A nonlinear Neumann problem for an indefinite superlinear elliptic equation

Klaus Pflüger Institut für Mathematik I Freie Universität Berlin Preprint Nr. 10 A 96

Abstract. We study the superlinear elliptic equation −Δu = a(x)u^{p} in a bounded domain Ω ⊂ IR^{n} with nonlinear Neumann boundary condition ∂nu = b(x)u^{q} on ∂Ω, where a and b may change sign. We derive existence and nonexistence results for positive solutions of this problem. In the case p = q we use a constrained maximization technique, while for p ≠ q we apply the mountain-pass lemma. The results are based on ideas from a recent paper of Berestycki, Capuzzo-Dolcetta and Nirenberg [3].

email: pflueger@math.fu-berlin.de

1 Introduction

In this paper we study existence of positive solutions to the following Neumann problem in a bounded domain Ω ⊂ IR^{n} (n ≥ 3)

−Δu = a(x)u^{p}  in Ω ,  (1)
∂nu = b(x)u^{q}  on Γ = ∂Ω,  (2)

where ∂nu denotes the outward normal derivative of u. We assume that a and b are continuous functions which may change sign, and the exponents p and q are subcritical, i. e. 1 < p < (n + 2)/(n − 2), 1 < q < n/(n − 2). Similar problems with homogeneous Dirichlet or Neumann conditions have been studied recently in [1] and [3]. Here we prove existence and nonexistence results for positve solutions in the case of nonlinear boundary conditions. The main results of this paper are the following.

Theorem 1 If 1 < p = q < n/(n − 2), then problem (1), (2) has a nontrivial positive solution if

∫_{Ω} a(x) dx + ∫_{Γ} b(x) dΓ < 0 and a(x) > 0 somewhere in Ω.  (3)

On the other hand, there exists no nontrivial positive solution if

∫_{Ω} a(x) dx + ∫_{Γ} b(x) dΓ ≥ 0 .  (4)

For the case p ≠ q we assume

α = ∫_{Ω} a(x) dx < 0 and a(x) > 0 somewhere,  (5)
if p > q then b(x) ≤ 0,  if  p < q then b(x) ≡ 0 .  (6)

Theorem 2 If p ≠ q, then problem (1), (2) has a nontrivial positive solution if conditions (5) and (6) are satisfied. There is no nontrivial positive solution if

b(x) ≥ 0 and ∫_{Ω} a(x) ≥ 0  or  a(x) ≥ 0 and ∫_{Γ} b(x)dΓ ≥ 0 .

The proof of these theorems will be given in the next sections.

2 Necessary conditions

To obtain necessary conditions for the existence of a nontrivial solution, we assume that u is a positive solution of (1), (2). By the maximum principle and the Hopf lemma, we claim u > 0 in \overline{Ω}. Multiplying (1) with u^{−p}, we obtain by partial integration

∫_{Ω} a(x) dx = ∫_{Ω} −Δu u^{−p}dx
= ∫_{Ω} ∇u∇(u^{−p}) dx − ∫_{Γ} b(x)u^{q}u^{−p} dΓ .

This shows

∫_{Ω} a(x) dx + ∫_{Γ} b(x)u^{q−p} dΓ < 0 .  (7)

In the same way, multiplying with u^{−q},we obtain

∫_{Ω} a(x)u^{p−q} dx + ∫_{Γ} b(x) dΓ < 0 .  (8)

Therefore ∫_{Ω} a(x) dx + ∫_{Γ} b(x) dΓ < 0 is a necessary condition in the case p = q and we have proved the second part of Theorem 1. From the inequalities above we derive that

if  b(x) ≥ 0  then  ∫_{Ω} a(x) dx < 0  (9)

is necessary, while

if  a(x) ≥ 0  then  ∫_{Γ} b(x) dΓ < 0  (10)

is a necessary condition for existence. This proves the second part of Theorem 2. On the other hand, direct integration of (1) leads to

∫_{Ω} a(x)u^{p} dx = − ∫_{Ω} Δu dx = − ∫_{Γ} ∂nu dΓ,

which shows

∫_{Ω} a(x)u^{p} dx + ∫_{Γ} b(x)u^{q} dΓ = 0 .  (11)

Therefore, a or b must be positive somewhere.

3 Sufficient conditions in the case p = q

To obtain a nontrivial solution of (1), (2), we consider the following constrained maximization problem. Let

I(u) = ∫_{Ω} a(x)|u|^{q+1} dx + ∫_{Γ} b(x)|u|^{q+1} dΓ .

Then we look for a solution of the variational problem

M = max{I(u) | u ∈ S} , S = {u ∈ H^{1}(Ω) | ∫_{Ω} |∇u|^{2} = 1} .  (12)

We assume that (3) from Theorem 1 holds. First of all we show that M = sup{I(u) | u ∈ S} is finite. For this we introduce a decomposition of H^{1}(Ω) similar to that in [3]: for u ∈ H^{1}(Ω) we set t = |Ω|^{−1} ∫_{Ω} u(x) dx, the avarage of u in Ω, and v = u − t. Then ∫_{Ω} v(x) dx = 0 and we set

H^{1}(Ω) = IR ⊕ V , where V = {v ∈ H^{1}(Ω) | ∫_{Ω} v dx = 0} .  (13)

From the Poincaré inequality ∥φ∥_{H^{1}}^{2} ≤ C (∫_{Ω} |∇φ|^{2}dx + | ∫_{Ω} φ dx|^{2}) for φ ∈ H^{1}(Ω), we see that

∥u∥^{2} = |t|^{2} + ∥v∥_{V}^{2} , with ∥v∥_{V}^{2} = ∫_{Ω} |∇v|^{2}dx

defines an equivalent norm on H^{1}(Ω). Under the assumption that a is positive somewhere, we get M > 0, since we can choose a function with compact support in {x ∈ Ω | a(x) > 0}, ∫_{Ω} |∇u|^{2}dx = 1. Now let u_{k} ∈ S be a maximizing sequence, i. e. I(u_{k}) → M, and let u_{k} = t_{k} + v_{k} be the decomposition according to (13). Since ∫_{Ω} |u_{k}|^{2}dx = ∫_{Ω} |v_{k}|^{2}dx = 1 and the embedding V ↪ L^{q+1}(Ω) and trace V → L^{q+1}(Γ) are compact, v_{k} converges weakly in H^{1}(Ω) and consequently has a strongly convergent subsequence in L^{q+1}(Ω) and L^{q+1}(Γ). Next we show that |t_{k}| is bounded. Assume that this is not true. Then we may assume that |t_{k}|^{−1}v_{k} → 0 in L^{q+1}(Ω) and L^{q+1}(Γ). Consequently, from (3) we get

∫_{Ω} a(x)|1 + |t_{k}|^{−1}v_{k}|^{q+1}dx + ∫_{Γ} b(x)|1 + |t_{k}|^{−1}v_{k}|^{q+1}dx ⟶
∫_{Ω} a(x) dx + ∫_{Γ} b(x)dΓ < 0.

From this we conclude

I(u_{k}) = |t_{k}|^{q+1} (∫_{Ω} a(x)|1 + |t_{k}|^{−1}v_{k}|^{q+1}dx + ∫_{Γ} b(x)|1 + |t_{k}|^{−1}v_{k}|^{q+1}dx) → −∞,

which is not possible. Therefore, t_{k} is bounded and contains a convergent subsequence. We may assume that u_{k} converges weakly in H^{1}(Ω) and strongly in L^{q+1}(Ω) and L^{q+1}(Γ). The limit function u satisfies I(u) = M and u is not a constant in view of (3). Hence 0 < ∫_{Ω} |∇u|^{2}dx ≤ 1 and there exists μ ≥ 1 such that μu ∈ S. Since I(μu) = μ^{q+1}M ≤ M, we have μ = 1. Further we claim u ≥ 0, since we can choose a maximizing sequence of positive functions. Consequently there exists a Lagrange multiplier λ ∈ IR such that

λ2 ∫_{Ω} ∇u∇φ = \frac{1}{q + 1} (∫_{Ω} a(x)u^{q}φ dx + ∫_{Γ} b(x)u^{q}φ dΓ)  (14)

for every φ ∈ H^{1}(Ω). Setting φ = u, we get

λ2(q + 1) ∫_{Ω} |∇u|^{2}dx = I(u) = M > 0 ,

which shows λ > 0. Let λ̃ = λ2(q + 1). From (14) we obtain

−λ̃Δu = a(x)u^{q} in Ω

and by partial integration

λ̃ ∫_{Γ} ∂nu φ dΓ = ∫_{Γ} b(x)u^{q}φ dΓ

for every φ ∈ H^{1}(Ω). Therefore w = λ̃^{−1/(q−1)}u is a solution of (1), (2) and the maximum principle implies u > 0 in Ω. This proves the first part of Theorem 1.

4 Sufficient conditions in the case p ≠ q

In the case p ≠ q we want to apply the mountain-pass lemma to obtain a nontrivial solution. Throughout this section we assume that (5)–(6) hold. Define

J(u) = ∫_{Ω} \frac{1}{2}|∇u|^{2} − \frac{1}{p + 1}a(x)|u|^{p+1}dx − ∫_{Γ} \frac{1}{q + 1}b(x)|u|^{q+1}dΓ .

We will show that J satisfies the assumptions of the mountain-pass lemma (cf. [2] or [5]), that is:

(i) J satisfies the Palais-Smale condition.

(ii) There are numbers ρ, δ > 0, such that J(u) ≥ δ for every u ∈ H^{1}(Ω) with ∥u∥ = ρ.

(iii) J(0) = 0 and there exists e ∈ H^{1}(Ω), ∥e∥ > ρ, such that J(e) < δ.

To this end we need the following lemma from [3]

Lemma 3 There exists a positive real number η > 0 such that for all t ∈ IR, v ∈ V satisfying ∥v∥_{V} ≤ η|t| the estimate

∫_{Ω} a(x)|u(x)|^{p+1}dx = ∫_{Ω} a(x)|t + v(x)|^{p+1}dx ≤ \frac{|t|^{p+1}}{2} ∫_{Ω} a(x)dx

holds.

To prove (i), let u_{k} be a Palais-Smale sequence in H^{1}(Ω) for J. Then for every ε > 0, k ≥ k_{ε} we have

|J(u_{k})| = |∫_{Ω} \frac{1}{2}|∇u_{k}|^{2} − \frac{1}{p + 1}a(x)|u_{k}|^{p+1}dx − ∫_{Γ} \frac{1}{q + 1}b(x)|u_{k}|^{q+1}dΓ| ≤ M ,  (15)
|⟨J′(u_{k}), u_{k}⟩| = |∫_{Ω} |∇u_{k}|^{2} − a(x)|u_{k}|^{p+1}dx − ∫_{Γ} b(x)|u_{k}|^{q+1}dΓ| ≤ ε∥u_{k}∥ .  (16)

Setting ε = 1, multiplying (16) with 1/(p + 1) and adding, we obtain

(\frac{1}{2} − \frac{1}{p + 1}) ∫_{Ω} |∇u_{k}|^{2}dx + (\frac{1}{p + 1} − \frac{1}{q + 1}) ∫_{Γ} b(x)|u_{k}|^{q+1}dΓ − ∥u_{k}∥ ≤ M .  (17)

By (6) the middle term on the left hand side is nonnegative. Setting u_{k} = t_{k} + v_{k} ∈ IR ⊕ V and ϑ = 1/2 − 1/(p + 1) > 0, we get

ϑ∥v_{k}∥_{V}^{2} − ∥t_{k} + v_{k}∥ ≤ M .  (18)

We now want to show that t_{k} and v_{k} are bounded. Without restrictions we may assume that ∥t_{k}∥ ≥ c > 0 for all k. Then (18) implies

ϑ\frac{∥v_{k}∥_{V}^{2}}{|t_{k}|} − (1 + \frac{∥v_{k}∥_{V}^{2}}{|t_{k}|^{2}} )^{1/2} ≤ M/|t_{k}| ,

or

ϑ\frac{∥v_{k}∥_{V}^{2}}{|t_{k}|} − \frac{∥v_{k}∥_{V}}{|t_{k}|} ≤ M_{1} .  (19)

Now suppose that t_{k} is not bounded. If ∥v_{k}∥_{V} → ∞, we obtain from (19)

∥v_{k}∥_{V} (ϑ\frac{∥v_{k}∥_{V}}{|t_{k}|} − \frac{1}{|t_{k}|}) ≤ M_{1}

and since 1/|t_{k}| → ∞, we see that ∥v_{k}∥_{V} /|t_{k}| → 0. That is, to every η > 0 there is a k_{η} such that ∥v_{k}∥_{V} ≤ η|t_{k}| holds for k ≥ k_{η}. On the other hand, if ∥v_{k}∥_{V} was bounded, the inequality ∥v_{k}∥_{V} ≤ η|t_{k}| is trivially satisfied for k large enough. In both cases we can apply Lemma 3 to obtain

J(u_{k}) ≥ \frac{1}{2}∥v_{k}∥_{V}^{2} − \frac{|t|^{p+1}}{2} \frac{α}{p + 1} .

Since α < 0, we get J(u_{k}) → ∞ which is a contradiction. Therefore, t_{k} must be bounded. Now from (18) it is easy to see that ∥v_{k}∥_{V} must also be bounded if t_{k} is. This shows that every Palais-Smale sequence for J is bounded. Now by standard arguments, using the compact embedding, resp. trace V ↪ L^{p+1}(Ω), V → L^{q+1}(Γ), we conclude that u_{k} has a convergent subsequence, thus (i) is proved. To show that (ii) holds, fix ρ > 0 and let η be as in Lemma 3. Choose u = t + v ∈ H^{1}(Ω) with ∥u∥ = ρ, i. e. t^{2} + ∥v∥_{V}^{2} = ρ^{2}. Assume first that ∥v∥_{V} ≤ η |t|. Then ρ^{2} ≤ t^{2}(1 + η^{2}) or t^{2} ≥ ρ^{2}(1 + η^{2})^{−1}. Setting α = (1/2) ∫_{Ω} a(x) dx. From Lemma 3 and (6) we get

J(u) ≥ ∫_{Ω} \frac{1}{2}|∇v|^{2} − \frac{1}{p + 1}a(x)|t + v|^{p+1}dx
≥ \frac{−α}{2} \frac{|t|^{p+1}}{p + 1} .

From (5) we see that −α > 0 and find

J(u) ≥ \frac{−α}{2(p + 1)} (\frac{ρ^{2}}{1 + η^{2}} )^{(p+1)/2} > 0 .

Now we consider the case ∥v∥_{V} ≥ η |t|. We have ∥u∥^{2} ≤ t^{2} + ∥v∥_{V}^{2} ≤ (\frac{1}{η^{2}} + 1)∥v∥_{V}^{2} . From the Sobolev embedding, resp. trace theorem we obtain

|∫_{Ω} a(x)|u|^{p+1}dx| ≤ C_{1}∥u∥^{p+1} ≤ C_{1}(\frac{1}{η^{2}} + 1)^{1/2}∥v∥_{V}^{p+1} ,
|∫_{Γ} b(x)|u|^{q+1}dx| ≤ C_{2}∥u∥^{q+1} ≤ C_{2}(\frac{1}{η^{2}} + 1)^{1/2}∥v∥_{V}^{q+1}

with constants C_{1}, C_{2} > 0. For v small enough, we infer

J(u) ≥ \frac{1}{2}∥v∥_{V}^{2} − \frac{C_{1}}{p + 1}∥v∥_{V}^{p+1} − \frac{C_{2}}{q + 1}∥v∥_{V}^{q+1} ≥ \frac{1}{4}∥v∥_{V}^{2} .

Since ∥u∥ = ρ and ∥v∥_{V} ≥ η∥u∥(1 + η^{2})^{−1/2}, we obtain

J(u) ≥ \frac{1}{4} \frac{η^{2}ρ^{2}}{1 + η^{2}} > 0

Hence the choice δ = min \{ \frac{1}{4} \frac{η^{2}ρ^{2}}{1+η^{2}} , \frac{−α}{2(p+1)} (\frac{ρ^{2}}{1+η^{2}} )^{(p+1)/2}\} proves (ii).

To prove (iii), we note that by (5) there is an open subset U ⊂ Ω, where a(x) > 0. Now choose a positive function w ∈ H^{1}(Ω) with compact support in U and observe that J(λw) → −∞ as λ → +∞. For λ large enough, we infer that e = λw satisfies (iii). Now the following variant of the mountain-pass lemma from [4] (cited in [3]) shows the existence of a positive solution.

Theorem 4 Let X be a Banach space and J ∈ C^{1}(X) a functional satisfying the conditions (i)–(iii) of the mountain-pass lemma. Assume that π : X → X is a continuous function satisfying J(π(u)) ≤ J(u) for every u ∈ X, π(0) = 0, π(e) = e (with e defined in (iii)). Then, there exists a critical point u ∈ \overline{π(X)} and J(u) ≥ δ.

Setting π(u) = |u|, we obtain a positive solution, which is strictly greater than 0 by the maximum principle. Thus, Theorem 2 is proved.

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