_{n} then, by Claim 2, we get n(u) > n; on the other hand, if (u, λ) ∈ Σ and u(0) = d̄ then, by Claim 1, we have n(u) < n^{∗}. Hence, being n^{∗} < n, condition (4.1) is satisfied.

As far as the boundedness of k^{−1}(n) is concerned, if (u, λ) ∈ k^{−1}(n) ⊂ Σ ∩ (B̄ \ A_{n}), then u(0) < d̄ and we deduce from Lemma 3.1 that u is bounded, hence (4.2) is fulfilled.

Now we have to choose an open set on which to compute the degree. From Theorem 2.2, we know that problem (2.1) has solutions with exactly n zeros in [0, R). These solutions enable us to determine an open bounded set Ω_{0} such that

(k^{−1}(n))_{0} ⊂ Ω_{0}.

We define

U_{0}^{n} = Ω_{0} ∩ (B̄ \ A_{n}).

Arguing as in [4], we can prove that the degree deg(I − N_{0}, U_{0}^{n}) is well defined and deg(I − N_{0}, U_{0}^{n}) ≠ 0.

Hence, an application of Theorem 4.2 provides the existence of a solution u_{n} of problem (3.1) with

n(u_{n}) = n and u_{n}(0) > 0.

Moreover, this solution u_{n} is such that ||u_{n}||_{1} ≤ ε_{0}.

A similar argument, considering the sets

B = {(u, λ) ∈ dom N : u(0) > −d̄}

and

A_{n} = {(u, λ) ∈ dom N : u(0) > −d_{n}}

shows that there exists at least one solution v_{n} of (3.1) such that

n(v_{n}) = n and v_{n}(0) < 0.

The last statement in Theorem 4.1 follows from the properties of n. □

Acknowledgments The first author wishes to thank the Organizers of the meeting “USA-Chile Workshop on Nonlinear Analysis” for the invitation and the hospitality.

References

[1] A. Ambrosetti - J. Garcia Azorero - I. Peral, Quasilinear equations with a multiple bifurcation, Differential Integral Equations, 10 (1997), pp. 37-50.

[2] G. J. Butler, Periodic solutions of sublinear second order differential equations, J. Math. Analysis Appl., 62 (1978), pp. 676-690.

[3] A. Capietto - W. Dambrosio, Boundary value problems with sublinear conditions near zero, NoDEA Nonlinear Differential Equations Appl., 6 (1999), pp. 149-172.

[4] A. Capietto - W. Dambrosio - F. Zanolin, Infinitely many radial solutions to a boundary-value problem in a ball, Quad. Dip. Mat. - Univ. Torino, 20 (1998). Ann. Mat. Pura Appl., to appear.

[5] A. Castro - A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc., 101 (1987), pp. 57-64.

[6] Y. Cheng, On the existence of radial solutions of a nonlinear elliptic equation on the unit ball, Nonlinear Analysis, 24 (1995), pp. 287-307.

[7] W. Dambrosio, Bounday value problems for second order strongly nonlinear differential equations, Ph. D. Thesis, University of Torino, 1998.

[8] H. Dang - R. Manásevich - K. Schmitt, Positive radial solutions of some nonlinear partial differential equations, Math. Nachr., 186 (1997), pp. 101-113.

[9] A. El Hachimi - F. De Thelin, Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball, J. Differential Eq., 128 (1996), pp. 78-102.

[10] M. Garcia-Huidobro - R. Manásevich - F. Zanolin, A Fredholm-like result for strongly nonlinear second order ODE’s, J. Differential Equations, 114 (1994), pp. 132-167.

[11] M. Garcia-Huidobro - R. Manásevich - F. Zanolin, Strongly nonlinear second order ODE’s with rapidly growing terms, J. Math. Analysis Appl., 202 (1996), pp. 1-26.

[12] M. Garcia-Huidobro - R. Manásevich - F. Zanolin, Infinitely many solutions for a Dirichlet problem with a non homogeneous p-Laplacian like operator in a ball, Adv. in Differential Eq., 2 (1997), pp. 203-230.

[13] M. Garcia-Huidobro - U. Ubilla, Multiplicity of solutions for a class of nonlinear second-order equations, Nonlinear Analysis, 28 (1997), pp. 1509-1520.

[14] J. Mawhin, Topological degree methods in Nonlinear Boundary Value Problems, CBMS Series, Amer. Math. Soc., Providence, RI, 1979.

[15] Z. Opial, Sur les périodes des solutions de l’équation différentielle x′′ + g(x) = 0, Ann. Polon. Math., 10 (1961), pp. 49-71.

[16] W. Reichel - W. Walter, Radial solutions of equations and inequalities involving the p-laplacian, J. Inequal. and Appl., 1 (1997), pp. 47-71.

Anna Capietto Dipartimento di Matematica - Università di Torino Via Carlo Alberto 10 - 10123 Torino - Italy e-mail: capietto@dm.unito.it Marielle Cherpion Département de Mathématique - Université Catholique de Louvain Chemin du Cyclotron, 2 - B-1348 Louvain-la-Neuve - Belgium e-mail: cherpion@amm.ucl.ac.be