A BIFURCATION PROBLEM GOVERNED BY THE BOUNDARY CONDITION I ^{∗}

Jorge García-Melián^{1}, Julio D. Rossi^{2} and José C. Sabina de Lis^{1} ^{†}

1 Dpto. de Análisis Matemático, Universidad de La Laguna, C/ Astrofísico Francisco Sánchez s/n, 38271 - La Laguna, SPAIN

2 Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, ARGENTINA

Abstract

We deal with positive solutions of Δu = a(x)u^{p} in a bounded smooth domain Ω ⊂ ℝ^{N} subject to the boundary condition ∂u/∂ν = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ_{1} where, roughly speaking, σ_{1} is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover we find the limit profile of the solution as λ → σ_{1}.

1. Introduction

It is the main concern of the present work the study of the following semilinear boundary value problem:

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = λu & x ∈ ∂Ω , \end{matrix}  (1.1)

where Ω ⊂ ℝ^{N} is a bounded smooth domain of class C^{2,α}, 0 < α < 1, with outward unit normal ν on ∂Ω and where λ ∈ ℝ is regarded as a perturbation parameter. It will be assumed that a(x) ∈ C^{α}(\overline{Ω}), a ≢ 0, is a nonnegative coefficient, while the exponent p will be kept in the range p > 1.

What constitutes the main feature of problem (1.1) is its dependence on the parameter λ precisely in the boundary condition. Our main objective here is just to study the variation regimes of λ giving existence of positive solutions to (1.1) (so, we have avoided employing the more formal |u|^{p} or |u|^{p−1}u instead of u^{p}) analyzing their uniqueness, dependence on λ and covering the asymptotic behavior when λ → ∞ in those cases where such solutions exist for large λ.

∗Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.

†E-mail addresses: jjgarmel@ull.es, jrossi@dm.uba.ar, josabina@ull.es

Such problem may be considered in some sense as “sublinear” due to the sign of the nonlinearity, controlled by the condition a(x) ≥ 0 in Ω. Accordingly, aside of considering the case a(x) > 0, x ∈ Ω, it is also of much interest to ascertain the effect exerted on the existence and behavior of positive solutions to (1.1) by the vanishing of a(x) somewhere in Ω. Specially, if a ≢ 0 is zero in a whole subdomain of Ω (see, for instance, [3], [20], [4], [7], [8], [18], [6] for this kind of features in the realm of not depending on parameters Dirichlet or Robin boundary conditions). In order to simplify the exposition and to avoid unnecessary technical complications it will be assumed that the set Ω ∩ {a = 0} is either empty, i.e. a(x) > 0 for all x ∈ Ω, or (its interior) constitutes a smooth subdomain Ω_{0} ⊂ Ω. As will be opportunely remarked later (see Remark 9), many other more common possibilities can be handled as variations of such reference situation. Moreover, when searching for weak solutions such requirements on Ω_{0} can be further relaxed (see Theorem 2).

In order to state our main results we are describing with more precision our hypotheses on the vanishing subdomain Ω_{0} ⊂ Ω. If a(x_{0}) = 0 at some x_{0} ∈ Ω it will be assumed that Ω ∩ \{a = 0\} = Ω \ \{a > 0\} = Ω ∩ \overline{Ω}_{0} where Ω_{0} ⊂ Ω is a C^{2,α} subdomain of Ω. As observed later, no essentially new phenomena arise if Ω_{0} consists of several connected pieces (see Remark 11). Being both ∂Ω, ∂Ω_{0} open and compact smooth n − 1 dimensional manifolds, and hence locally connected, they can only exhibit finitely many connected pieces all of them also being pairwise disjoint closed n − 1 dimensional manifolds. Since a ≢ 0 then Ω_{0} ≠ ∅ implies Ω ∩ ∂Ω_{0} ≠ ∅. Again for the sake of brevity it will be assumed in most part of the work the following requirement on ∂Ω_{0}.

(H) “Writing ∂Ω_{0} = Γ_{1} ∪ Γ_{2}, with Γ_{1} = ∂Ω ∩ ∂Ω_{0} and Γ_{2} = Ω ∩ ∂Ω_{0}, Γ_{2} satisfies Γ_{2} ⊂ Ω”.

Notice that Γ_{2} ≠ ∅ whenever Ω_{0} ≠ ∅. Such hypothesis says that those possible components of ∂Ω_{0} touching ∂Ω are “separated” from the ones meeting Ω (which are required in (H) to lie entirely in Ω). This is obviously the case if, for instance, either Ω_{0} = ∅ or Ω_{0} ⊂ \overline{Ω}_{0} ⊂ Ω. Nevertheless, in the part of this work devoted to weak solutions we are also dealing with a more general setting allowing that a component of ∂Ω_{0} simultaneously meets ∂Ω and Ω.

In the λ-regime for the existence of positive solutions to (1.1) the relative position of ∂Ω with respect to the null set Ω_{0} turns out to be crucial. If Ω_{0} is far apart from ∂Ω, i.e. Γ_{1} = ∅ or plainly Ω_{0} = ∅, then positive solutions exist for λ arbitrarily large. On the contrary, such an existence is limited above for λ by a threshold value λ = σ_{1} which is the principal eigenvalue of the mixed Dirichlet-Steklov eigenvalue problem:

\{ \begin{matrix}Δφ = 0 & x ∈ Ω_{0} \\ \frac{∂φ}{∂ν} = σφ & x ∈ Γ_{1} \\ φ = 0 & x ∈ Γ_{2}. \end{matrix}  (1.2)

The required discussion on the existence and properties of the principal eigenvalue σ_{1} for such an unusual problem and other eigenvalue problems will be provided in Section 2. For the moment, we are already in position of stating our main results.

Theorem 1. Assume that a ∈ C^{α}(\overline{Ω}), 0 < α < 1 is nonnegative while either a(x) > 0 for all x ∈ Ω or Ω ∩ \{a = 0\} = Ω ∩ \overline{Ω}_{0} with Ω_{0} a C^{2,α} subdomain of Ω satisfying (H). Then the following properties hold true.

i) Problem (1.1) admits a positive solution if and only if:

0 < λ < σ_{1},  (1.3)

where 0 < σ_{1} < ∞ is the principal eigenvalue of (1.2) provided Γ_{1} is nonempty, σ_{1} = ∞ otherwise. Moreover, such a positive solution is unique.

ii) If u_{λ} stands for the positive solution to (1.1) then u_{λ} ∈ C^{2,α}(\overline{Ω}) for every λ satisfying (1.3), being the mapping λ ↦ u_{λ} increasing and continuous regarded as valued in C^{2,α}(\overline{Ω}). Moreover:

lim_{λ→0+} u_{λ} = 0,

in C^{2,α}(Ω), i.e. u_{λ} bifurcates from u = 0 at λ = 0.

iii) Observing (1.1) as the stationary problem associated to:

\{ \begin{matrix}\frac{∂u}{∂t} = Δu − a(x)u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = λu & x ∈ ∂Ω , \end{matrix}  (1.4)

then u_{λ} is asymptotically stable for all λ satisfying (1.3). Moreover, it is globally attractive for the positive solutions to (1.4).

Remark 1. It should be remarked that a(x) in Theorem 1 is free of vanishing somewhere on ∂Ω (even being identically zero on ∂Ω).

If solutions to (1.1) are permitted being merely weak solutions then the conditions on a(x), Ω and Ω_{0} can be considerably relaxed. Regarding existence, next theorem describes a particular choice for such a more weak setting which will be only used in the context of such result.

Theorem 2. Let Ω ⊂ ℝ^{N} be a Lipschitz bounded domain, a(x) ∈ L^{∞}(Ω) satisfying either a(x) > 0 a.e. in Ω, or a(x) > 0 a. e. in Ω \ Ω_{0}, a(x) = 0 a.e. in Ω_{0}, Ω_{0} a C^{1} subdomain of Ω. If Γ_{1} := ∂Ω ∩ ∂Ω_{0} defines a C^{1} n − 1 dimensional submanifold of ∂Ω with boundary, when nonempty, then problem (1.1) admits a positive weak solution u ∈ W^{1,2}(Ω) ∩ C^{1,β}(Ω) for every 0 < β < 1 provided that 0 < λ < σ_{1}, where σ_{1} stands for the weak principal eigenvalue of (1.2) if Γ_{1} ≠ ∅, σ_{1} = ∞ otherwise.

Remark 2. The meeting region ∂Ω ∩ ∂Ω_{0} if nonempty is required to be a not too small part of ∂Ω (indeed a submanifold with boundary of ∂Ω). Cases of “smaller” intersections, for instance ∂Ω ∩ ∂Ω_{0} a n − 2 submanifold of ∂Ω or even a smaller object) may be handled as a perturbation of the first case. However, that is beyond our objectives in this work.

The behavior of positive solutions u_{λ} to (1.1) described in Theorem 1 is similar to the corresponding solutions to the logistic problem

\{ \begin{matrix}−Δu = λu − a(x)u^{r} & x ∈ Ω \\ u = 0 & x ∈ ∂Ω, \end{matrix}  (1.5)

r > 1, a(x) being as in that theorem and λ_{1}(Ω) < λ < λ_{1}(Ω_{0}) (λ_{1}(Ω), λ_{1}(Ω_{0}) the first Dirichlet eigenvalues in Ω and Ω_{0}, respectively). See [8] for details.

Theorem 1 asserts that positive solutions u_{λ} to (1.1) abruptly cease to exist when λ crosses σ_{1} if such a value is finite. This raises the question on what kind of singularization undergoes u_{λ} when λ → σ_{1} and where does it occur. Next result gives a full answer to this question (check [8], [18] for the corresponding case in the logistic problem (1.5)).

Theorem 3. Under the hypotheses of Theorem 1 assume in addition that a ∈ C^{1}(\overline{Ω}). Set Ω^{+} := Ω ∩ {a > 0} and suppose that Γ_{1} ≠ ∅ and so σ_{1} < ∞. The profile of the positive solution u_{λ} to (1.1) as λ ↗ σ_{1} is then described in the following terms.

i) Firstly,

lim_{λ↗σ_{1}} u_{λ}(x) = ∞,

uniformly in \overline{Ω}_{0}.

ii) If Γ^{+} := ∂Ω^{+} ∩ ∂Ω ≠ ∅. Then,

u_{λ}(x) → z_{M,Ω^{+}}(x),

as λ ↗ σ_{1} in C^{2,α}(Ω^{+} ∪ Γ^{+}), where z_{M,Ω^{+}} ∈ C^{2,α}(Ω^{+} ∪ Γ^{+}) stands for the minimum solution to the singular mixed boundary value problem:

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω^{+} \\ u = ∞, & x ∈ ∂Ω^{+} ∩ Ω \\ \frac{∂u}{∂ν} = σ_{1}u, & x ∈ Γ^{+}. \end{matrix}  (1.6)

iii) If, on the contrary, ∂Ω^{+} ⊂ Ω then

u_{λ}(x) → z_{D,Ω^{+}}(x),

as λ ↗ σ_{1} in C^{2,α}(Ω^{+}), where z_{D,Ω^{+}}(x) ∈ C^{2,α}(Ω^{+}) is the minimum solution of the singular Dirichlet problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω^{+} \\ u = ∞, & x ∈ ∂Ω^{+}. \end{matrix}  (1.7)

Remark 3. Observe that from (H), Ω = Ω_{0} ∪ Γ_{2} ∪ Ω^{+} and so ∂Ω^{+} = Γ^{+} ∪ Γ_{2}, all the sets involved being pair-wise disjoint.

As a counterpart of Theorem 3 the following result describes the asymptotic profile of u_{λ} if σ_{1} = ∞.

Theorem 4. Suppose that σ_{1} = ∞ in Theorem 1. Then,

u_{λ}(x) → z_{D}(x),

as λ → ∞ in C^{2,α}(Ω) where z_{D}(x) ∈ C^{2,α}(Ω) is now the minimum solution to the problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ u = ∞, & x ∈ ∂Ω. \end{matrix}  (1.8)

Moreover:

lim inf_{λ→∞} (λ^{−2/(p−1)} sup_{x∈Ω} u_{λ}(x)) ≥ |a|_{∞,Ω}^{−1/(p−1)}.  (1.9)

Remark 4. It can be shown that the minimum solution to (1.8) coincides with the corresponding minimum solution of the singular Neumann problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω, \\ \frac{∂u}{∂ν} = ∞, & x ∈ ∂Ω. \end{matrix}

On the other hand, the uniqueness of a positive solution to the singular problem (1.6) holds if a(x) suitably decays to zero at Γ_{2} = ∂Ω^{+} \ ∂Ω. Notice that a must be identically zero there. The uniqueness for (1.8) holds even under less restrictive assumptions on the behavior of a at ∂Ω. See [9], [19] for further details and references.

This paper is organized as follows. Section 2 contains a detailed analysis of some auxiliary special eigenvalue problems. Existence and uniqueness results are treated in Section 3 while the discussion of asymptotic profiles of the positive solutions as λ → σ_{1} are considered in Section 4.

2. Preliminaries

In the present section two different kinds of –to some extent– non standard eigenvalue problems are considered. The first one –treated in our next result– was studied in detail in [15] (see Section 2) and the material required for the present work is thus collected in a single statement. An outline of its proof is enclosed for later use and also in order to properly highlight the smoothness properties of the relevant eigenfunctions.

Proposition 5. Let Ω ⊂ ℝ^{N} be a smooth C^{2,α} domain, 0 < α < 1, and consider the eigenvalue problem:

\{ \begin{matrix}Δϕ = μϕ & x ∈ Ω \\ \frac{∂ϕ}{∂ν} = λϕ & x ∈ ∂Ω, \end{matrix}  (2.1)

where μ stands for the eigenvalue while λ ∈ ℝ has the “status” of a parameter. Then, the following properties hold. i) For every λ ∈ ℝ, (2.1) admits a unique principal eigenvalue μ_{1}, i. e. an eigenvalue

associated to a one-signed eigenfunction, which is simple with any associated eigenfunction ϕ_{1} ∈ C^{2,α}(\overline{Ω}).

ii) As a function of λ ∈ ℝ, μ_{1} is smooth and increasing with μ_{1} = 0 at λ = 0 and μ_{1} → ∞

as λ → ∞. Moreover:

lim inf_{λ→∞} \frac{μ_{1}}{λ^{2}} ≥ 1.  (2.2)

Remark 5. Problem (2.1) becomes non standard when λ > 0 which is just the case of interest for the present work (with λ < 0, λ = 0 (2.1) is the Robin, Neumann eigenvalue problem, respectively).

Sketch of the proof: We are only dealing with the anomalous sign λ > 0. Define M = {u ∈ H^{1}(Ω) : ∫_{Ω} u^{2} = 1} and consider J : M → ℝ defined as:

J(u) = ∫_{Ω} |∇u|^{2} − λ ∫_{∂Ω} u^{2}.

The functional J is sequentially weakly lower semicontinuous. It is in addition coercive in M. In fact, by using the equivalent norm |u|_{H^{1}(Ω)}^{2} = ∫_{Ω} |∇u|^{2} + ∫_{∂Ω} u^{2}, the existence of a sequence u_{n} ∈ M, |u_{n}|_{H^{1}(Ω)} → ∞ with J(u_{n}) ≤ K for some K > 0 implies that ∫_{∂Ω} u_{n}^{2} → ∞. Then, after picking a subsequence if necessary, v_{n} := u_{n}/|u_{n}|_{L^{2}(∂Ω)} weakly converges to some v ∈ H^{1}(Ω) being v_{n} → v both in L^{2}(Ω) and L^{2}(∂Ω). That is not possible since |v_{n}|_{L^{2}(Ω)} = |u_{n}|_{L^{2}(∂Ω)}^{−1} → 0 while v ≠ 0. Thus, J is coercive and reaches an infimum in M ([22]) which must be negative. Setting −μ_{1} = inf_{M} J then:

−μ_{1} = inf_{u∈H^{1}(Ω)\\{0\}} \frac{∫_{Ω} |∇u|^{2} − λ ∫_{∂Ω} u^{2}}{∫_{Ω} u^{2}} := inf_{u∈H^{1}(Ω)\\{0\}} Q(u),

and any ϕ ∈ H^{1}(Ω) satisfying Q(ϕ) = −μ_{1} defines a weak eigenfunction to (2.1) associated to μ_{1}, i.e.,

∫_{Ω} ∇ϕ∇v + μ_{1} ∫_{Ω} ϕv = λ ∫_{∂Ω} ϕv,  (2.3)

for all v ∈ H^{1}(Ω). To show that ϕ ∈ C^{2,α}(\overline{Ω}) consider the auxiliary problem:

\{ \begin{matrix}Δv − Mv = f & x ∈ Ω \\ \frac{∂v}{∂ν} − λv = 0 & x ∈ ∂Ω, \end{matrix}  (2.4)

with M > 0, f := (μ_{1} − M)ϕ. Problem (2.4) can be transformed into a C^{α} coefficients Neumann problem by setting w = e^{λh(x)}v, h being any C^{2,α} extension to \overline{Ω} of dist (x, ∂Ω) a bit further away from ∂Ω. As a first consequence, ϕ ∈ H^{2}(Ω). On the other hand, it follows from [1] that (2.4) becomes uniquely solvable in W^{2,p}(Ω) for any f ∈ L^{p}(Ω) and p ≥ 1 if M is selected conveniently large. Bootstrapping in ϕ we get ϕ ∈ W^{2,p}(Ω) for all p ≥ 1, hence ϕ ∈ C^{1,β}(\overline{Ω}) for all 0 < β < 1. Being (2.4) also uniquely solvable in C^{2,α}(\overline{Ω}) for M large (see [1], [12]) we finally get the searched regularity ϕ ∈ C^{2,α}(\overline{Ω}).

That any eigenfunction ϕ to μ_{1} must be one-signed follows from the fact that, say ϕ^{+} is also an eigenfunction provided ϕ^{+} ≢ 0 since Q(ϕ^{+}) = −μ_{1}. By the maximum principle ϕ^{+} > 0 in \overline{Ω}, hence ϕ^{−} = 0 and ϕ is positive.

Check [15] for a detailed account of the remaining and other additional interesting properties of μ_{1}.

Remark 6. As a consequence of the estimate (2.2) some rough information on the behavior of the –normalized in some way– principal eigenfunction ϕ_{1,λ} > 0 as λ → ∞ can be given. In fact, observe that from the maximum principle ϕ_{1,λ}(x) < sup_{∂Ω} ϕ_{1,λ} for all x ∈ Ω. Assume that ϕ_{1,λ} has been normalized with sup_{∂Ω} ϕ_{1,λ} = 1. Then by integrating in (2.1) we arrive at:

0 < ∫_{Ω} ϕ_{1,λ} < \frac{λ}{μ_{1}} |∂Ω|.

Thus from (2.2) it follows that ϕ_{1,λ} → 0 in L^{1}(Ω) as λ → ∞. A more detailed account of this convergence will be given later in Section 4 (cf. Remark 10).

In our next result an eigenvalue problem similar than (1.2) is considered. In fact a Schrödinger potential term q(x) is included for later use in Section 4 (see the proof of Theorem 3).

Theorem 6. Let Ω_{0} ⊂ ℝ^{N} be a class C^{2,α} bounded domain whose boundary splits up in two sets Γ_{1}, Γ_{2} of connected pieces (thus, each of them being a closed compact n − 1 dimensional manifold) while the potential q(x) ∈ C^{α}(\overline{Ω}_{0}). Then, the eigenvalue problem,

\{ \begin{matrix}Δφ − q(x)φ = 0 & x ∈ Ω_{0} \\ \frac{∂φ}{∂ν} = σφ & x ∈ Γ_{1} \\ φ = 0 & x ∈ Γ_{2} \end{matrix}  (2.5)

admits a principal eigenvalue, i.e. an eigenvalue with a single sign eigenfunction, if and only if

λ_{1}(q) > 0,  (2.6)

where λ_{1}(q) stands for the principal Dirichlet eigenvalue of the operator −Δ + q(x) in the domain Ω_{0}. Moreover, such an eigenvalue σ_{1}(q) is the unique principal eigenvalue, simple and the smaller among all possible eigenvalues. In addition, σ_{1}(q) is increasing with respect to q. Finally, any associated eigenfunction φ_{1} belongs to C^{2,α}(\overline{Ω}_{0}).

Remark 7. As a consequence of Theorem 6 observe that the eigenvalue problem (1.2) admits a principal eigenvalue σ_{1} (σ_{1} will be used with the meaning of σ_{1}(q)_{|q=0}) since λ_{1}(q)_{|q=0}, the first Dirichlet eigenvalue of −Δ in Ω_{0}, is positive. Furthermore, σ_{1} is positive. In fact, if φ_{1} is a positive associated eigenfunction it follows from the maximum principle that φ_{1} > 0 on Γ_{1} while direct integration gives:

σ_{1} = \frac{∫_{Ω_{0}} |∇φ_{1}|^{2}}{∫_{Γ_{1}} φ_{1}^{2}} .

The same holds true for σ_{1}(q) if either q(x) ≥ 0 in Ω_{0} or inf_{Ω_{0}} q is not too large (in absolute value) in case of being negative.

Proof of Theorem 6. We begin by noticing that the existence of a principal eigenvalue σ with associated positive eigenfunction φ provides a positive strict supersolution to the equation,

−Δu + qu = 0,

in Ω_{0} since φ must be positive on Γ_{1}. This is well known to imply that λ_{1}(q) > 0 (see [17]).

Let us show now the sufficiency of (2.6) to ensure the existence of a principal eigenvalue. Define now H = H_{Γ_{2}}(Ω_{0}) = {u ∈ H^{1}(Ω_{0}) : u_{|Γ_{2}} = 0} equipped with the norm |u|_{H}^{2} = ∫_{Ω_{0}} |∇u|^{2} + ∫_{Γ_{2}} u^{2} = ∫_{Ω_{0}} |∇u|^{2}, which is equivalent to the standard one in H^{1}(Ω), M = {u ∈ H : ∫_{Γ_{1}} u^{2} = 1} being J : M → ℝ defined as

J(u) = ∫_{Ω_{0}} |∇u|^{2} + ∫_{Ω_{0}} qu^{2}.

We are next proving that (2.6) implies the coercivity of J. Assume on the contrary that u_{n} ∈ M exists such that |u_{n}|_{H} → ∞ with J(u_{n}) ≤ C for some constant C. Since q is bounded this implies that t_{n} := |u_{n}|_{L^{2}(Ω_{0})} → ∞. Setting v_{n} = u_{n}/t_{n} we find that v_{n} is bounded in H^{1}(Ω_{0}) then v_{n} ⇀ v weakly in H^{1}(Ω_{0}) and strongly in L^{2}(Ω_{0}) and L^{2}(Γ_{1}). Observe that v = 0 in Γ_{1} (sense of traces) and hence v ∈ H_{0}^{1}(Ω_{0}) while |v|_{L^{2}(Ω_{0})} = 1. Finally, from the boundedness of J(u_{n}) we obtain,

λ_{1}(q) ∫_{Ω_{0}} v^{2} ≤ ∫_{Ω_{0}} |∇v|^{2} + ∫_{Ω_{0}} qv^{2} ≤ 0,

and this implies v = 0 which is impossible. Thus, J is coercive in M.

On the other hand, since J is also sequentially weakly lower semicontinuous, J attains a minimum value σ_{1} in M ([22]),

σ_{1} = inf_{u∈H\\{u|_{Γ_{1}}≡0\}} \frac{∫_{Ω_{0}} |∇u|^{2} + ∫_{Ω_{0}} qu^{2}}{∫_{Γ_{1}} u^{2}} .  (2.7)

Such variational representation simultaneously ensures that σ_{1} is an eigenvalue, and the smaller among all other possible eigenvalues σ to (2.5). The discussion of both the smoothness and the one sign character of any associated eigenfunction φ_{1} is the same as in Proposition 5. In fact, observe that the analysis in [1] of smoothness up to the boundary can be separately performed on each Γ_{i}, i = 1, 2. On the other hand, the simplicity of σ_{1} follows from the fact that two possible independent eigenfunctions lead to two L^{2} orthogonal eigenfunctions what in this case is impossible (this also explains why σ_{1} is the unique principal eigenvalue). Finally, the monotonicity of σ_{1}(q) with respect q follows from the variational representation (2.7).

Remark 8. The proof of existence of a unique principal eigenvalue to problem (2.5) can be extended without changes to more general domains Ω_{0}. For instance, Ω_{0} a Lipschitz bounded domain where Γ_{1} ⊂ ∂Ω is a C^{1} n − 1 dimensional compact manifold with boundary and Γ_{2} := ∂Ω\Γ_{1} nonempty. Notice that Γ_{1} and Γ_{2} meet at the n−2 dimensional boundary of Γ_{1}.

In our next result, the dependence of the eigenvalue problem (2.5) with respect to certain perturbations of the domain Ω_{0} (Ω_{0} under the hypotheses of Theorem 6) and the potential q is considered. The perturbed domains are,

Ω_{0,t} = Ω_{0} ∪ {x ∈ ℝ^{N} : dist(x, Γ_{2}) < |t|},

0 < |t| < ε being the perturbation parameter, ε > 0 small. Notice that Ω_{0} is deformed an amount |t| in the outward normal field only with respect the boundary Γ_{2}. Remark also that ∂Ω_{0,t} = Γ_{1} ∪ Γ_{2,t}. The perturbed version of (2.5) under consideration is given by

\{ \begin{matrix}Δφ = q(t)φ & x ∈ Ω_{0,t} \\ \frac{∂φ}{∂ν} = σφ & x ∈ Γ_{1} \\ φ = 0 & x ∈ Γ_{2,t}, \end{matrix}  (2.8)

where |t| < ε, q ∈ C^{1}(−ε, ε), q(0) = 0. As observed in Remark 7, problem (2.8) admits a principal eigenvalue σ_{1,t}. By means of the holomorphic families of type (A) approach (see [13]) and arguing as in [15, 18] –coefficient q in addition real analytic in this case– or by the direct variational approach used in [11] it can be shown that σ_{1,t} is differentiable with respect to t. More importantly, an expression for its derivative can be produced and is enclosed in the next statement. For the sake of brevity its proof is omitted. Nevertheless it can be reconstructed following the course of the one given in either [11] or [18].

Proposition 7. Under the conditions of Theorem 6 let σ_{1} be the principal eigenvalue of (1.2) with associated eigenfunction φ_{1}. Then the derivative of the principal eigenvalue σ_{1,t} to the perturbed problem (2.8) at t = 0 is given by the expression

\frac{d}{dt}(σ_{1,t})_{|t=0} = \frac{q_{1} ∫_{Ω_{0}} φ_{1}^{2} − ∫_{Γ_{2}} |∇φ_{1}|^{2}}{∫_{Γ_{1}} φ_{1}^{2}} ,  (2.9)

where q_{1} = \frac{dq}{dt}_{|t=0}.

3. Existence and uniqueness results

This section is devoted to the proof of our results concerning existence and uniqueness of positive solutions to the problem (1.1).

3.1. Proof of Theorem 2

Proof of Theorem 2: Firstly, we are introducing the concept of weak solution to be used. It is said that u ∈ H^{1}(Ω) is a weak solution of (1.1) provided:

∫_{Ω} a|u|^{p+1} < ∞,

together with the usual relation:

∫_{Ω} ∇u∇v + ∫_{Ω} a|u|^{p−1}uv = λ ∫_{∂Ω} uv,  (3.1)

for all v ∈ C^{1}(\overline{Ω}). In the case u ≥ 0 a.e. in Ω, the test function v may be supposed to belong to H^{1}(Ω) by a standard approximation argument. In particular, notice that positive weak solutions u ∈ H^{1}(Ω) are only possible provided λ > 0 since from (3.1) with v = u:

∫_{Ω} |∇u|^{2} + ∫_{Ω} au^{p+1} = λ ∫_{∂Ω} u^{2}.

Let us consider in H^{1}(Ω) the equivalent norm |u|_{H^{1}(Ω)} = (∫_{Ω} |∇u|^{2} + ∫_{∂Ω} |u|^{2})^{1/2} and introduce the functional,

Φ(u) = \frac{1}{2} ∫_{Ω} |∇u|^{2} + \frac{1}{p + 1} ∫_{Ω} a(x)|u|^{p+1} − \frac{λ}{2} ∫_{∂Ω} |u|^{2},

which may be infinity at some u ∈ H^{1}(Ω). We will find a weak solution of (1.1) by showing that Φ reaches its minimum in H^{1}(Ω).

As a preliminary remark notice that for a large constant K we have,

Φ(K) = \frac{K^{p+1}}{p + 1} ∫_{Ω} a(x) − \frac{λ|∂Ω|}{2}K^{2} < 0,

and a possible minimum u of Φ will satisfy Φ(u) < 0, being such minimum nontrivial. Replacing u by |u|, we may assume in addition that u is nonnegative.

To see that there exists a minimum of Φ in H^{1}(Ω) let us prove that Φ is coercive and weakly lower semicontinuous (cf. [3], [22]). Let us first check that Φ is coercive. Proceeding by contradiction, assume that there exists a sequence u_{n} ∈ H^{1}(Ω) such that

|u_{n}|_{H^{1}(Ω)} → ∞ and Φ(u_{n}) ≤ C.

Hence

\frac{1}{2} ∫_{Ω} |∇u_{n}|^{2} + \frac{1}{p + 1} ∫_{Ω} a(x)|u_{n}|^{p+1} ≤ C + \frac{λ}{2} ∫_{∂Ω} |u_{n}|^{2},  (3.2)

and therefore, since |u_{n}|_{H^{1}(Ω)} → ∞ we have that

∫_{∂Ω} |u_{n}|^{2} → ∞.

Normalizing according to v_{n} = u_{n}/|u_{n}|_{L^{2}(∂Ω)} it follows from (3.2)

\frac{1}{2} ∫_{Ω} |∇v_{n}|^{2} + \frac{1}{(p + 1)(∫_{∂Ω} |u_{n}|^{2})^{1−p}} ∫_{Ω} a(x)|v_{n}|^{p+1} ≤ \frac{C}{∫_{∂Ω} |u_{n}|^{2}} + \frac{λ}{2}.  (3.3)

Therefore v_{n} is a bounded sequence in H^{1}(Ω) and thus there exists a subsequence (that we will still call v_{n}) such that v_{n} ⇀ v weakly in H^{1}(Ω) and v_{n} → v strongly in L^{2}(∂Ω). Hence |v|_{L^{2}(∂Ω)} = 1. On the other hand, it follows from (3.3) that

\frac{1}{(p + 1)(∫_{∂Ω} |u_{n}|^{2})^{1−p}} ∫_{Ω} a(x)|v_{n}|^{p+1} ≤ \frac{C}{∫_{∂Ω} |u_{n}|^{2}} + \frac{λ}{2}.

Therefore

\frac{1}{(p + 1)} ∫_{Ω} a(x)|v_{n}|^{p+1} ≤ C (∫_{∂Ω} |u_{n}|^{2})^{1−p} → 0, n → ∞.

Hence

∫_{Ω} a(x)|v|^{p+1} = 0,

and we conclude that v ≡ 0 in Ω^{+} = {a > 0}. Notice that in the case σ_{1} = +∞ this implies v = 0 on ∂Ω, which is not possible since |v|_{L^{2}(∂Ω)} = 1.

If, on the contrary, σ_{1} < +∞, we get, using (3.3) again, that

\frac{1}{2} ∫_{Ω} |∇v|^{2} ≤ \frac{λ}{2},

with |v|_{L^{2}(∂Ω)} = 1 and v = 0 on Γ_{2}, and taking in account (2.7) that is not compatible with the assumption λ < σ_{1}.

Finally, let us see that Φ is weakly lower semicontinuous. Assume that u_{n} ⇀ u weakly in H^{1}(Ω). By standard compactness results we have that u_{n} → u both strongly in L^{2}(∂Ω) and L^{2}(Ω). Thus to show that

Φ(u) ≤ lim inf_{n→∞} Φ(u_{n}),

it is only necessary to prove that

∫_{Ω} a|u|^{p+1} ≤ lim inf_{n→∞} ∫_{Ω} a|u|_{n}^{p+1}.  (3.4)

However, Fatou’s lemma implies that from any subsequence of u_{n} a new one can be selected so that (3.4) holds. Thus the relation holds true for the whole sequence.

We have just shown that Φ achieves an absolute minimum at some nonnegative ũ ∈ H^{1}(Ω). It can be checked now that Φ can be differentiated at any u ∈ H^{1}(Ω), Φ(u) < ∞, in any direction v ∈ C^{1}(\overline{Ω}) with:

\frac{d}{dt}(Φ(u + tv))_{t=0} = ∫_{Ω} ∇u∇v + ∫_{Ω} a|u|^{p}uv − λ ∫_{∂Ω} uv.  (3.5)

This means that ũ defines a weak solution to (1.1).

To conclude the proof observe that any weak nonnegative solution to (1.1) satisfies −Δu ≤ 0 weakly in Ω. Thus u ∈ L_{loc}^{∞}(Ω) (see Chapter 8 in [12]) what provides by bootstrap u ∈ C^{1,β}(Ω), 0 < β < 1.

3.2. Proof of Theorem 1

We are next providing a proof of Theorem 1. In the following lemma a proof of uniqueness for classical solutions is given. For later use we are dealing with slightly more general problems than (1.1). After that we will find the range for existence in a second lemma to finally proceed with the main course of the proof.

Lemma 8. Suppose that Ω ⊂ ℝ^{N} is a bounded C^{2,α} domain such that ∂Ω splits in two disjoint groups of connected pieces Γ_{N} and Γ_{D}. Consider the problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = λu & x ∈ Γ_{N} \\ u = g(x) & x ∈ Γ_{D}, \end{matrix}

and let u_{1}, u_{2} ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}) be two classical nonnegative and nontrivial solutions. Then u_{1} = u_{2}.

Proof: It can be shown by means of Hopf’s maximum principle that any nonnegative classical solution u ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}), u ≢ 0, satisfies inf_{Ω} u > 0.

We next use ideas from [3]. Assume that u_{1}, u_{2} are positive solutions and consider the relation,

∫_{Ω} (−\frac{Δu_{1}}{u_{1}} + \frac{Δu_{2}}{u_{2}} ) (u_{1}^{2} − u_{2}^{2}) = − ∫_{Ω} a(x)(u_{1}^{p−1} − u_{2}^{p−1})(u_{1}^{2} − u_{2}^{2}) ≤ 0.  (3.6)

Integrating by parts we get,

− ∫_{Ω} Δu_{1} \frac{(u_{1}^{2} − u_{2}^{2})}{u_{1}} = − ∫_{Γ_{N}} \frac{∂u_{1}}{∂ν} \frac{(u_{1}^{2} − u_{2}^{2})}{u_{1}} + ∫_{Ω} ∇u_{1}∇ (\frac{u_{1}^{2} − u_{2}^{2}}{u_{1}} )

= −λ ∫_{Γ_{N}} (u_{1}^{2} − u_{2}^{2}) + ∫_{Ω} ∇u_{1}∇ (\frac{u_{1}^{2} − u_{2}^{2}}{u_{1}} ) .

Analogously

− ∫_{Ω} Δu_{2} \frac{(u_{1}^{2} − u_{2}^{2})}{u_{2}} = −λ ∫_{Γ_{N}} (u_{1}^{2} − u_{2}^{2}) + ∫_{Ω} ∇u_{2}∇ (\frac{u_{1}^{2} − u_{2}^{2}}{u_{2}} ) .

Thus,

∫_{Ω} (−\frac{Δu_{1}}{u_{1}} + \frac{Δu_{2}}{u_{2}} ) (u_{1}^{2} − u_{2}^{2}) = ∫_{Ω} |∇u_{1} − \frac{u_{1}}{u_{2}} ∇u_{2}|^{2} + |∇u_{2} − \frac{u_{2}}{u_{1}} ∇u_{1}|^{2}

= ∫_{Ω} u_{1}^{2} |∇ (\frac{u_{2}}{u_{1}} )|^{2} + u_{2}^{2} |∇ (\frac{u_{1}}{u_{2}} )|^{2} .

In conclusion, both integrals in (3.6) vanish. From the first one we get u_{2} = cu_{1} for some constant c, while the second implies u_{1} = u_{2} in {a > 0}. Therefore u_{1} = u_{2}.

Lemma 9. Positive classical solutions to (1.1) are only possible if λ satisfies (1.3):

0 < λ < σ_{1}.

Proof: If u is a positive solution of (1.1), direct integration gives:

∫_{Ω} au^{p} = λ ∫_{∂Ω} u.

As already shown, u is bounded away from zero in \overline{Ω} and hence λ > 0.

As for the complementary estimate assume again that u is a positive solution. From Green’s identity:

∫_{∂Ω_{0}} φ_{1} \frac{∂u}{∂ν} − u\frac{∂φ_{1}}{∂ν} = 0,

where φ_{1} is a principal positive eigenfunction to (1.2). Thus,

(λ − σ_{1}) ∫_{Γ_{1}} uφ_{1} = ∫_{Γ_{2}} u\frac{∂φ_{1}}{∂ν} .

As

\frac{∂φ_{1}}{∂ν} (x) < 0, x ∈ Γ_{2},

we then obtain that necessarily λ < σ_{1}.

Proof of Theorem 1. To obtain a classical positive solution we are employing the method of sub and supersolutions (cf. [2]). A nonnegative \underline{u} ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}) is said to be a subsolution of (1.1) if

Δ\underline{u} ≥ a(x)\underline{u}^{p} x ∈ Ω,

together with \frac{∂\underline{u}}{∂ν} ≤ λ\underline{u} on ∂Ω (a supersolution is defined by reversing the inequalities).

For every λ in the existence range (1.3) if ϕ_{1} is the positive eigenfunction associated to the principal eigenvalue μ_{1} = μ_{1}(λ) of (2.1), normalized as sup_{Ω} ϕ_{1} = 1, then it can be checked that

\underline{u} := Aϕ_{1},

defines a subsolution of (1.1) provided,

0 < A ≤ (\frac{μ_{1}}{|a|_{∞,Ω}} )^{1/(p−1)} ,  (3.7)

what means A not too large. Finding a comparable supersolution is however a more subtle task. We are considering by turn the cases a > 0 in ∂Ω and a = 0 at some parts of ∂Ω (possibly at the whole ∂Ω).

Under the first assumption, if a > 0 in \overline{Ω} (in particular Ω_{0} = ∅),

\overline{u} := Bϕ_{1},

is readily seen to be a supersolution if B is large enough as to satisfy:

B ≥ (\frac{μ_{1}}{inf_{Ω} a (inf_{Ω} ϕ_{1})^{p−1}} )^{1/(p−1)} .

Recall that inf_{Ω} ϕ_{1} > 0. Thus assume a > 0 on ∂Ω but Ω_{0} ≠ ∅. To construct a supersolution an approach from [16] is used. Let δ > 0 be chosen so that B(Ω_{0}, δ) := {x ∈ Ω : dist(x, Ω_{0}) < δ} ⊂⊂ Ω (B_{δ} for short). Let ψ_{1} be the principal positive eigenfunction of the Dirichlet problem:

\{ \begin{matrix}−Δψ = σψ & x ∈ B_{δ} \\ ψ = 0 & x ∈ ∂B_{δ}, \end{matrix}

associated to the principal eigenvalue σ = σ_{1}(B_{δ}), satisfying sup_{B_{δ}} ψ_{1} = 1 (recall that σ_{1}(B_{δ}) > 0). The restriction of ψ_{1} to \overline{B(Ω_{0}, δ/2)} being positive can be extended to the whole of \overline{Ω} as a positive C^{2,α} function ψ̂_{1} such that,

\frac{∂ψ̂_{1}}{∂ν} = λψ̂_{1},

on ∂Ω. For instance, it suffices that ψ̂_{1} = ce^{λd(x)} near ∂Ω with d(x) = dist (x, ∂Ω) and c a positive constant. Then,

\overline{u} := Bψ̂_{1},

defines a C^{2,α} supersolution to (1.1) if B > 0 is so large as:

\{\frac{sup_{D_{δ}} |Δψ̂_{1}|}{inf_{D_{δ}} a \{inf_{D_{δ}} ψ̂_{1}\}^{p}} \}^{1/(p−1)} ≤ B,

being D_{δ} = \overline{Ω} \ B(Ω_{0}, δ/2).

As a final conclusion, in the case a > 0 at ∂Ω a pair \underline{u}, \overline{u} of a sub and a supersolution can be constructed so that \underline{u} ≤ \overline{u}, hence a positive solution u ∈ C^{2,α}(\overline{Ω}) exists in the functional interval [\underline{u}, \overline{u}] (cf. [2]).

Let us deal now with the case a = 0 at some points of ∂Ω. Under hypothesis (H) we will first consider the very particular case a(x) = 0 for all x ∈ ∂Ω. To construct a classical supersolution \overline{u} consider,

Ω_{δ} := [B(Ω_{0}, δ) ∪ {dist (x, ∂Ω) < δ}] ∩ Ω
= {x ∈ Ω : dist (x, Ω_{0}) < δ or dist (x, ∂Ω) < δ}.

Observe that ∂Ω_{δ} = ∂Ω ∪ [∂Ω_{δ} ∩ Ω] and that Ω_{δ} approaches Ω_{0} and ∂Ω_{δ} ∩ Ω approaches (∂Ω \ Γ_{1}) ∪ Γ_{2} as δ → 0+ (although Ω_{δ} may not be connected, the results in Section 2 concerning the eigenvalue problems still hold true). The auxiliary problem:

\{ \begin{matrix}Δφ = 0 & x ∈ Ω_{δ} \\ \frac{∂φ}{∂ν} = σφ & x ∈ ∂Ω \\ φ = 0 & x ∈ ∂Ω_{δ} ∩ Ω, \end{matrix}

admits (Theorem 6) a principal eigenvalue σ_{1,δ} with a positive associated eigenfunction φ_{1} (its dependence on δ being omitted at this moment). We claim that σ_{1,δ} ≤ σ_{1} while in addition lim_{δ↘0} σ_{1,δ} = σ_{1}. Assumed this fact, if λ < σ_{1} a construction similar as the previous one can be performed with the positive eigenfunction φ_{1} associated to σ_{1,δ}, provided δ > 0 is small as to verify,

λ < σ_{1,δ} ≤ σ_{1}.

Namely, to extend its restriction to Ω_{δ/2} to the whole \overline{Ω} as a positive smooth function φ̂_{1}. Remark that now we define φ̂_{1} away from ∂Ω and have no interference with the boundary condition in (1.1). The desired supersolution is thus provided by,

\overline{u} := Bφ̂_{1},

with B > 0 large enough. Once more we get the existence of a positive solution to (1.1) in this case.

For the general case of a introduce a_{δ}(x) = η(x)a(x) where η is smooth, supp η = {dist (x, ∂Ω) ≥ δ}, η = 1 in {dist (x, ∂Ω) ≥ 2δ}, 0 ≤ η ≤ 1. Assuming δ > 0, the problem

\{ \begin{matrix}Δu = a_{δ}(x)u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = λu & x ∈ ∂Ω , \end{matrix}

as just seen before admits a positive solution u_{λ,δ} if λ < σ_{1,δ} < σ_{1} and defines in turn a supersolution to (1.1). It can be also enlarged by multiplying by a constant, i.e.,

\overline{u} = Bu_{λ,δ},

with B > 0. This concludes the proof of existence.

Let us come back to show the claim regarding σ_{1,δ}. To fix notation set Γ_{δ} = ∂Ω_{δ} ∩ Ω = {x ∈ Ω : dist (x, ∂Ω) = δ or dist (x, Ω_{0}) = δ}, H_{Γ_{δ}}^{1} = {u ∈ H^{1}(Ω_{δ}) : u_{|Γ_{δ}} = 0}. Then, if for u ∈ H_{Γ_{δ}}^{1} and δ > 0 we define,

Q_{δ}(u) = \frac{∫_{Ω_{δ}} |∇u|^{2}}{∫_{∂Ω} u^{2}} ,

whereas, for u ∈ H_{Γ_{2}}^{1}

Q_{0}(u) = \frac{∫_{Ω_{0}} |∇u|^{2}}{∫_{Γ_{1}} u^{2}} ,

then, for every δ ≥ 0 small

σ_{1,δ} = inf_{u∈H_{Γ_{δ}}^{1} \{0}} Q_{δ}(u),

wherein we understand Γ_{δ} = Γ_{2} and σ_{1,δ} = σ_{1} when δ = 0. Select now any eigenfunction associated to σ_{1}, φ_{1} ∈ H_{Γ_{2}}^{1}(Ω_{0}). Its extension φ̄_{1} as zero to Ω_{δ} belongs to H_{Γ_{δ}}^{1}(Ω_{δ}) while:

σ_{1} = Q_{0}(φ_{1}) = Q_{δ}(φ̄_{1}) ≥ σ_{1,δ}.

If, similarly φ_{1,δ_{i}} are eigenfunctions associated to σ_{1,δ_{i}}, i = 1, 2, 0 < δ_{1} < δ_{2}, φ_{1,δ_{1}} extended as zero to Ω_{δ_{2}} (φ̄_{1,δ_{1}} the extension) lies in H_{Γ_{δ_{2}}}^{1} and,

σ_{1,δ_{1}} = Q_{δ_{1}}(φ_{1,δ_{1}}) = Q_{δ_{2}}(φ̄_{1,δ_{1}}) ≥ σ_{δ_{2}}.

In conclusion σ_{1,δ} does not decrease as δ ↘ 0 and σ_{1,δ} ≤ σ_{1}.

We are next showing that lim inf_{δ↘0} σ_{1,δ} = σ_{1} (i.e., that σ_{1,δ} → σ_{1} as δ ↘ 0). In fact, for each δ > 0 small normalize φ_{1,δ} so that ∫_{∂Ω} φ_{1,δ}^{2} = 1, designate as φ̄_{1,δ} ∈ H^{1}(Ω) its extension by zero to Ω and use |u|_{1} = (∫_{Ω} |∇u|^{2} + ∫_{∂Ω} u^{2})^{1/2} as a equivalent norm for H^{1}(Ω). Since,

|φ_{1,δ}|_{1}^{2} = 1 + σ_{1,δ} ≤ 1 + σ_{1},

φ̄_{1,δ} ⇀ φ̄ weakly in H^{1}(Ω) as δ ↘ 0, hence φ̄_{1,δ} → φ̄ in L^{2}(Ω), L^{2}(∂Ω) and L^{2}(Γ_{2}). As a first conclusion, φ̄ = 0 a. e. in Ω \ Ω_{0} and thus φ̄ = 0 both on Γ_{2} and ∂Ω \ Γ_{1} in the sense of traces. In particular,

∫_{Γ_{1}} φ̄^{2} = 1.

On the other hand,

σ_{1} ≤ ∫_{Ω_{0}} |∇φ̄|^{2} = ∫_{Ω} |∇φ̄|^{2} ≤ lim inf_{δ→0} ∫_{Ω} |∇φ̄_{1,δ}|^{2} = lim inf_{δ→0} σ_{1,δ} ≤ σ_{1}.

This implies both that φ̄ as observed in Ω_{0} defines an eigenfunction associated to σ_{1} and the desired limit. The proof of the claim is concluded.

We are next showing the remaining parts of Theorem 1. As for ii) observe that problem (1.1)_{λ} always admits subsolutions as smaller as desired (see (3.7)) while the positive solution u_{λ_{1}} to (1.1) with λ_{1} replacing λ defines a strict supersolution to (1.1)_{λ} if λ_{1} > λ. Thus,

u_{λ}(x) < u_{λ_{1}}(x) x ∈ Ω,

for λ < λ_{1}. To achieve the continuity of u_{λ} in λ assume that λ_{n} → λ, λ > 0, and set \underline{λ} = inf λ_{n} (it can be assumed \underline{λ} > 0), λ̄ = sup λ_{n}. Then u\underline{λ} ≤ u_{λ_{n}} ≤ uλ̄ and u_{λ_{n}} is bounded in L^{∞}. Now the relevant estimate in [2] (see Lemma 3.2 there) implies the boundedness of u_{λ_{n}} in W^{1,p}(Ω) for all p > 1 what, in turn and by standard arguments, successively provides its boundedness in W^{2,p}(Ω) for all p > 1 and in C^{1,β}(\overline{Ω}) for all 0 < β < 1. Next, the Schauder estimates ([12]) provide a subsequence u_{λ_{n′}} → u_{λ} in C^{2,α}(\overline{Ω}). Finally, the uniqueness and the same argument permit extending such convergence to the whole sequence u_{λ_{n}}. By the same token, if λ_{n} → 0 in ℝ^{+}, u_{λ_{n}} converges in C^{2,α}(\overline{Ω}) to a nonnegative solution of (1.1) with λ = 0 what means that such limit is zero.

To show iii) let u_{0} ∈ C^{1,α}(\overline{Ω})^{+}, u_{0} ≢ 0 be an initial datum for (1.4) (due to the parabolic regularization effect, such initial smoothness can be considerably relaxed). Then, (1.4) admits a nonnegative a globally defined for t > 0 solution u(x, t), u(·, t) ∈ C^{2,α}(\overline{Ω}) for all t. In fact u_{−}(x, t) = 0 serves as a subsolution while u_{+}(x, t) = cϕ_{1}e^{μ_{1}(λ)t}, with ϕ_{1} a positive principal eigenfunction of (2.1), defines a positive supersolution for a suitably chosen constant c > 0. On the other hand, the parabolic strong maximum principle implies that for all t > 0, u(x, t) > 0 in \overline{Ω}. This means that for t_{0} > 0 small as desired,

\underline{u}(x) ≤ u(x, t_{0}) ≤ \overline{u}(x), x ∈ Ω,

if the constants A, B, modulating respectively the subsolution \underline{u} and the supersolution \overline{u} are properly chosen. It is then well known (cf. [21]) that this implies

u(x, t) → u_{λ}(x) x ∈ Ω,

as t → ∞ (exponentially) in C^{2,α}(\overline{Ω}) and the proof of Theorem 1 is finished.

Remark 9. It is implicit in the proof of Theorem 1 the existence for all λ > 0 of a unique positive classical solution u ∈ C^{2,α}(\overline{Ω}) to (1.1) if, for instance, dist (\{a = 0\} ∩ Ω, ∂Ω) > 0 even being a = 0 at points of ∂Ω. Thus a ∈ C^{α}(\overline{Ω}) may vanish in Ω provided there is a gap between {a = 0} and the boundary ∂Ω.

4. Behavior as λ → σ_{1}

In this section we are providing the proofs of Theorems 3 and 4. We are first considering the latter.

Proof of Theorem 4: First of all, it is well known that for a function a ∈ C^{α}(\overline{Ω}), positive in Ω, problem (1.8)

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ u = ∞ & x ∈ ∂Ω, \end{matrix}

admits a minimum positive solution z_{D}(x) ∈ C^{2,α}(Ω) attaining the boundary condition in the sense lim_{d(x)↘0} u(x) = ∞, d(x) = dist (x, ∂Ω) (see the references in [5, 9, 19]). It can be also shown that,

z_{D}(x) = sup u(x) x ∈ Ω,

being the supremum extended to all classical nonnegative solutions u ∈ C^{2}(Ω) ∩ C(\overline{Ω}) of Δu = au^{p}. We claim that the same facts hold also true in the case where a vanishes in a whole subdomain Ω_{0} ⊂⊂ Ω just under the conditions of the present statement (see a brief proof below). Assumed this and if u_{λ} ∈ C^{2,α}(\overline{Ω}) stands for the positive solution to (1.1) then,

u_{λ}(x) < z_{D}(x) x ∈ Ω,

for all λ > 0. Being u_{λ} increasing in λ this implies that,

z(x) := sup_{λ>0} u_{λ}(x) = lim_{λ→∞} u_{λ}(x) ≤ z_{D}(x) x ∈ Ω.

Moreover, employing the L^{p} estimates and bootstrap in the standard way (check the proof of Theorem 1) we get z ∈ C^{2,α}(Ω) being the limit valid in C^{2,α}(Ω). Therefore, z solves the equation in (1.8).

Our next issue is elucidating the boundary behavior of z. It will be in fact shown that z and z_{D} coincide. As a first remark observe that,

z(x) = sup_{m>0} v_{m}(x) x ∈ Ω,

where for m > 0, v_{m} ∈ C^{2,α}(\overline{Ω}) is the unique positive solution to,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = m & x ∈ ∂Ω. \end{matrix}

In fact, notice that v_{m} increases in m while, for λ > λ_{1} > 0,

\frac{∂u_{λ}}{∂ν} = λu_{λ} > λu_{λ_{1}} → ∞,

uniformly on ∂Ω as λ → ∞.

On the other hand, in order to work with the constant coefficients case observe that for each m > 0 we get by comparison ṽ_{m} < v_{m} where ṽ_{m} is the positive solution of,

\{ \begin{matrix}Δu = |a|_{∞}u^{p} & x ∈ Ω \\ \frac{∂u}{∂ν} = m & x ∈ ∂Ω. \end{matrix}

Thus, being z̃ = sup_{m} ṽ_{m} we have z̃ ≤ z and so to get z = z_{D} we only need to show that z̃ → ∞ as d → 0+. Using now ideas from [5, 9] it is not too hard to prove the following assertions. Consider,

γ = \frac{2}{p − 1} A = \{\frac{γ(γ + 1)}{|a|_{∞}}\}^{1/(p−1)} ,

together with,

h(d) = (A − η)d^{−γ} d(x) = dist (x, ∂Ω),

0 < η < A. Then, for η as small as desired positive constants δ, τ_{0} can be found such that,

w_{τ}(x) = h(d(x) + τ),

defines a C^{2,α} subsolution of Δu = |a|_{∞}u^{p} in the region D_{δ} = {x ∈ Ω : 0 < d(x) < δ} for every 0 ≤ τ < τ_{0}. In addition, a positive constant k can be chosen so that,

0 < w_{τ}(x) − k < z̃(x),  (4.1)

for every x with d(x) = δ. Notice that w_{τ} − k is again a subsolution in the same conditions as w_{τ}. On the other hand, ṽ_{m} solves the next mixed problem in D_{δ},

\{ \begin{matrix}Δu = |a|_{∞}u^{p} & x ∈ D_{δ} \\ \frac{∂u}{∂ν} = m & x ∈ ∂Ω \\ u = ṽ_{m} & d = δ. \end{matrix}

For 0 < τ < τ_{0} fixed we conclude in view of (4.1) and the finiteness of \frac{∂w_{τ}}{∂ν} at ∂Ω that for m large enough,

w_{τ}(x) − k ≤ ṽ_{m}(x) x ∈ D_{δ}.

Making first m → ∞ and then τ ↘ 0 we get,

h(d) − k ≤ z̃,

which gives z̃ → ∞ as d ↘ 0, as desired.

Let us prove now the claim. Namely, that the singular boundary value problem (1.8) has a minimum classical positive solution z ∈ C^{2,α}(Ω) if a ∈ C^{α}(\overline{Ω}) vanishes in a C^{2,α} subdomain Ω_{0} ⊂ \overline{Ω}_{0} ⊂ Ω (see [10] for a more general setting). Following a standard approach (cf. [9], [14]) z is given by the limit, whose existence must be proved,

z(x) = lim u_{m}(x) x ∈ Ω,  (4.2)

where u_{m} ∈ C^{2,α}(Ω) is the positive solution to the problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω \\ u = m & x ∈ ∂Ω. \end{matrix}

The existence of u_{m} is achieved, say by sub and supersoltions, while u_{m} increases in m and the limit (4.2) is actually a supremum.

If z_{D,Ω^{+}} ∈ C^{2,α}(Ω^{+}), Ω^{+} = {a(x) > 0} ∩ Ω, stands for the minimum positive solution to

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω^{+} \\ u = ∞ & x ∈ ∂Ω^{+}, \end{matrix}

then,

u_{m}(x) < z_{D,Ω^{+}}(x) x ∈ Ω^{+},

and so by the bootstrap argument invoked before we get the existence and validity of the limit (4.2) in C^{2,α}(Ω^{+}). In order to cover \overline{Ω}_{0} observe that u_{m} solves the Dirichlet problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ B(Ω_{0}, δ) \\ u = u_{m} & x ∈ ∂B(Ω_{0}, δ), \end{matrix}

where B(Ω_{0}, δ) = {x ∈ Ω : dist (x, Ω_{0}) < δ}, δ > 0 small. On the other hand, since ∂B(Ω_{0}, δ) ⊂ Ω^{+} then,

u_{m}(x) < z_{D,Ω^{+}}(x) x ∈ ∂B(Ω_{0}, δ).

Being every u_{m} subharmonic this means that u_{m} is bounded in B(Ω_{0}, δ) and again, this implies the existence of the limit (4.2) in C^{2,α}(B(Ω_{0}, δ)), therefore in C^{2,α}(Ω). This finish the proof of the claim.

Finally, the asymptotic rate (1.9) follows from (see (3.7)),

sup_{Ω} u_{λ} ≥ (\frac{μ_{1}}{|a|_{∞}} )^{1/(p−1)}

together with the estimate (2.2) for μ_{1} as λ → ∞.

Remark 10. It was pointed out in Remark 6 that the principal positive eigenfunction ϕ_{1,λ}, normalized as sup_{Ω} ϕ_{1,λ} = 1, decays to zero a. e. in Ω as λ → ∞. As a consequence of the last part of the proof of Theorem 4 it further follows that ϕ_{1,λ}(x) → 0 as λ → ∞ for all x ∈ Ω, faster than any polynomial. In fact (3.7) implies, using p > 1 as a parameter, that

λ^{2/(p−1)}ϕ_{1,λ}(x) ≤ (\frac{λ^{2}}{μ_{1}} )^{1/(p−1)} |a|_{∞,Ω}^{1/(p−1)}u_{λ}(x) x ∈ Ω.

Hence, from (2.2):

lim sup_{λ→∞} (λ^{2/(p−1)}ϕ_{1,λ}(x)) ≤ |a|_{∞,Ω}^{1/(p−1)}z(x).

Thus, the conclusion follows.

Proof of Theorem 3. Let us first deal with the behavior of u_{λ} in Ω_{0} with special emphasis on the boundary behavior. To construct a suitable subsolution to (1.1) consider an special choice of the perturbed eigenvalue problem (2.5). Namely,

\{ \begin{matrix}Δφ = c_{1}δ φ & x ∈ B(Ω_{0}, δ) \\ \frac{∂φ}{∂ν} = σφ & x ∈ Γ_{1} \\ φ = 0 & x ∈ Γ_{2,δ}, \end{matrix}  (4.3)

where B(Ω_{0}, δ) = {x ∈ Ω : dist (x, Ω_{0}) < δ}, Γ_{2,δ} = {x ∈ Ω : dist (x, Ω_{0}) = δ} and δ > 0 is small (notice that ∂B(Ω_{0}, δ) = Γ_{1} ∪ Γ_{2,δ}). If φ_{1} is any principal eigenfunction to (1.2), c_{1} is chosen a positive constant such that,

c_{1} ∫_{Ω_{0}} φ_{1}^{2} < ∫_{Γ_{2}} |∇φ_{1}|^{2}.

Thus, by using the expression (2.9) for the derivative dσ_{1,t}/dt_{|t=0} (now t = δ), the principal eigenvalue σ_{1,δ} > 0 to (4.3) increases when δ decays to zero. Moreover, with the same arguments as in the proof of Theorem 1 (now in a simpler case) it is shown that lim_{δ↘0} σ_{1,δ} = σ_{1}. On the other hand, if φ̃_{δ} stands for the principal eigenfunction to (4.3) normalized according to sup_{Ω_{0}} φ̃_{δ} = 1, then the L^{∞} boundedness of {φ̃_{δ}} and the estimates in [2], [1] yield φ̃_{δ} → φ_{1} in C^{2,α}(\overline{Ω}_{0}), where now φ_{1} is the principal eigenfunction of (1.2) normalized as φ̃_{δ}.

For λ ↗ σ_{1} we take δ = δ(λ) ↘ 0 such that σ_{1,δ} < λ < σ_{1} and look for a weak subsolution to (1.1) in the form

\underline{u} = A φ̃_{δ},

where φ̃_{δ} is extended as zero to \overline{Ω}\B(Ω_{0}, δ) and A = A(δ). We claim that such a subsolution can be constructed so that,

δ A(δ) → ∞ δ → 0 + .  (4.4)

Assumed this fact and since as large as desired positive supersolutions \overline{u} to (1.1) can be found, then the positive solution u_{λ} satisfies

\underline{u}(x) ≤ u_{λ}(x) ≤ \overline{u}(x) x ∈ Ω.

Being φ̃_{δ} → φ_{1} in C^{2,α}(\overline{Ω}_{0}) and φ_{1} > 0 in Ω_{0} ∪ Γ_{1}, (4.4) implies that u_{λ} → ∞ in Ω_{0} ∪ Γ_{1}. Moreover as,

inf_{Γ_{2}} φ_{δ} ∼ C_{1}δ δ ↘ 0,  (4.5)

(see [18]) by (4.4) we conclude that u_{λ} → ∞ as λ ↗ σ_{1} uniformly on Γ_{2}. This proves i).

To show the claim we only need to check that an A satisfying (4.4) can be found with the additional requirement,

A φ̃_{δ}(x) ≤ \{\frac{c_{1}}{sup_{0<d(x)<δ}(a(x)/δ)}\}^{1/(p−1)}  0 < d(x) < δ,  (4.6)

with d(x) = dist (x, Ω_{0}). This election is actually possible due to (4.5) and the fact a(x) = o(d(x)) as d(x) ↘ 0.

Observe now that iii) is an immediate consequence of the previous discussion. In fact, Ω^{+} ⊂⊂ Ω means that ∂Ω^{+} = Γ_{2}. Since u_{λ}_{|Γ_{2}} → ∞ uniformly on Γ_{2} we then conclude that u_{λ} → z_{D,Ω^{+}} in C^{2,α}(Ω^{+}) (check the proof of Theorem 4).

As for ii) we are proving the existence of the minimum solution to (1.6). Accordingly, assume that z_{1}(x) ∈ C^{2,α}(Ω^{+} ∪ Γ^{+}) is any of its possible positive solutions. For each m ∈ ℕ the auxiliary problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω^{+} \\ \frac{∂u}{∂ν} = σ_{1}u & x ∈ Γ^{+} \\ u = m & x ∈ Γ_{2}, \end{matrix}  (4.7)

admits at most a unique positive solution in C^{2,α}(\overline{Ω^{+}}) (Lemma 8) and has supersolutions as large as required (proof of Theorem 1). Therefore it possesses a positive solution u_{m} ∈ C^{2,α}(\overline{Ω^{+}}) which by comparison is seen to satisfy,

u_{m}(x) ≤ z_{1}(x).

In fact, being Ω^{+,δ} := Ω \ \overline{B(Ω_{0}, δ)}, such inequality holds on Ω^{+,δ} ∪ Γ^{+}, for every δ → 0+. On the other hand u_{m} is increasing in m while u_{m}(x) ≤ z_{D,Ω^{+}}(x) in Ω^{+}. Hence,

lim u_{m}(x) = sup u_{m}(x) := z(x) ≤ z_{D,Ω^{+}}(x) x ∈ Ω^{+},  (4.8)

being such a limit valid in C^{2,α}(Ω^{+}). Hence, z solves Δu = a(x)u^{p} in Ω^{+}. To cover Γ^{+} observe that ∂Ω^{+,δ} = Γ^{+} ∪ Γ_{2,δ}, Γ_{2,δ} = {x ∈ Ω : dist (x, Ω_{0}) = δ} while for δ > 0 small the auxiliary problem,

\{ \begin{matrix}Δu = a(x)u^{p} & x ∈ Ω^{+,δ} \\ \frac{∂u}{∂ν} = σ_{1}u & x ∈ Γ^{+} \\ u = z_{D,Ω^{+}} & x ∈ Γ_{2,δ}, \end{matrix}  (4.9)

also admits a unique positive solution u_{δ}(x) ∈ C^{2,α}(\overline{Ω^{+,δ}}) (just the same reasons as in (4.7)). Being every u_{m} a subsolution to (4.9), and possessing such problem supersolutions large as necessary we obtain,

u_{m}(x) ≤ u_{δ}(x) x ∈ \overline{Ω^{+,δ}}.

By using the estimates in [2] and [1] we achieve that the limit (4.8) holds in C^{2,α}(\overline{Ω^{+,δ}}) for δ > 0 arbitrarily small. Thus the limit is actually valid in C^{2,α}(Ω^{+} ∪ Γ^{+}) and in particular

\frac{∂z}{∂ν} = σ_{1}z x ∈ Γ^{+}.

Since in addition z(x) ≤ z_{1}(x), z is the minimum solution to (1.6) which is denoted as z_{M,Ω^{+}}. On the other hand, the solution u_{λ} to (1.1) satisfies,

u_{λ}(x) ≤ z_{M,Ω^{+}}(x) x ∈ Ω^{+} ∪ Γ^{+},

since u_{λ}(x) ≤ u_{m}(x) in \overline{Ω^{+}} for m large. Reasoning as before

z_{2}(x) := sup_{λ↗σ_{1}} u_{λ}(x) = lim_{λ↗σ_{1}} u_{λ}(x) ≤ z_{M,Ω^{+}}(x) x ∈ Ω^{+} ∪ Γ^{+},

defines a classical solution of (1.6) being the limit valid in C^{2,α}(Ω^{+} ∪ Γ^{+}). Therefore, z_{2} = z_{M,Ω^{+}} and the proof is concluded.

Remark 11. Let us briefly discuss what happens if Ω_{0} ⊂ Ω consists of M connected pieces Ω_{0,1}, …, Ω_{0,M}, all of them C^{2,α} and satisfying (H) in the sense that for each 1 ≤ i ≤ M, Γ_{1,i} = ∂Ω_{0,i} ∩ ∂Ω is nonempty and consists of connected pieces of ∂Ω, Γ_{2,i} = ∂Ω_{0,i} ∩ Ω.

If Ω_{0,i} ⊂⊂ Ω for each i ∈ {1, …, M} then a unique positive classical solution u_{λ} to (1.1) exits for all λ > 0 which exhibits all the features of both Theorems 1 and 4.

If on the contrary, Γ_{1,j} ≠ ∅ for some 1 ≤ j ≤ M define σ^{∗} = min{σ_{1,i}} where σ_{1,i} stands for the principal eigenvalue to problem (2.1) in Ω_{0,i}, assumed that Γ_{1,i} ≠ ∅. We then also achieve the existence of a unique positive solution u_{λ} if and only if 0 < λ < σ^{∗} together with the remaining properties of Theorem 1 with σ_{1} replaced by σ^{∗}. As for Theorem 3 u_{λ}(x) → ∞, x ∈ ∪_{i}\overline{Ω_{0,i}}, uniformly as λ ↗ σ^{∗}, the union being extended to those connected pieces with exactly σ_{1,i} = σ^{∗}. Regarding ii) and iii), Ω^{+} must be now replaced by Ω^{∗} = Ω \ ∪_{i}\overline{Ω_{0,i}}, Γ^{+} by Γ^{∗} = ∂Ω^{∗} ∩ ∂Ω and σ_{1} by σ^{∗}. Then the same conclusions hold true again. In particular, observe that u_{λ} remains finite as λ ↗ σ^{∗} on the closure \overline{Ω_{0,l}} of every component, if any, with σ_{1,l} > σ^{∗}.

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