ANTIMAXIMUM PRINCIPLE FOR QUASILINEAR PROBLEMS

DAVID ARCOYA AND JULIO D. ROSSI

Abstract. In this paper, for a bounded smooth domain Ω ⊂ ℝ^{N}, h ∈ L^{r}(Ω), r > N/2, g ∈ L^{s}(∂Ω), s > N − 1, we prove the maximum and antimaximum principle for the quasilinear b.v.p −div(A(u)∇u) + u = h in Ω, A(u)∇u · η = λf(u) + g, on ∂Ω, where A is elliptic and bounded and f is asymptotically linear. The sharpness of this result (r > N/2 and s > N − 1) is discussed for the linear b.v.p., −Δu + u = h in Ω, ∂u/∂η = λu + g on ∂Ω.

1. Introduction.

In [7] it is proved the well-known maximum/antimaximum principle for the linear Laplace operator with zero Dirichlet boundary condition. Denoting by μ_{1} the first eigenvalue of the Laplace operator (with zero Dirichlet boundary condition), it states that ”given any positive h ∈ L^{r}(Ω), r > N, there exists ε = ε(h) > 0 such that every solution u of −Δu = μu + h in Ω with u |_{∂Ω}= 0 satisfies either u < 0 in Ω provided that μ_{1} < μ < μ_{1} + ε (antimaximum principle), or u > 0 in Ω if μ < μ_{1}” (maximum principle).

The main goal of this paper is to put in evidence the nonlinear nature of this principle. Indeed, we extend the principle to the case of elliptic quasilinear equations with a nonlinear boundary condition. Since we also want to point out some differences between the case of Dirichlet boundary condition and Neumann one, we study here the associated b.v.p with a nonlinear Neumann boundary condition. Specifically, we consider the following quasilinear problem

(1.1)  \{ \begin{matrix}−div(A(x, u)∇u) + u = h, in Ω, \\ A(x, u)∇u · η = λf(x, u) + g, on ∂Ω, \end{matrix}

where Ω is a bounded domain in ℝ^{N} with smooth boundary (C^{1} is enough for our purposes), η is the outer normal and h(x) ∈ L^{r}(Ω) (r > N/2), g(x) ∈ L^{s}(∂Ω) (s > N − 1) are given functions. We also assume that the Carathéodory symmetric matrix A(x, s) = (a_{ij}(x, s)) satisfies for some positive constants α, β that

(1.2) \{ \begin{matrix}|A(x, s)| ≤ β, & A(x, s)ξ · ξ ≥ α|ξ|^{2}, \\ |A(x, s) − A(x, t)| ≤ ω(|s − t|), & lim_{s→+∞} A(x, s) = A(x, +∞), \end{matrix}

with ω a nondecreasing function with ω(0) = 0 and ∫_{0^{+}} (1/ω(s))ds = +∞. A function

ω of this kind is usually called an Osgood function (see [4] where this assumption is introduced). On the Carathéodory nonlinearity f(x, s), we impose that for some C_{1}, C_{2} > 0,

Key words and phrases. Antimaximum principle, Neumann boundary conditions, eigenvalues.

2000 Mathematics Subject Classification. 35P05, 35J60, 35J55.

(1.3)  \{ \begin{matrix}lim_{s→+∞} f(x, s)/s = f′(x, +∞) > 0, \\ |f(x, s)| ≤ C_{1}|s| + C_{2}, \end{matrix}

To simplify the notation, in what follows we drop the explicit dependence of A and f on x. Hence we will denote A(u) for A(x, u), A(+∞) for A(x, +∞), etc.

Under the above hypotheses, it arises the following linear eigenvalue problem at infinity,

(1.4)  \{ \begin{matrix}−div(A(+∞)∇u) + u = 0, in Ω, \\ A(+∞)∇u · η = λf′(+∞)u, on ∂Ω. \end{matrix}

This kind of eigenvalue problems with the eigenvalue λ in the boundary condition appear in a natural way as one considers the Sobolev trace inequality. In fact, taking into account the compact immersion H^{1}(Ω) ↪ L^{2}(∂Ω), there exits a positive constant λ_{1} such that λ_{1}^{1/2}∥u∥_{L^{2}(∂Ω)} ≤ ∥u∥_{H^{1}(Ω)} for every u ∈ H^{1}(Ω). The extremals u (functions for which the above inequality is actually an equality) are solutions of the eigenvalue problem with λ = λ_{1}, A(+∞) = Id and f′(+∞) = 1. Non-homogeneous Neumann boundary conditions also appear in the study of conformal deformations on Riemannian manifolds with boundary, see for example [6].

Using that f′(+∞) > 0 and our assumptions on A(+∞) we obtain, using the same arguments of [9], that there exists a first positive eigenvalue λ_{1} of (1.4) which can be variationally characterized by

(1.5)  λ_{1} = inf_{u∈H^{1}(Ω)} {∫_{Ω} A(+∞)∇u · ∇u + |u|^{2} / ∫_{∂Ω} f′(+∞)|u|^{2} = 1} .

In addition, it is isolated and simple, and it has an associated eigenfunction φ_{1} which is strictly positive in \overline{Ω}. With this notation, the main result of the paper is the following theorem.

Theorem 1.1. Let h ∈ L^{r}(Ω) and g ∈ L^{s}(∂Ω) with r > N/2 and s > N − 1 and assume (1.2) and (1.3). Then there exist R, ε > 0 such that if u is a solution of (1.1) with ∥u∥ ≥ R, we have

1. If A(s)ξ·ξ ≤ A(+∞)ξ·ξ for every ξ ∈ ℝ^{N}, f(s) ≥ f′(+∞)s for s ≥ s_{0} >> 0 and

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} > 0,

then λ_{1} < λ < λ_{1} + ε implies u < 0 in \overline{Ω}.

2. Conversely, if A(s)ξ · ξ ≥ A(+∞)ξ · ξ for every ξ ∈ ℝ^{N}, f(s) ≤ f′(+∞)s

for s ≥ s_{0} >> 0 and

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} < 0,

then λ_{1} − ε < λ < λ_{1} implies u < 0 in \overline{Ω}.

A dual result proving the positivity of the solutions (a local maximum principle) is also true if we replace the conditions about the behavior at +∞ with the reciprocal assumptions at −∞, see Corollary 3.1 below.

Also, let us remark explicitly that we do not require to g and h to have definite sign. Our assumptions involve an integral balance with φ_{1} as a weight. This allows us to take, for example, a negative h and to obtain, for instance, the antimaximum result if g is large enough.

With respect to the proof of this theorem, we emphasize that the main idea is to see the problem (1.1) from a bifurcation (from infinity) point of view like in [2].

As a direct consequence, we deduce for the linear b.v.p.

(1.6)  \{ \begin{matrix}−Δu + u = h, in Ω, \\ \frac{∂u}{∂η} = λu + g, on ∂Ω \end{matrix}

the following (see also [3] for a similar result).

Corollary 1.1. Let h ∈ L^{r}(Ω) and g ∈ L^{s}(∂Ω) with r > N/2 and s > N − 1.

(1) If

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} > 0,

then there exists ε > 0 such that every solution of (1.6) with λ_{1}−ε < λ < λ_{1} verifies u > 0 in Ω and every solution of (1.6) with λ_{1} < λ < λ_{1} +ε verifies u < 0 in \overline{Ω}.

(2) If

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} < 0,

then there exists ε > 0 such that every solution of (1.6) with λ_{1}−ε < λ < λ_{1} verifies u < 0 in Ω and every solution of (1.6) with λ_{1} < λ < λ_{1} +ε verifies u > 0 in \overline{Ω}.

(3) If

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} = 0,

then every solution of (1.6) with λ ≠ λ_{1} changes sign in \overline{Ω}.

Note that for the Neumann boundary condition we are just assuming that h is in L^{r}(Ω) with r > N/2. This is in contrast with the case of Dirichlet boundary condition in [7] where it is assumed r > N. This strengthened assumption is necessary for the Dirichlet case. Indeed, it is proved in [12] that there exists a positive function h ∈ L^{N}(Ω) for which the the antimaximum principle does not hold; i.e. the space L^{r}(Ω), with r > N is sharp for this principle.

As far as we are concerned, we also prove here that L^{N/2}(Ω) and L^{N−1}(∂Ω) are sharp for the antimaximum principle in the case of Neumann boundary conditions. First, let us see that antimaximum principle does not hold, in general, for h ∈ L^{N/2}(Ω).

Theorem 1.2. There exists a positive function h ∈ L^{N/2}(Ω) such that the solution of (1.6) with g ≡ 0 and λ > λ_{1} is positive somewhere in Ω.

Next, we prove also that the hypothesis g ∈ L^{N−1}(∂Ω) is not sufficient to obtain the antimaximum principle.

Theorem 1.3. There exist a bounded domain Ω and a positive function g ∈ L^{N−1}(∂Ω) such that the solution of (1.6) with h ≡ 0 and λ > λ_{1} is positive somewhere in Ω.

The paper is organized as follows: in Section 2 we prove some regularity results that we need to prove the Theorem 1.1. This theorem and the first consequences are given in Section 3. To conclude, Section 4 is devoted to show that the integrability conditions on h and g are sharp.

2. Some Regularity results

In order to prove Theorem 1.1, we will need the C^{ν}-regularity of the solutions for ν ∈ (0, 1). Our argument for this is based in [10].

Theorem 2.1. Let a(x) = (a_{ij}(x))_{i,j=1,2,...N} be a measurable and bounded symmetric matrix in Ω. Assume that there exist positive constants α, β such that

(2.1)  α|ξ|^{2} ≤ a(x)ξ · ξ ≤ β|ξ|^{2}, ∀ξ ∈ ℝ^{N}, ∀x ∈ Ω.

Let also h ∈ L^{r}(Ω) and g ∈ L^{s}(∂Ω) with r > N/2 and s > N − 1, then for every ν ∈ (0, 1), there exists C > 0 which depends only on α and β such that every solution v of the Neumann boundary value problem

(2.2)  \{ \begin{matrix}−div (a(x)∇v) + v = h,  x ∈ Ω \\ a(x)∇v · η = g, on ∂Ω. \end{matrix}

satisfies that v ∈ C^{ν}(\overline{Ω}) with

∥v∥_{C^{ν}(\overline{Ω})} ≤ C (∥h∥_{L^{r}(Ω)}∥ + ∥g∥_{L^{s}(∂Ω)}) .

Proof: Define Ψ_{v} : C^{1}(\overline{Ω}) ⟶ ℝ by

Ψ_{v}(φ) = ∫_{Ω} h φ + ∫_{∂Ω} g φ, φ ∈ C^{1}(\overline{Ω}).

From Hölder inequality, we have that for every φ ∈ C^{1}(\overline{Ω})

|Ψ_{v}(φ)| ≤ ∥h∥_{L^{r}(Ω)}∥φ∥_{L^{r′} (Ω)} + ∥g∥_{L^{s}(∂Ω)}∥φ∥_{L^{s′} (∂Ω)}.

Let q = max {Nr/(N − r), Ns/(N − 1)} and notice that the assumptions r > N/2 and s > N − 1 mean that q > N. Hence the conjugate q′ of q is smaller than N and thus the embeddings

W^{1,q′}(Ω) ↪ L^{r′}(Ω)  and  W^{1,q′}(Ω) ↪ L^{s′}(∂Ω)

hold. By using these, we obtain for some positive C_{1} that

|Ψ_{v}(φ)| ≤ C_{1} (∥h∥_{L^{r}(Ω)}∥ + ∥g∥_{L^{s}(∂Ω)}) ∥φ∥_{W^{1,q′} (Ω)}, ∀φ ∈ C^{1}(\overline{Ω}).

Recalling that C^{1}(\overline{Ω}) is dense in W^{1,q′}(Ω) we can extend Ψ_{v} to W^{1,q′}(Ω) with

(2.3)  ∥Ψ_{v}∥ ≤ C_{1} (∥h∥_{L^{r}(Ω)}∥ + ∥g∥_{L^{s}(∂Ω)}) ,

in other words, Ψ_{v} is a continuous functional on the dual space of W^{1,q′}(Ω). Consequently, by applying the characterization of the dual of the Sobolev space, there exist w, w_{1}, w_{2}, ⋯, w_{N} ∈ L^{q}(Ω) and a constant K = K(N, q, Ω) > 0 such that

Ψ_{v}(φ) = ∫_{Ω} ∑_{i=1}^{N} w_{i} \frac{∂φ}{∂x_{i}} + ∫_{Ω} w φ, ∀φ ∈ W^{1,q′}(Ω),

with

(2.4)  K (∑_{i=1}^{N} ∥w_{i}∥_{L^{q}(Ω)} + ∥w∥_{L^{q}(Ω)}) ≤ ∥Ψ_{v}∥.

In this way, if v is a solution of (2.2), we get

∫_{Ω} a(x)∇v ∇φ + ∫_{Ω} v φ = ∫_{Ω} ∑_{i=1}^{N} w_{i} \frac{∂φ}{∂x_{i}} + ∫_{Ω} w φ, ∀φ ∈ H^{1}(Ω).

with w, w_{1}, w_{2}, ⋯, w_{N} ∈ L^{q}(Ω), q > N. Our final idea is to use Theorems 4.1 and 4.2 in [10]. In order to do that and following the notations in that paper, we take V = H^{1}(Ω) and observe that by the remark following [10, Definition 2.1]

K^{+}(y) = K^{−}(y) = (−∞, +∞), ∀y ∈ ∂Ω,

and, in addition, the function \overline{ρ}(y) appearing in that definition can be chosen independently of y ∈ ∂Ω, i.e. \overline{ρ}(y) ≡ \overline{ρ} > 0. On the other hand, since the boundary of Ω is C^{1}, it follows that Ω is λ-admisible with respect to H^{1}(Ω) for every λ ∈ (0, 1) [10, Definition 3.1]. Indeed, by the arguments in [10, page 28], every bounded domain with C^{1}-boundary is an open of the kind given in the Example 11.1 of [10] with the function ρ̃ independent of y ∈ ∂Ω. Therefore, assertion at the end of the page 20 in [10] works and we deduce from Theorem 4.1 in that paper that v ∈ C^{ν}(\overline{Ω}) and the existence of a constant C_{2} > 0 which just depends on α and β (and not in the particular function a(x) that we are considering) such that

∥v∥_{C^{ν}(\overline{Ω})} ≤ C_{2} (∥w∥_{L^{q}(Ω)} + ∑_{i=1}^{N} ∥w_{i}∥_{L^{q}(Ω)}) .

Taking into account (2.4) and (2.3) we conclude the proof. □

The above result will be combined with the following L^{p}(∂Ω)-regularity. Instead of giving here the sharpest result, we just state here the version that we will need in our arguments.

Lemma 2.1. Let again a(x) = (a_{ij}(x))_{i,j=1,2,...N} be a measurable and bounded symmetric matrix in Ω satisfying (2.1). If h ∈ L^{N/2}(Ω) and g ∈ L^{s}(∂Ω) with 2(N − 1)/N < s ≤ N − 1, then every solution v of the Neumann boundary value problem (2.2) satisfies that

v ∈ L^{p}(∂Ω), ∀p ∈ [1, ϕ(s)),

with

ϕ(s) = \{ \begin{matrix}(\frac{N − 2}{N − 1} − 1 + \frac{1}{s})^{−1} , & if s < N − 1, \\ ∞, & if s = N − 1. \end{matrix}

Proof. We follow the Moser technique. For every k > 0, let us denote by T_{k} the truncature function given by

T_{k}(t) = \{ \begin{matrix}−k, if t < −k, \\ t, if − k ≤ t ≤ k, \\ k, if k < t. \end{matrix}

Notice that v_{k} := |T_{k}(v)|^{2γ}T_{k}(v) ∈ H_{0}^{1}(Ω), for every v ∈ H_{0}^{1}(Ω) and k > 0. Thus, if v is a solution of (2.2), we deduce by using v_{k} as test function that

(2γ + 1) ∫_{Ω} a(x)|T_{k}(v)|^{2γ}∇T_{k}(v) · ∇T_{k}(v) + ∫_{Ω} v|T_{k}(v)|^{2γ}T_{k}(v) = ∫_{Ω} hv_{k} + ∫_{∂Ω} gv_{k}.

Taking into account that from (2.1)

∫_{Ω} a(x)|T_{k}(v)|^{2γ}∇T_{k}(v) · ∇T_{k}(v) ≥ α ∫_{Ω} |T_{k}(v)|^{2γ}|∇T_{k}(v)|^{2}

= \frac{α}{(γ + 1)^{2}} ∫_{Ω} |∇[|T_{k}(v)|^{γ}T_{k}(v)]|^{2}

and

∫_{Ω} v|T_{k}(v)|^{2γ}T_{k}(v) = ∫_{Ω} ||T_{k}(v)|^{γ}T_{k}(v)|^{2} + ∫_{Ω} [v − T_{k}(v)] |T_{k}(v)|^{2γ}T_{k}(v)

(t − T_{k}(t) ≥ 0, ∀t ∈ ℝ) ≥ ∫_{Ω} ||T_{k}(v)|^{γ}T_{k}(v)|^{2} ,

we get for some positive constant C_{1}

∫_{Ω} |∇[|T_{k}(v)|^{γ}T_{k}(v)]|^{2} + ∫_{Ω} ||T_{k}(v)|^{γ}T_{k}(v)|^{2} ≤ C_{1} (∫_{Ω} hv_{k} + ∫_{∂Ω} gv_{k}) .

≤ C_{1}∥h∥_{L^{N/2}(Ω)} ∫_{Ω} |T_{k}(v)|^{(2γ+1) \frac{N}{N−2}}

+C_{1}∥g∥_{L^{s}(∂Ω)} ∫_{∂Ω} |T_{k}(v)|^{(2γ+1)s′}

where s′ = s/(s − 1). Suppose now that v ∈ L^{μ}(Ω) ∩ L^{m}(∂Ω) with \frac{m}{s′} ≤ μ\frac{N−2}{N}, then by choosing

γ = \frac{1}{2} (\frac{m}{s′} − 1) ≤ \frac{1}{2} (\frac{μ(N − 2)}{N} − 1)

we have

(2γ + 1)s′ = m, (2γ + 1)\frac{N}{N − 2} = \frac{m}{s′} \frac{N}{N − 2} ≤ μ

and thus

∫_{Ω} |∇[|T_{k}(v)|^{γ}T_{k}(v)]|^{2} + ∫_{Ω} ||T_{k}(v)|^{γ}T_{k}(v)|^{2} ≤ C_{2} (∥h∥_{L^{N/2}(Ω)}∥v∥_{L^{μ}(Ω)}
+ ∥g∥_{L^{s}(∂Ω)}∥v∥_{L^{m}(∂Ω)})

for C_{2} > 0. Taking the limit as k tends to infinity and using Fatou Lemma,

∫_{Ω} |∇[|v|^{γ}v]|^{2} + ∫_{Ω} ||v|^{γ}v|^{2} ≤ C_{2} (∥h∥_{L^{N/2}(Ω)}∥v∥_{L^{μ}(Ω)} + ∥g∥_{L^{s}(∂Ω)}∥v∥_{L^{m}(∂Ω)}) .

Consequently, denoting

(2.5)

2^{∗} = \{ \begin{matrix}2N/(N − 2), & if N ≥ 3, \\ ∞, & if N = 2, \end{matrix}

\overline{2} = \{ \begin{matrix}2(N − 1)/(N − 2), & if N ≥ 3, \\ ∞, & if N = 2, \end{matrix}

we have

|v|^{γ}v ∈ H^{1}(Ω) ↪ L^{2^{∗}}(Ω) ∩ L\overline{2}(∂Ω),

which implies that

v ∈ L^{2^{∗}(γ+1)}(Ω) ∩ L^{\overline{2}(γ+1)}(∂Ω).

In this way we have shown the implication

(2.6)  \begin{matrix}v ∈ L^{μ}(Ω) ∩ L^{m}(∂Ω) \\ \frac{m}{s′} ≤ μ\frac{N − 2}{N} \end{matrix} \} ⟹ v ∈ L^{φ(m)}(Ω) ∩ L^{ψ(m)}(∂Ω),

with

φ(m) = \frac{2^{∗}}{2} (\frac{m}{s′} + 1) , ψ(m) = \frac{\overline{2}}{2} (\frac{m}{s′} + 1) .

Now, let us define by induction the sequences

\begin{matrix}μ_{0} = 2^{∗}, & m_{0} = \overline{2}, \\ μ_{n+1} = φ(m_{n}), & m_{n+1} = ψ(m_{n}), & n ≥ 0. \end{matrix}

Since φ(m), ψ(m) > m, for all m > 0, the sequences {μ_{n}}_{n≥1} and {m_{n}}_{n≥0} are increasing. Applying (2.6) we deduce that

v ∈ L^{φ(m_{n})}(Ω) ∩ L^{ψ(m_{n})}(∂Ω)

and the lemma is easily concluded by observing that, if s < N − 1, m_{n} increasingly converges to the fixed point of ψ, i.e. to the number

ϕ(s) = \frac{1}{\frac{N − 2}{N − 1} − 1 + \frac{1}{s}} ,

and that, if s = N − 1, converges to +∞. □

3. Local Maximum and Anti-maximum Principle

We begin by proving the quasilinear local anti-maximum principle.

Proof of Theorem 1.1. Let us begin by item 1. We argue by contradiction assuming that there exist a sequence (λ_{n}, u_{n}) with λ_{n} ↘ λ_{1}, ∥u_{n}∥ ⟶ +∞ and u_{n} being a solution of (1.1) for λ = λ_{n} which is not a negative function in \overline{Ω} (i.e., it is nonnegative somewhere). Notice that this means that λ = λ_{1} is a bifurcation point from infinity. Using that u_{n} is a weak solution of (1.1) we get

(3.1) ∫_{Ω} A(u_{n})∇u_{n}∇φ + ∫_{Ω} u_{n}φ = ∫_{Ω} hφ + ∫_{∂Ω} gφ + λ_{n} ∫_{∂Ω} f(u_{n})φ, ∀φ ∈ H^{1}(Ω).

If we consider w_{n} = u_{n}/∥u_{n}∥_{H^{1}(Ω)} we get

∫_{Ω} A(u_{n})∇w_{n}∇φ + ∫_{Ω} w_{n}φ = ∫_{Ω} \frac{h}{∥u_{n}∥_{H^{1}(Ω)}} φ + ∫_{∂Ω} \frac{g}{∥u_{n}∥_{H^{1}(Ω)}} φ

+λ_{n} ∫_{∂Ω} \frac{f(u_{n})}{∥u_{n}∥_{H^{1}(Ω)}} φ, ∀φ ∈ H^{1}(Ω).

From which it follows that u_{n}/∥u_{n}∥_{H^{1}(Ω)} → ±φ_{1} in H^{1}(Ω), see [1] for the details. By using the embedding of H^{1}(Ω) into L\overline{2}(∂Ω) (\overline{2} is given by (2.5)) and (1.3) we have that f(u_{n})/∥u_{n}∥_{H^{1}(Ω)} ∈ L\overline{2}(∂Ω). A bootstrap argument applying Lemma 2.1 implies that u_{n} and thus f(u_{n})/∥u_{n}∥_{H^{1}(Ω)} are in some L^{s}(∂Ω) for some s > N − 1. Indeed, it can be occur two possibilities: either \overline{2} > N − 1 or \overline{2} ≤ N − 1. In the first case, we have shown the assertion. In the second one, (i.e. if N ≥ 4), Lemma 2.1 gives us that u_{n}, f(u_{n})/∥u_{n}∥_{H^{1}(Ω)} ∈ L^{ϕ(\overline{2})}(∂Ω). Again, there are two possibilities for the number ϕ(\overline{2}): either ϕ(\overline{2}) > N −1 or ϕ(\overline{2}) ≤ N −1. If the second inequality holds, a new application of Lemma 2.1 yields u_{n}, f(u_{n})/∥u_{n}∥_{H^{1}(Ω)} ∈ L^{ϕ(ϕ(\overline{2}))}(∂Ω). It is easily deduced that in a finite number of steps we get that u_{n}, f(u_{n})/∥u_{n}∥_{H^{1}(Ω)} ∈ L^{s}(∂Ω), for some s > N − 1 and the claim is proved. Now, by Theorem 2.1, we have a uniform bound for the sequence u_{n}/∥u_{n}∥_{H^{1}(Ω)} in C^{ν}(\overline{Ω}) therefore, passing to a subsequence if necessary, we can assume that convergence is uniform (indeed, we may assume Cν̃-convergence for any ν̃ < ν). Since u_{n} ≮ 0 in \overline{Ω}, we get that u_{n}/∥u_{n}∥_{H^{1}(Ω)} → φ_{1}. Hence u_{n} is strictly positive in \overline{Ω} for large n. Inspired by [5], we take as test function in (3.1) φ = φ_{1}^{2}/u_{n}, and we obtain

∫_{Ω} A(u_{n})∇u_{n} (\frac{2φ_{1}}{u_{n}} ∇φ_{1} − \frac{φ_{1}^{2}}{u_{n}^{2}} ∇u_{n}) + ∫_{Ω} φ_{1}^{2}

= ∫_{Ω} h\frac{φ_{1}^{2}}{u_{n}} + ∫_{∂Ω} g \frac{φ_{1}^{2}}{u_{n}} + λ_{n} ∫_{∂Ω} f(u_{n})\frac{φ_{1}^{2}}{u_{n}} .

Using the variational characterization (1.5) of λ_{1}, we get that φ_{1} satisfies

∫_{Ω} φ_{1}^{2} = − ∫_{Ω} A(+∞)∇φ_{1} · ∇φ_{1} + λ_{1} ∫_{∂Ω} f′(+∞)φ_{1}^{2}.

Hence, we get

− ∫_{Ω} (A(+∞) − A(u_{n}))|∇φ_{1}|^{2} − ∫_{Ω} A(u_{n}) (∇φ_{1} − \frac{φ_{1}}{u_{n}} ∇u_{n}) (∇φ_{1} − \frac{φ_{1}}{u_{n}} ∇u_{n})

= ∫_{Ω} h\frac{φ_{1}^{2}}{u_{n}} + ∫_{∂Ω} g \frac{φ_{1}^{2}}{u_{n}} − λ_{1} ∫_{∂Ω} f′(+∞)φ_{1}^{2} + λ_{n} ∫_{∂Ω} f(u_{n})\frac{φ_{1}^{2}}{u_{n}} .

As (A(+∞) − A(u_{n}))ξ · ξ ≥ 0 and f(u_{n}) ≥ f′(+∞)u_{n} we have,

0 ≥ ∫_{Ω} h\frac{φ_{1}^{2}}{u_{n}} + ∫_{∂Ω} g \frac{φ_{1}^{2}}{u_{n}} − (λ_{1} − λ_{n}) ∫_{∂Ω} f′(+∞)φ_{1}^{2}.

Using that u_{n}/∥u_{n}∥_{H^{1}(Ω)} → φ_{1} we obtain, for n large enough,

sgn [∫_{Ω} h\frac{φ_{1}^{2}}{u_{n}} + ∫_{∂Ω} g \frac{φ_{1}^{2}}{u_{n}} ] = sgn [∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1}]

and we conclude that λ_{n} < λ_{1}, for n large. This is a contradiction proving that the statement of the item 1. holds.

To prove item 2., we assume again by contradiction that there exist a sequence (λ_{n}, u_{n}) with λ_{n} ↗ λ_{1}, ∥u_{n}∥ ⟶ +∞ and u_{n} being a solution of (1.1) for λ = λ_{n} which is positive somewhere in \overline{Ω}. Similar arguments to the previous ones show that u_{n}/∥u_{n}∥ → φ_{1} uniformly in \overline{Ω} and, hence, that u_{n} > 0 in \overline{Ω} for n large enough. Let us take φ = u_{n} as test function in (3.1) to obtain

(3.2)  ∫_{Ω} A(u_{n})∇u_{n} · ∇u_{n} + ∫_{Ω} |u_{n}|^{2} = ∫_{Ω} hu_{n} + ∫_{∂Ω} gu_{n} + λ_{n} ∫_{∂Ω} f(u_{n})u_{n}.

Using that A(u_{n})ξ · ξ ≥ A(+∞)ξ · ξ and the characterization of λ_{1}, (1.5), we have

∫_{Ω} A(u_{n})∇u_{n} · ∇u_{n} + ∫_{Ω} |u_{n}|^{2} ≥ ∫_{Ω} A(+∞)∇u_{n} · ∇u_{n} + ∫_{Ω} |u_{n}|^{2}

≥ λ_{1} ∫_{∂Ω} f′(+∞)|u_{n}|^{2}.

Hence, using that f(u_{n}) ≤ f′(+∞)u_{n}, we obtain

0 ≤ ∫_{Ω} hu_{n} + ∫_{∂Ω} gu_{n} + (λ_{n} − λ_{1}) ∫_{∂Ω} f′(+∞)|u_{n}|^{2}.

As in the previous case we use that u_{n}/∥u_{n}∥_{H^{1}(Ω)} → φ_{1} to obtain,

sgn [∫_{Ω} hu_{n} + ∫_{∂Ω} gu_{n}] = sgn [∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1}] .

Therefore, λ_{n} > λ_{1}, for n large, a contradiction which states item 2. □

Under our hypotheses on A and f, (1.2), (1.3), following the arguments in [1] and [2] one can check that, if we restrict ourselves to positive solutions, then there exists a bifurcation from +∞ only at λ = λ_{1}. Notice that, from the point of view of the bifurcation theory, the meaning of the first item in the theorem is that the bifurcation of positive solutions from +∞ is subcritical (i.e. to the left of λ_{1}).

Similarly, if we replace the assumptions at +∞ with the following ones

lim_{s→+∞} A(x, s) = A(x, −∞),

lim_{s→−∞} f(x, s)/s = f′(x, −∞) > 0,

and we denote by λ_{−} the first eigenvalue of the eigenvalue problem (1.4) with A(−∞) and f(−∞) instead of A(+∞) and f′(+∞), we have a quasilinear version of the local maximum principle:

Corollary 3.1. Let h ∈ L^{r}(Ω) and g ∈ L^{s}(∂Ω) with r > N/2 and s > N −1. Then there exist R, ε > 0 such that if u is a solution of (1.1) with ∥u∥ ≥ R, we have

1. If A(s)ξ·ξ ≤ A(−∞)ξ·ξ for every ξ ∈ ℝ^{N}, f(s) ≤ f′(−∞)s for s ≤ s_{0} << 0 and

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} > 0,

then λ_{−} < λ < λ_{−} + ε implies u > 0 in \overline{Ω}.

2. Conversely, if A(s)ξ · ξ ≥ A(−∞)ξ · ξ for every ξ ∈ ℝ^{N}, f(s) ≥ f′(−∞)s

for s ≥ s_{0} << 0 and

∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} < 0,

then λ_{−} − ε < λ < λ_{−} implies u > 0 in \overline{Ω}.

We leave the details of the proof to the reader. With respect to the proof of the Corollary 1.1, remark that problem (1.6) is a particular case of (1.1) with A(s) = Id (= A(+∞) = A(−∞)) and f(s) = s. Hence Theorem 1.1 and Corollary 3.1 work and items 1. and 2. of the corollary are clearly deduced. Therefore, to conclude the proof we just have to discuss item 3.

Proof of Corollary 1.1 − 3. Let u be any solution of (1.6) with λ ≠ λ_{1} and take φ = φ_{1} as test function in the equation satisfied by u, to get

(λ − λ_{1}) ∫_{∂Ω} uφ_{1} = ∫_{Ω} hφ_{1} + ∫_{∂Ω} gφ_{1} = 0.

Since φ_{1} > 0 in \overline{Ω}, we conclude that u must change sign in ∂Ω. □

4. Counterexamples

Proof of Theorem 1.2. First let us find a positive function h ∈ L^{N/2}(Ω) such that the solution of

(4.1)  \{ \begin{matrix}−Δz + z = h, in Ω, \\ \frac{∂z}{∂η} = 0, on ∂Ω, \end{matrix}

is unbounded in Ω. To see that, let E_{N}(x), (x ∈ ℝ^{N}), the fundamental solution of the Helmhotz equation

−ΔE_{N}(x) + E_{N}(x) = 0.

It can be seen in [11, pag. 245] (see also [8, Proposition 27, pag. 739]) that E_{N} has a singularity at zero. Specifically, it holds

lim_{|x|→0} |x|^{N−2}E_{N}(x) = C > 0, for N ≥ 3,

and

lim_{|x|→0} \frac{E_{2}(x)}{− log |x|} = C > 0 (for N = 2).

As a consequence of it, for every h ∈ L^{N/2}(Ω) the Green function G_{N}(x, y) associated to the b.v.p. (4.1) satisfies that

lim_{y→x} |x − y|^{N−2}G_{N}(x, y) = C > 0, for N ≥ 3,

and

lim_{y→x} \frac{G_{2}(x, y)}{− log |x − y|} = C > 0 (for N = 2).

Hence

G_{N}(x, y) = \{ \begin{matrix}\frac{C}{|x − y|^{N−2}} + bounded, if N ≥ 3, \\ −C log |x − y| + bounded, if N = 2. \end{matrix}

The solution z(x) of the b.v.p. (4.1) can be written in terms of h in the following way:

z(x) = ∫_{Ω} G_{N}(x, y)h(y) dy.

Thus, by choosing x_{0} ∈ \overline{Ω} and h > 0 such that for some r ∈ (0, 1),

• h is bounded in Ω\Ω_{r}, with Ω_{r} = {x ∈ Ω / |x − x_{0}| < r},

• for x ∈ Ω_{r},

h(x) = \{ \begin{matrix}\frac{1}{|x − x_{0}|^{2} (− log |x − x_{0}|)^{a}} , for N ≥ 3, with \frac{2}{N} < a < 1, \\ \frac{1}{|x − x_{0}|^{2} (− log |x − x_{0}|)^{2}} , for N = 2, \end{matrix}

we have h ∈ L^{N/2}(Ω) (since a > 2/N in the case N ≥ 3). Then for |x − x_{0}| < r/2, the integrability of the function h together to the boundedness of G_{N}(x, ·) in Ω\Ω_{r} imply

∫_{Ω} G_{N}(x, y)h(y) dy = { ∫_{Ω\Ω_{r}} + ∫_{Ω_{r}} } G_{N}(x, y)h(y) dy

= bounded + ∫_{Ω_{r}} G_{N}(x, y)h(y) dy.

Since

∫_{Ω_{r}} G_{N}(x − y)h(y)dy = \{ \begin{matrix}bounded + C ∫_{Ω_{r}} \frac{|x − y|^{2−N}}{|y − x_{0}|^{2}(− log |y − x_{0}|)^{a}} dy, if N ≥ 3, \\ bounded − C ∫_{Ω_{r}} \frac{log |x − y|}{|y − x_{0}|^{2}(− log |y − x_{0}|)^{2}} dy, if N = 2, \end{matrix}

from Fatou lemma, we deduce that

lim_{x→x_{0}} ∫_{Ω_{r}} G_{N}(x − y)h(y) dy ≥ \{ \begin{matrix}C ∫_{Ω_{r}} \frac{1}{|y − x_{0}|^{N}(− log |y − x_{0}|)^{a}} dy, if N ≥ 3, \\ C ∫_{Ω_{r}} \frac{1}{|y − x_{0}|^{2}(− log |y − x_{0}|)} dy, if N = 2, \end{matrix}

= +∞

where, in the case N ≥ 3, we are using that a < 1.

Consequently, the solution z(x) of (4.1) satisfies

lim_{x→x_{0}} z(x) = +∞,

and, in particular, it is unbounded.

Now let u be any solution of (1.6) with λ > λ_{1}, h and g ≡ 0. Let us consider w = u − z which satisfies

\{ \begin{matrix}−Δw + w = 0 in Ω, \\ \frac{∂w}{∂η} = λu, on ∂Ω. \end{matrix}

Arguing like in the proof of Theorem 1.1, by using Lemma 2.1, u ∈ L^{s}(∂Ω) with s > N − 1. Now, from Theorem 2.1, we get that w ∈ C(\overline{Ω}) and then w is bounded in Ω. Hence there exists a constant C_{1} > 0 such that

w = u − z ≥ −C_{1},

and therefore

lim sup_{x→x_{0}} u(x) ≥ lim_{x→x_{0}} z(x) − C_{1} = +∞.

This proves that there exists points in Ω where u is positive and hence the antimaximum principle does not hold for this function h. □

Remark 4.1. We remark explicitly that, for the case of Neumann boundary conditions, the sharpness of the space L^{r}(Ω) with r > N/2 (instead of r > N like in the Dirichlet case) is due to the positivity of the first eigenfunction φ_{1} on ∂Ω.

Proof of Theorem 1.3. Let us assume that for some x_{0} ∈ ∂Ω and r ∈ (0, 1), we have

{x ∈ ∂Ω / |x − x_{0}| < r} = {x ∈ ℝ^{N} / |x − x_{0}| < r, x_{N} = 0} .

By using similar arguments to the previous ones in the proof of Theorem 1.2, we just have to prove the existence of a positive function g for which the solution of the b.v.p.

(4.2)  \{ \begin{matrix}−Δz + z = 0, in Ω, \\ \frac{∂z}{∂η} = g, on ∂Ω, \end{matrix}

be unbounded in Ω. As before, note that z is given by

z(x) = ∫_{∂Ω} G_{N}(x, y)g(y) dσ(y),

where dσ(y) denotes the surface measure on ∂Ω. We consider g > 0 in ∂Ω such that

• g is bounded in ∂Ω\A_{r}, with A_{r} = {x ∈ ∂Ω / |x − x_{0}| < r},

• for x ∈ A_{r},

g(x) = \{ \begin{matrix}\frac{1}{|x − x_{0}| (− log |x − x_{0}|)^{a}} , for N ≥ 3, with \frac{1}{N − 1} < a < 1, \\ \frac{1}{|x − x_{0}| (− log |x − x_{0}|)^{2}} , for N = 2. \end{matrix}

We have g ∈ L^{N−1}(Ω) (since a > 1/(N − 1) in the case N ≥ 3). Then for x ∈ ∂Ω such that |x − x_{0}| < r/2, the integrability of the function g together to the boundedness of G_{N}(x, ·) in ∂Ω\A_{r} imply

∫_{∂Ω} G_{N}(x, y)g(y)dσ(y) = { ∫_{∂Ω\A_{r}} + ∫_{A_{r}} } G_{N}(x, y)g(y)dσ(y)

= bounded + ∫_{A_{r}} G_{N}(x, y)g(y)dσ(y).

Since

∫_{A_{r}} G_{N}(x−y)g(y)dσ(y) = \{ \begin{matrix}bounded + C ∫_{A_{r}} \frac{|x − y|^{2−N}}{|y − x_{0}|(− log |y − x_{0}|)^{a}} dσ(y), (N ≥ 3) \\ bounded − C ∫_{A_{r}} \frac{log |x − y|}{|y − x_{0}|(− log |y − x_{0}|)^{2}} dσ(y), (N = 2) \end{matrix}

from Fatou lemma, we deduce that

lim_{x→x_{0}} ∫_{A_{r}} G_{N}(x − y)g(y)dσ(y) ≥ \{ \begin{matrix}C ∫_{A_{r}} \frac{|y − x_{0}|^{1−N}}{(− log |y − x_{0}|)^{a}} dσ(y), (N ≥ 3) \\ C ∫_{A_{r}} \frac{1}{|y − x_{0}|(− log |y − x_{0}|)}dσ(y), (N = 2) \end{matrix}

= +∞

where, in the case N ≥ 3, we are using that a < 1.

Consequently, the solution z(x) of (4.2) satisfies

lim_{x→x_{0}} z(x) = +∞,

and, in particular, it is unbounded.

Let now u be a solution of (1.6) with λ > λ_{1}, g and h ≡ 0. We can proceed as before considering w = u − z and obtaining from the regularity results that w is bounded in Ω. The proof concludes similarly to the previous one. □

Acknowledgements. D. Arcoya is partially supported by D.G.I.C.Y.T. (Spain), Proyecto BFM2003-03772. J. D. Rossi is partially supported by ANPCyT PICT No. 03-00000-00137, Fundación Antorchas and CONICET. We want to thank J. L. Gámez, J. García-Azorero, S. Martínez and I. Peral for several interesting discussions.

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Departamento de Análisis Matemático, Universidad de Granada, Granada 18071, Spain

E-mail address: darcoya@ugr.es

Departamento de Matemática, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile.

E-mail address: jrossi@mat.puc.cl