Compact Traces in Weighted Sobolev Spaces

Klaus Pflüger Institut für Mathematik I Freie Universität Berlin Preprint Nr. 29 A 95

Abstract. We study trace operators in weighted Sobolev spaces W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(∂Ω; w) for sufficiently regular unbounded domains Ω ⊂ IR^{N} with noncompact boundary. We show that under certain conditions on the weight functions v_{0}, v_{1}, w, this operator is compact. This result is then used to prove existence of a nontrivial solution of a semilinear elliptic equation with nonlinear boundary condition in an unbounded domain.

email: pflueger@math.fu-berlin.de 1. INTRODUCTION

In the study of partial differential equations in a domain Ω ⊂ IR^{N} it is often useful to know that embeddings of the Sobolev space W^{m,p}(Ω) into the Lebesgue space L^{q}(Ω) are compact. In particular for nonlinear equations of variational form such a compactness property is often used to show that the energy functional for this equation satisfies the Palais-Smale condition. In the case when Ω is not bounded, compactness of the embedding above fails in general. So it seems to be natural to study more general function spaces, for instance weighted Sobolev spaces, where compact embeddings can be obtained for suitable weight functions (see e. g. [1], [4], [6]). Such an approach using weighted norms was used by several authors for the study of semilinear elliptic equations in IR^{N} (see e. g. [2], [7], [12]). Now in the study of nonhomogeneous (or nonlinear) Neumann boundary value problems, there appears an additional difficulty since then a boundary integral appears in the energy functional. If Ω is an unbounded domain with noncompact boundary one is confronted with two problems: first, there is no general result for embeddings W^{m,p}(Ω; V) ↪ L^{q}(Ω; w) if the boundary ∂Ω is not compact (here V denotes an admissible set of weight functions acting on the derivatives of functions from W^{m,p}), and second there is no result which guarantees that the trace operator W^{m,p}(Ω; V) ↪ L^{q}(∂Ω; w) is compact. The aim of the present paper is to fill this gap and to give an application of these embedding, resp. trace theorems to a semilinear elliptic problem with nonlinear boundary condition, which was studied before by the author in [8] with different methods. This paper is organized as follows. In Section 2 we introduce the necessary notations and state a recent result on S.-K. Chua concerning the existence of a continuous extension operator in weighted Sobolev spaces. Then we prove a basic lemma, which is used several times in the paper. At the end of this section we state our main results (Theorems 3 and 4). In Section 3 we study traces on (N −k)–dimensional planes in IR^{N} (Theorem 5) and obtain as a corollary (Corollary 6) a compact embedding result for fairly general domains Ω. In Section 4 the proof of Theorems 3 and 4 is given, and in Section 5 we state some concrete examples for admissible weight functions. In the last Section 6 a nonlinear elliptic boundary value problem is studied, where Theorem 4 and Corollary 6 are essentially used. For the proof of our trace theorems we use an approach similar to that of Kufner/Opic in [6] for embeddings of weighted Sobolev spaces.

2. NOTATIONS AND MAIN RESULTS

For a measurable function u and a positive measurable function w, defined in a domain Ω ⊂ IR^{N}, we define the weighted p-norm (1 ≤ p < ∞)

∥u∥_{p,Ω,w} = (∫_{Ω} |u(x)|^{p}w(x) dx)^{1/p}

and denote by L^{p}(Ω; w) the space of all measurable functions u such that ∥u∥_{p,Ω,w} is finite. Correspondingly for a submanifold Γ in IR^{N}, we define L^{q}(Γ; w) to be the space of all measurable functions u such that ∥u∥_{q,Γ,w} is finite. The weighted Sobolev space W^{m,p}(Ω; v_{0}, …, v_{m}) is defined as the set of all functions u ∈ L^{p}(Ω, v_{0}) such that all derivatives D^{α}u (|α| ≤ m) belong to L^{p}(Ω, v_{|α|}). The corresponding norm is defined by

∥u∥_{m,p,Ω,v_{α}} = (∫_{Ω} ∑_{|α|=0}^{m} |D^{α}u(x)|^{p}v_{|α|}(x) dx)^{1/p} .

A domain Ω ⊂ IR^{N} is called an (ε, δ)-domain if for all x, y ∈ Ω satisfying |x − y| < δ there exists a rectifiable curve ω in Ω connecting x and y, such that

ℓ(ω) < \frac{|x − y|}{ε}  and  d(z) > \frac{ε|x − z| |y − z|}{|x − y|} for every z ∈ ω

hold. Here, ℓ(ω) is the length of the curve ω and d(z) denotes the distance of z from the boundary of Ω. The Muckenhoupt class A_{p} is defined as the set of all positive functions v in IR^{N} which satisfy

\frac{1}{|Q|} (∫_{Q} v dx)^{1/p} (∫_{Q} v^{\frac{−1}{p−1}} dx)^{(p−1)/p} ≤ C  (1 < p < ∞)
\frac{1}{|Q|} ∫_{Q} v dx ≤ C ess inf_{x∈Q} v(x)  (p = 1)

for all cubes Q in IR^{N}. In the present paper we shall often make use of the following result of S.-K. Chua: \underline{THEOREM 1.} (Chua [3]) Let Ω be an unbounded (ε, ∞)-domain, let m ∈ IN, 1 ≤ p < ∞, v_{i} ∈ A_{p} (i = 0, …, m) be given. Then there exists a bounded linear extension operator

Λ : W^{m,p}(Ω; v_{0}, …, v_{m}) ⟶ W^{m,p}(IR^{N}; v_{0}, …, v_{m})

satisfying Λf = f a. e. on Ω. The norm of Λ depends only on ε, v_{i}, m, p, N. We assume in the sequel that our weight functions v_{i} and w are defined in all of IR^{N} and belong to A_{p}. Since we consider traces, we also assume that w is continuous so that its trace on a submanifold is defined. The theorem above is then used to transfer our trace theorems in IR^{N} to more general domains Ω. In order to avoid too much notation, we denote a function u defined in a domain Ω, its extension Λu on IR^{N} and its trace γu on a submanifold Γ by the same symbol u. Our main tool in the proof of the trace theorems below is the following. \underline{LEMMA 2.} Let D_{n} be an increasing sequence of bounded domains in IR^{N} such that IR^{N} = ∪_{n}D_{n}. Let Γ be a smooth (N − k)-dimensional submanifold of IR^{N} and denote Γ_{n} = D_{n} ∩ Γ, Γ^{n} = Γ \ \overline{Γ}_{n}. Assume that

W^{1,p}(D_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w)  is compact for every n,  (1)
lim_{n→∞} sup_{∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}≤1} ∥u∥_{q,Γ^{n},w} = 0 .  (2)

Then the trace operator W^{1,p}(IR^{N}; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is compact. On the other hand, if the trace operator is compact, then (2) holds. If

W^{1,p}(D_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w)  is continuous for every n,  (3)
lim_{n→∞} sup_{∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}≤1} ∥u∥_{q,Γ^{n},w} < ∞ ,  (4)

then the trace operator W^{1,p}(IR^{N}; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is continuous. If the trace operator is continuous, then (4) holds. \underline{Proof.} First of all we notice that (2) is equivalent to the statement that to every ε > 0 there exists an ñ, such that

∥u∥_{q,Γ,w}^{q} ≤ ε∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} + ∥u∥_{q,Γñ,w}^{q} for every u ∈ W^{1,p}(IR^{N}; v_{0}, v_{1}) .  (5)

Indeed, let (2) hold; then for every ε > 0 there is some ñ ∈ IN such that for every n ≥ ñ we have

∥u∥_{q,Γ^{n},w}^{q} ≤ ε∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} .

Since

∥u∥_{q,Γ,w}^{q} = ∥u∥_{q,Γ_{n},w}^{q} + ∥u∥_{q,Γ^{n},w}^{q} ,

we obtain (5). On the other hand, if (5) is satisfied, for every ε > 0 we can choose ñ, such that

∥u∥_{q,Γñ,w}^{q} + ∥u∥_{q,Γñ,w}^{q} ≤ ε∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} + ∥u∥_{q,Γñ,w}^{q},

which implies

∥u∥_{q,Γ^{n},w}^{q} ≤ ε∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} for every n ≥ ñ

and (2) follows. Now let u_{j} be a bounded sequence in W^{1,p}(IR^{N}; v_{0}, v_{1}) , ∥u_{j}∥_{1,p,IR^{N} ,v_{0},v_{1}} ≤ C and let ε > 0 be given. From (1) it follows that there is a subsequence which is a Cauchy sequence in L^{q}(Γñ; w). Therefore we can find n_{ε} such that for every i, j ≥ n_{ε}

∥u_{j} − u_{i}∥_{q,Γ,w}^{q} ≤ ε∥u_{j} − u_{i}∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} + ∥u_{j} − u_{i}∥_{q,Γñ,w}^{q}
≤ (2^{q}C^{q} + 1)ε .

Consequently, u_{j} contains a Cauchy sequence in L^{q}(Γ; w) and the trace operator is compact. Now let W^{1,p}(IR^{N}; v_{0}, v_{1}) ↪ L^{q}(Γ; w) be compact and assume that (5) does not hold. Then there exist an ε > 0 and a sequence (u_{n})_{n} in W^{1,p}(IR^{N}; v_{0}, v_{1}), such that

∥u_{n}∥_{q,Γ,w}^{q} > ε∥u_{n}∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} + ∥u_{n}∥_{q,Γ^{n},w}^{q} .

This shows a_{n} = ∥u_{n}∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} > 0 and for ũ_{n} = u_{n}/a_{n} we obtain

∥ũ_{n}∥_{q,Γ,w}^{q} > ε + ∥ũ_{n}∥_{q,Γ^{n},w}^{q} .  (6)

Since ũ_{n} is bounded in W^{1,p}(IR^{N}; v_{0}, v_{1}), there is a subsequence converging to ũ in L^{q}(Γ; w). Now taking the limit in (6), we get

∥ũ∥_{q,Γ,w}^{q} > ε + ∥ũ∥_{q,Γ,w}^{q} ,

which is a contradiction. The second part of Lemma 2 can be proved in a similar way; one just has to note that (4) implies ∥u∥_{q,Γ^{n},w} ≤ C_{1}∥u∥_{1,p,IR^{N} ,v_{0},v_{1}} and (3) means that ∥u∥_{q,Γ_{n},w} ≤ C_{2}∥u∥_{1,p,D_{n},v_{0},v_{1}}. It follows that ∥u∥_{q,Γ,w} ≤ C∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}. □

To apply this lemma to certain classes of weight functions, we need some estimates for the weights. First of all we define our covering by the family of cubes with edge lenght n

D_{n} = {x ∈ IR^{N} | |x| < n} , D^{n} = int (IR^{N} \ D_{n})

(|·| denotes the maximum norm in IR^{N}). In particular we have Γ_{n} = {x ∈ Γ | |x| < n}, Γ^{n} = {x ∈ Γ | |x| > n}. Then we assume the existence of positive, continuous functions r, b_{0}, b_{1} defined in IR^{N} and constants K_{0}, c_{r} such that for some ñ ≥ 2 the following conditions hold

r(x) ≤ |x|/3 for every x ∈ Dñ ,  (7)
c_{r}^{−1} ≤ r(x)/r(y) ≤ c_{r} for every x ∈ Dñ , y ∈ Q(x, r(x))  (8)
v_{1}(x)r^{−p}(x) ≤ K_{0}v_{0}(x) for a. e. x ∈ Dñ  (9)
w(y) ≤ b_{0}(x) , b_{1}(x) ≤ v_{1}(y) for every x ∈ Dñ , a. e. y ∈ Q(x, r(x))  (10)

Examples for such functions are given in Section 5. We denote by Q(x, r(x)) the cube in IR^{N} centered at x with edge lenght r(x). Condition (7) implies that if Q(x, r(x)) ∩ D^{3n} ≠ ∅ and n ≥ ñ, then |x| > 2n. Consequently, if y ∈ Q(x, r(x)), then |y| ≥ |x| − r(x) ≥ (2/3)|x| > n, which shows that Q(x, r(x)) ⊂ D^{n} if Q(x, r(x)) ∩ D^{3n} ≠ ∅. Finally we define

B_{n,k} = sup_{x∈D^{n}} \frac{b_{0}^{1/q}(x)}{b_{1}^{1/p}(x)} r^{\frac{N−k}{q} − \frac{N}{p} +1}(x) .

In this paper we always suppose that Ω is an unbounded, connected domain with sufficiently smooth boundary Γ = ∂Ω (the exact assumptions on Γ are formulated in Section 4). In particular, Ω satisfies the assumptions of Theorem 1. We define Ω_{n} = D_{n} ∩ Ω, Γ_{n} = D_{n} ∩ Γ. Then Ω = ⋃ Ω_{n}, Γ = ⋃ Γ_{n} and we have the following results:

\underline{THEOREM 3.} Let 1 ≤ p ≤ q < ∞ and \frac{N − 1}{q} − \frac{N}{p} +1 ≥ 0. Assume that W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w) is continuous for every n and r, v_{0}, v_{1}, w satisfy (7)–(10). If lim_{n→∞} B_{n,1} < ∞, then there exists a continuous trace operator W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(∂Ω; w).

\underline{THEOREM 4.} Let 1 ≤ p ≤ q < ∞ and \frac{N − 1}{q} − \frac{N}{p} +1 ≥ 0. Assume that W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w) is compact for every n and r, v_{0}, v_{1}, w satisfy (7)–(10). If lim_{n→∞} B_{n,1} = 0 , then the trace operator W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(∂Ω; w) is compact.

The proof of these theorems is given in Section 4.

3. COMPACT TRACES ON PLANES

The aim of this section is to prove a special case of Theorems 3 and 4, namely

\underline{THEOREM 5.} Let 1 ≤ p ≤ q < ∞ and \frac{N − k}{q} − \frac{N}{p} + 1 ≥ 0 and Γ be the intersection of the domain Ω with an (N − k)− dimensional plane. Denote Ω_{n} = D_{n} ∩ Ω, Γ_{n} = Γ ∩ Ω_{n}, Γ^{n} = Γ \ \overline{Γ}_{n} and let r, v_{0}, v_{1}, w satisfy (7)–(10) Assume that W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w) is compact for every n. If lim_{n→∞} B_{n,k} = 0, then W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is compact. If W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Γ_{n}; w) is continuous for every n and lim_{n→∞} B_{n,k} < ∞, then W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is continuous.

\underline{Proof.} Without restriction we may assume that Γ has the form

Γ = Ω ∩ IR^{N−k} = {x ∈ Ω | x = (x′, 0), x′ ∈ IR^{N−k}} .

First we treat the case Ω = IR^{N}, i. e. Γ = IR^{N−k}. Let Γ_{R}^{n} = {x ∈ Γ^{n} | |x| < R}, where |x| denotes the maximum norm in IR^{N−k}. Since Γñ ; R is bounded, there exists a locally finite covering with cubes

Q^{γ}(x_{j}, r(x_{j})) = {x ∈ IR^{N−k} | |x − x_{j}| < r(x_{j})} (j ∈ IN) ,

such that

∑_{j} χ_{Q^{γ}(x_{j},r(x_{j}))}(η) ≤ κ for every η ∈ IR^{N−k} ,

where χ_{Q^{γ}} is the characteristic function on Q^{γ} and the constant κ only depends on the dimension N − k. This result, known as Besicovitch covering lemma can be found in [5]. We denote by Q(x_{j}, r(x_{j})) the corresponding cube in IR^{N} with same center and edge length.

\overline{}Γñ Q_{j}  Q_{j}^{γ}

Since y ∈ Q(x_{j}, r(x_{j})) implies that the projection π(y) on IR^{N−k} is in Q^{γ}(x_{j}, r(x_{j})), we get

∑_{j} χ_{Q(x_{j},r(x_{j}))}(y) ≤ κ for every y ∈ IR^{N}

with the same κ as above, where χ_{Q} is the characteristic function on Q. We define

I = {j ∈ IN | Q(x_{j}, r(x_{j})) ∩ D^{3ñ} ≠ ∅}.

From (7) it follows that Q(x_{j}, r(x_{j})) ⊂ Dñ if j ∈ I and we get

Γ_{R}^{n} ⊂ ⋃_{j∈I} Q(x_{j}, r(x_{j})) ⊂ Dñ for every n ≥ 3ñ .

This shows that we can use the estimates (8)–(10) for n ≥ 3ñ and j ∈ I. Now let K_{1} denote the Sobolev constant for the trace operator in the unit cube Q(0, 1) centered at the origin, i. e.

∥u∥_{q,Q^{γ}(0,1)} ≤ K_{1} ∥u∥_{1,p,Q(0,1)} .

Then by translation to x and dilation by r we obtain

( ∫_{Q^{γ}(x,r)} |u(η)|^{q} dη)^{1/q} ≤ r^{\frac{N−k}{q}} K_{1} ∥u∥_{1,p,Q(0,1)}

= K_{1}r^{\frac{N−k}{q}} (r^{−N} ∫_{Q(x,r)} |u(y)|^{p} dy + r^{−N+p} ∫_{Q(x,r)} |∇u(y)|^{p} dy)^{1/p}

and from (8)–(10) we get for j ∈ I (denoting r_{j} = r(x_{j}), Q_{j} = Q(x_{j}, r_{j}))

∫_{Q_{j}^{γ}} |u(η)|^{q}w(η) dη ≤ (b_{0}^{1/q}(x_{j})K_{1}r_{j}^{\frac{N−k}{q} − \frac{N}{p} +1})^{q} (r_{j}^{−p} ∫_{Q_{j}} |u(y)|^{p} dy + ∫_{Q_{j}} |∇u(y)|^{p} dy)^{q/p}
≤ K_{1}^{q} (\frac{b_{0}^{1/q}(x_{j})}{b_{1}^{1/p}(x_{j})}r_{j}^{\frac{N−k}{q} − \frac{N}{p} +1})^{q} ·

(c_{r}^{p} ∫_{Q_{j}} |u(y)|^{p} \frac{v_{1}(y)}{r^{p}(y)} dy + ∫_{Q_{j}} |∇u(y)|^{p}v_{1}(y) dy)^{q/p}
≤ K_{1}^{q} B_{n,k}^{q} (c_{r}^{p}K_{0} ∫_{Q_{j}} |u(y)|^{p}v_{0}(y) dy + ∫_{Q_{j}} |∇u(y)|^{p}v_{1}(y) dy)^{q/p}
≤ K B_{n,k}^{q} (∫_{Q_{j}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p} ,

where K = K_{1}^{q} max{c_{r}^{q}K_{0}^{q/p}; 1}. Taking the sum over j ∈ I in the above estimate, we obtain for n ≥ 3ñ (since q/p ≥ 1)

∥u∥_{q,Γ_{R}^{n},w}^{q} ≤ KB_{n,k}^{q} (∑_{j} ∫_{Q_{j}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p}

≤ κ^{q/p}KB_{n,k}^{q} ∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} .

Since the right hand side does not depend on R, we can take the limit R → ∞ and get

∥u∥_{q,Γ^{n},w}^{q} ≤ C B_{n,k}^{q} ∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} .

If B_{n,k} → 0 as n → ∞, we see that condition (2) in Lemma 2 is fulfilled and the compactness of the trace operator follows. If lim_{n→∞} B_{n,k} < ∞, then (4) holds and we obtain continuity of the trace. The general case Ω ⊂ IR^{N} now follows from Theorem 1, since

W^{1,p}(Ω; v_{0}, v_{1}) ⟶^{Λ} W^{1,p}(IR^{N}; v_{0}, v_{1}) ↪^{γ} L^{q}(Γ; w) .

is a composition of a continuous operator with γ compact, resp. continuous. □

If we take in Theorem 5 the intersection of Ω with IR^{N}, we obtain the following

\underline{COROLLARY 6.} Let 1 ≤ p ≤ q < ∞, \frac{N}{q} − \frac{N}{p} + 1 ≥ 0 and r, v_{0}, v_{1}, w satisfy (7)–(10).

If W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Ω_{n}; w) is compact for every n and lim_{n→∞} B_{n,0} = 0 , then W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Ω; w) is compact. If W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Ω_{n}; w) is continuous for every n and lim_{n→∞} B_{n,0} < ∞, then W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Ω; w) is continuous. □

Remark that we have obtained this result without further restrictions on the domain Ω. In particular for weight functions in A_{p} this corollary is a generalization of a result of Opic and Gurka [4].

4. PROOF OF THEOREMS 3 AND 4

Let Γ = ∂Ω be a C^{1}- smooth submanifold of dimension (N − 1) in IR^{N}. We assume that there exists a locally finite covering of Γ with open subsets U_{i} ⊂ IR^{N} having the following properties:

U1 There is a global constant θ such that ∑_{i} χ_{U_{i}}(y) ≤ θ for every y ∈ IR^{N}.

U2 There exist C^{1}- diffeomorphisms φ_{i} : U_{i} → B_{i} ⊂ IR^{N} such that 0 ∈ B_{i} and φ_{i}^{−1}(IR^{N−1} × {0}) = U_{i} ∩ Γ.

U3 The partial derivatives of the coordinate functions φ_{i} and φ_{i}^{−1} are uniformly bounded by a constant K (not depending on i).

Remark. Conditions U1 and U3 are necessary to control the behaviour of Γ at infinity. It is easy to see that these conditions are satisfied for any compact C^{1}− submanifold of IR^{N}. To give an example of an unbounded domain, the boundary of which satisfies the above assumptions, consider an infinite cylinder Q × IR, where Q ⊂ IR^{N−1} is smooth and bounded. Then there exists a finite covering {U_{i}, φ_{i}}_{i=1}^{m} of the compact manifold ∂Q × [0, 1] with properties U1–U3. By translation of U_{i} and φ_{i} in the unbounded direction x_{N}, we obtain a covering of the whole boundary ∂Q × IR, which obviously satisfies U1–U3. Denote B_{i}^{γ} = B_{i} ∩ IR^{N−1} × {0}, and let ρ = r ∘ φ_{i}^{−1} be the weight on B_{i} induced by r. Then there exists a locally finite covering of B_{i}^{γ} with cubes in IR^{N−1}, Q_{j}^{γ} = Q^{γ}(ξ_{j}, ρ(ξ_{j})), ξ_{j} ∈ B_{i}^{γ}. As in the last section we denote Q_{j} = Q(ξ_{j}, ρ(ξ_{j})) the corresponding cube in IR^{N} and we have

∑_{j} χ_{Q_{j}}(z) ≤ κ for every z ∈ IR^{N} .

where κ only depends on N. Next we write ṽ_{0}, ṽ_{1}, w̃, b̃_{0}, b̃_{1} for the transformed functions on B_{i}, i. e. ṽ_{0} = v_{0} ∘ φ_{i}^{−1} and so on. From assumption U3 we obtain the following mapping properties for φ_{i}.

\underline{LEMMA 7.} If U3 holds, then

(i) The Jacobi determinants of φ_{i} and φ_{i}^{−1} are uniformly bounded by K_{2} = K^{N}N! . (ii) For every x ∈ U_{i} ∩ Γñ we have

φ_{i}^{−1}[Q(ξ, ρ(ξ))] ⊂ Q(x, K_{3}r(x)) , where ξ = φ_{i}(x) , K_{3} = NK .

\underline{Proof.} Since the first statement is obvious, we only prove (ii). Denote f = φ_{i}^{−1} and let f_{k}, k = 1, …, N be the components of f. By translation we may assume that ξ = 0 and x = f(0) = 0. For z ∈ Q(0, ρ(0)) we obtain by the mean value theorem

f_{k}(z) = \frac{∂f_{k}}{∂ξ_{1}} (ζ)z_{1} + … + \frac{∂f_{k}}{∂ξ_{N}} (ζ)z_{N} ,

where ζ lies on the straight line between 0 and z. From U3 we get

|f_{k}(z)| ≤ K(|z_{1}| + … + |z_{N}|) ≤ NK |z| ,

where |z| denotes the maximum norm of z. Consequently, |f(z)| ≤ NK ρ(0) = NK r(0) for every z ∈ Q(0, ρ(0)) . □

In the sequel we may assume without restriction that the constant K_{3} in Lemma 7 is equal to 1. For if φ_{i} is a family of coordinate functions with properties U1–U3, then ψ_{i}(x) = K_{3}φ_{i}(x) has the same properties (with different constants K and K_{2}) and ψ_{i}^{−1}(z) = φ_{i}^{−1}(z/K_{3}) satisfies |ψ_{i}^{−1}(z)| ≤ ρ(0) = r(0) for every z ∈ Q(0, ρ(0)). Lemma 7 guarantees that the following estimates analogous to (8)–(10) are valid (with constants c_{r} and K_{0} from (8) and (9)).

c_{r}^{−1} ≤ ρ(ξ)/ρ(z) ≤ c_{r} for every ξ = φ_{i}(x), x ∈ U_{i} ∩ Γñ, z ∈ Q(ξ, ρ(ξ))  (11)
ṽ_{1}(z)ρ^{−p}(z) ≤ K_{0}ṽ_{0}(z) for a. e. z = φ_{i}(x), x ∈ U_{i} ∩ Dñ  (12)
w̃(z) ≤ b̃_{0}(ξ) , b̃_{1}(ξ) ≤ ṽ_{1}(z) for every ξ = φ_{i}(x), x ∈ U_{i} ∩ Γñ, a.e. z ∈ Q(ξ, ρ(ξ)) (13)

Remark that the cubes used in the proof below are always centered at points in φ_{i}(U_{i} ∩ Γñ), so that in our situation it is sufficient that (11) and (13) are satisfied for x ∈ Γñ (and not for all x ∈ Dñ). As first step we want prove the following estimate (for n ≥ 3ñ):

∫_{U_{i}∩Γ_{R}^{n}} |u(η)|^{q}w(η) dη ≤ C B_{n,1}^{q} (∫_{U_{i}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p} .  (14)

Denoting ρ_{j} = ρ(ξ_{j}), we obtain as in Section 3 for n ≥ 3ñ

∫_{U_{i}∩Γ_{R}^{n}} |u(η)|^{q}w(η) dη ≤ K_{2} ∑_{j} ∫_{Q_{j}^{γ}} |u(φ_{i}^{−1}(ξ))|^{q}w̃(ξ)) dξ

≤ K_{2}K_{1}^{q} ∑_{j} (\frac{b̃_{0}^{1/q}(ξ_{j})}{b̃_{1}^{1/p}(ξ_{j})}ρ_{j}^{\frac{N−1}{q} − \frac{N}{p} +1})^{q} ·

(c_{r}^{p}K_{0} ∫_{Q_{j}} |u(φ_{i}^{−1}(z))|^{p}ṽ_{0}(z) dz + ∫_{Q_{j}} |∇(u ∘ φ_{i}^{−1})(z)|^{p}ṽ_{1}(z) dz)^{q/p}

Since b̃_{0}(z_{j}) = b_{0}(x_{j}), b̃_{1}(z_{j}) = b_{1}(x_{j}), ρ(z_{j}) = r(x_{j}) with x_{j} ∈ U_{i} ∩ Γ_{R}^{n} ⊂ D_{R}^{n}, we finally obtain, setting y = φ_{i}^{−1}(z),

∫_{U_{i}∩Γ_{R}^{n}} |u(η)|^{q}w(η) dη ≤ K_{2}K_{1}^{q} B_{n,1}^{q} max{K_{0}c_{r}^{p}; 1}κ^{q/p} ·

(∫_{B_{i}} |u(φ_{i}^{−1}(z))|^{p}ṽ_{0}(z) + |∇(u ∘ φ_{i}^{−1})(z)|^{p}ṽ_{1}(z) dz)^{q/p}
≤ C B_{n,1}^{q} (∫_{U_{i}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p} ,

where we have used U3 to estimate the derivatives of φ_{i}^{−1} and the constant C depends on K, K_{0}, K_{1}, K_{2}, c_{r} and κ. This proves (14). Now let I = {i ∈ IN | U_{i} ∩ Γñ ≠ ∅}; then from U1 and (14) we obtain

∫_{Γ_{R}^{n}} |u(η)|^{q}w(η) dη ≤ ∑_{i∈I} ∫_{U_{i}∩Γ_{R}^{n}} |u(η)|^{q}w(η) dη
≤ C B_{n,1}^{q} ∑_{i∈I} (∫_{U_{i}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p}

≤ C B_{n,1}^{q} (∑_{i∈I} ∫_{U_{i}} |u(y)|^{p}v_{0}(y) + |∇u(y)|^{p}v_{1}(y) dy)^{q/p}
≤ θ^{q/p} C B_{n,1}^{q}∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q} .

In the limit R → ∞ we get for n ≥ 3ñ

∥u∥_{q,Γ^{n},w}^{q} ≤ K B_{n,1}^{q}∥u∥_{1,p,IR^{N} ,v_{0},v_{1}}^{q}

and we can again apply Lemma 2 to the trace operator

W^{1,p}(IR^{N}; v_{0}, v_{1}) ↪ L^{q}(Γ; w) .

The assertions of Theorems 3 and 4 for general domains Ω ⊂ IR^{N} then follow again from Theorem 1. □

5. EXAMPLES

Let B_{1} = {x ∈ IR^{N} | |x| < 1} be the unit ball in IR^{N}. We define

v_{0}(x) = |x|^{β−p} , v_{1}(x) = |x|^{β} , w(x) = |x|^{α} for x ∈ IR^{N} \ B_{1} , α, β ∈ IR

and v_{0} = v_{1} = w ≡ 1 in B_{1}. Then v_{0}, v_{1} and w belong to A_{p} (1 < p < ∞) if α, β and β − p belong to the interval (−N, N(p − 1)); they are in A_{1} if α, β and β − p are in (−N, 0] (cf. [10]). We can now choose r(x) = |x|/3 and b_{0}(x) = (4/3)^{α}w(x), b_{1}(x) = (2/3)^{β}v_{1}(x). If

\frac{α}{q} − \frac{β}{p} + \frac{N − 1}{q} − \frac{N}{p} + 1 ≤ 0

then lim_{n→∞} B_{n,1} < ∞ and Theorem 3 shows that, if

\frac{N − 1}{q} − \frac{N}{p} + 1 ≥ 0 ,

then W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is continuous. If these inequalities are strict, i. e.

\frac{α}{q} − \frac{β}{p} + \frac{N − 1}{q} − \frac{N}{p} + 1 < 0  and  \frac{N − 1}{q} − \frac{N}{p} + 1 > 0 ,

then we conclude from Theorem 4 that W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Γ; w) is compact. Observe that the second inequality means that q is strictly smaller than the critical Sobolev exponent p(N − 1)/(N − p) for the usual trace operator in W^{1,p}.

6. AN APPLICATION TO NONLINEAR BOUNDARY VALUE PROBLEMS

We consider the following semilinear elliptic problem in an unbounded domain Ω ⊂ IR^{N} (N ≥ 3) with smooth boundary Γ

Au ≡ −Δu + b(x)u = f(x, u)  in Ω ,  (15)
∂nu = φ(ξ, u)  on Γ ,  (16)

where b is a positive measurable function and ∂n denotes the outer normal derivative on Γ. We define a bilinear form associated with A by

⟨u, v⟩_{A} = ∫_{Ω} ∇u∇v + b(x)uv dx .

A weak solution of (15)–(16) is then a critical point of the functional

J(u) = \frac{1}{2}⟨u, u⟩_{A} − ∫_{Ω} F (x, u) dx − ∫_{Γ} Φ(ξ, u) dξ ,

where F (x, u) = ∫_{0}^{u} f(x, t) dt and Φ(ξ, u) = ∫_{0}^{u} φ(ξ, t) dt. In this section we always assume that the weight functions v_{0}, v_{1}, w are choosen such that we can apply Theorem 4 and Corollary 6 to ensure that the embedding W^{1,2}(Ω; v_{0}, v_{1}) ↪ L^{p}(Ω; w) and the trace W^{1,2}(Ω; v_{0}, v_{1}) ↪ L^{q}(Γ; w) are compact for 2 < p < 2N/(N − 2), 2 < q < 2(N −1)/(N −2) and continuous for 2 ≤ p ≤ 2N/(N −2), 2 ≤ q ≤ 2(N −1)/(N −2), respectively. Then we have the estimates ∥u∥_{p,Ω,w} ≤ C∥u∥_{v_{0},v_{1}} and ∥u∥_{q,Γ,w} ≤ C∥u∥_{v_{0},v_{1}}, where we used the abbreviation ∥u∥_{v_{0},v_{1}} = ∥u∥_{1,2,Ω,v_{0},v_{1}}. Furthermore, we make the following assumptions:

1^{o} We assume that A defines a continuous bilinear form ⟨·, ·⟩_{A} on W^{1,2}(Ω; v_{0}, v_{1}) and satisfies the ellipticity condition (with some constant K > 0)

⟨u, u⟩_{A} ≥ 2K∥u∥_{v_{0},v_{1}}^{2} for every u ∈ W^{1,2}(Ω; v_{0}, v_{1}) .

2^{o} f and φ are Caratheodory functions on Ω × IR and Γ × IR, respectively, f(x, 0) = φ(ξ, 0) = 0, lim_{t→0} f(x, t)/t = lim_{t→0} φ(ξ, t)/t = 0 uniformly in x and ξ and there exist an open, nonempty subset O ⊂ Ω and a number R > 0 such that F (x, u) > 0 for every u ≥ R , x ∈ O.

3^{o} |f(x, t)| ≤ f_{0}(x) + f_{1}(x)|t|^{p−1} for x ∈ Ω, t ∈ IR, where 2 < p < 2N/(N − 2) and f_{0}, f_{1} are positive functions satisfying

f_{0} ∈ L^{p/(p−1)}(Ω; w^{1/(1−p)}) , f_{0} ≤ Cw , f_{1} ≤ Cw

and analogously |φ(ξ, t)| ≤ φ_{0}(ξ) + φ_{1}(ξ)|t|^{q−1} for ξ ∈ Γ, t ∈ IR, 2 < q < 2(N − 1)/(N − 2), φ_{0}, φ_{1} are positive functions satisfying

φ_{0} ∈ L^{q/(q−1)}(Γ; w^{1/(1−q)}) , φ_{0} ≤ Cw , φ_{1} ≤ Cw

with appropriate constant C.

4^{o} There exists a θ ∈ IR, 0 ≤ θ < 1/2 such that F (x, t) ≤ θf(x, t)t, Φ(ξ, t) ≤ θφ(ξ, t)t for a. e. x ∈ Ω, ξ ∈ Γ and every t ∈ IR.

We denote by N_{f}, N_{F}, N_{φ}, N_{Φ} the corresponding Nemytskii operators. Under the assumptions above we have the following results.

\underline{LEMMA 8.} The operators N_{f} : L^{p}(Ω; w) → L^{p/(p−1)}(Ω; w^{1/(1−p)}), N_{F} : L^{p}(Ω; w) → L^{1}(Ω), N_{φ} : L^{q}(Γ; w) → L^{q/(q−1)}(Γ; w^{1/(1−q)}) and N_{Φ} : L^{q}(Γ; w) → L^{1}(Γ) are bounded and continuous. \underline{Proof.} We only prove the assertions for N_{f} and N_{F}, since the arguments for N_{φ} and N_{Φ} are identical. For u ∈ L^{p}(Ω; w) we get (setting p′ = p/(p − 1))

∫_{Ω} |f(x, u)|^{p′}w^{1/(1−p)}dx ≤ C (∫_{Ω} f_{0}(x)^{p′}w^{1/(1−p)}dx + ∫_{Ω} f_{1}(x)^{p′}|u|^{p′(p−1)}w^{1/(1−p)}dx)
≤ C + C ∫_{Ω} |u|^{p}w^{p/(p−1)}w^{1/(1−p)}dx
= C + C∥u∥_{p,Ω,w}^{p} .

Therefore, N_{f} is bounded. The continuity now follows from standard properties of Nemytski operators (cf. [11]). In the same way we obtain (using Hölder’s inequality)

∫_{Ω} |F (x, u)|dx ≤ ∫_{Ω} f_{0}(x)|u|dx + ∫_{Ω} f_{1}(x)|u|^{p}dx
≤ (∫_{Ω} f_{0}(x)^{p′}w^{1/(1−p)}dx)^{1/p′} (∫_{Ω} |u|^{p}w dx)^{1/p} + C ∫_{Ω} |u|^{p}w dx
≤ C∥u∥_{p,Ω,w} + C∥u∥_{p,Ω,w}^{p} .

This completes the proof. □

\underline{LEMMA 9.} The functional J is Fréchet-differentiable in W^{1,2}(Ω; v_{0}, v_{1}), and the derivative is given by

⟨J′(u), v⟩ = ⟨u, v⟩_{A} − ∫_{Ω} f(x, u)v dx − ∫_{Γ} φ(ξ, u)v dξ .  (17)

\underline{Proof.} Since (17) is the Gateaux derivative of J in u in direction v, we only have to show the continuity of J′, i. e. the continuity of the mapings u ↦ ⟨u, ·⟩_{A}, u ↦ ∫_{Ω} F (x, u) · dx and u ↦ ∫_{Γ} φ(ξ, u) · dx in the dual norm. The first mapping is clearly continuous since we assumed that ⟨·, ·⟩_{A} is a continuous bilinear form. The second mapping is regarded as composition of the following mappings:

W^{1,2}(Ω; v_{0}, v_{1}) ↪^{i} L^{p}(Ω; w) ⟶^{N_{f}} L^{p/(p−1)}(Ω; w^{1/(1−p)}) ⟶^{j} (W^{1,2}(Ω; v_{0}, v_{1}))′ ,

where ⟨j(g), v⟩ = ∫_{Ω} gv dx. Since we know that i and N_{f} are continuous, it suffices to show that j is a bounded operator. By Hölder’s inequality we get

∫_{Ω} |gv|dx ≤ (∫_{Ω} |g|^{p′}w^{1/(1−p)}dx)^{1/p′} (∫_{Ω} |v|^{p}w dx)^{1/p}
≤ C∥g∥_{p/(p−1),Ω,w^{1/(1−p)}}∥v∥_{v_{0},v_{1}} .

Analogously we write the third mapping as

W^{1,2}(Ω; v_{0}, v_{1}) ↪^{γ} L^{q}(Γ; w) ⟶^{N_{φ}} L^{q/(q−1)}(Γ; w^{1/(1−q)}) ⟶^{ℓ} (W^{1,2}(Ω; v_{0}, v_{1}))′ .

Again γ and N_{φ} are continuous and ⟨ℓ(h), v⟩ = ∫_{Γ} h γ(v) dξ satisfies

∫_{Γ} |h γ(v)|dx ≤ (∫_{Γ} |h|^{q′}w^{1/(1−q)}dx)^{1/q′} (∫_{Γ} |γ(v)|^{q}w dx)^{1/q}
≤ C∥h∥_{q/(q−1),Γ,w^{1/(1−q)}}∥v∥_{v_{0},v_{1}} ,

which proves the lemma. □

\underline{LEMMA 10.} J satisfies the Palais-Smale condition on W^{1,2}(Ω; v_{0}, v_{1}).

\underline{Proof.} First we show that every Palais-Smale sequence is bounded. Let u_{k} ∈ W^{1,2}(Ω; v_{0}, v_{1}) be such a sequence, i. e. J′(u_{k}) → 0, |J(u_{k})| ≤ M. From |⟨J′(u_{k}), u_{k}/∥u_{k}∥_{v_{0},v_{1}}⟩| ≤ ε for k large, we obtain

−⟨u_{k}, u_{k}⟩_{A} + ∫_{Ω} f(x, u_{k})u_{k} dx + ∫_{Γ} φ(ξ, u_{k})u_{k} dξ ≤ ε∥u_{k}∥_{v_{0},v_{1}} .

Multiplying this equation with θ from 4^{o} and adding this to the inequality \frac{1}{2}⟨u_{k}, u_{k}⟩_{A} − ∫_{Ω} F (x, u_{k}) dx − ∫_{Γ} Φ(ξ, u_{k}) dξ ≤ M, we obtain

(\frac{1}{2} − θ)⟨u_{k}, u_{k}⟩_{A} + ∫_{Ω} θf(x, u_{k})u_{k} − F (x, u_{k}) dx

+ ∫_{Γ} θφ(ξ, u_{k})u_{k} − Φ(ξ, u_{k}) dξ − θε∥u_{k}∥_{v_{0},v_{1}} ≤ M .  (18)

From 4^{o} we get

(\frac{1}{2} − θ)⟨u_{k}, u_{k}⟩_{A} − θε∥u_{k}∥_{v_{0},v_{1}} ≤ M .

Now 1^{o} implies that ⟨u, u⟩_{A}/∥u∥_{v_{0},v_{1}} → ∞ as ∥u∥_{v_{0},v_{1}} → ∞, which shows that ∥u_{k}∥_{v_{0},v_{1}} must be bounded.

From Theorem 4 and Corollary 6 we get a subsequence (again denoted by u_{k}) which is convergent in L^{p}(Ω; w) with traces convergent in L^{q}(Γ; w). For this subsequence we get

⟨u_{n} − u_{k}, u_{n} − u_{k}⟩_{A} ≤ |⟨u_{n}, u_{n} − u_{k}⟩_{A}| |⟨u_{k}, u_{n} − u_{k}⟩_{A}|
≤ 2ε∥u_{n} − u_{k}∥_{v_{0},v_{1}} +
|∫_{Ω} f(x, u_{n})(u_{n} − u_{k})dx| + |∫_{Ω} f(x, u_{k})(u_{n} − u_{k})dx| +
|∫_{Γ} φ(ξ, u_{n})(u_{n} − u_{k})dξ| + |∫_{Γ} φ(ξ, u_{k})(u_{n} − u_{k})dξ| .

By Hölder’s inequality we get

|∫_{Ω} f(x, u_{n})(u_{n} − u_{k})dx| ≤ ∫_{Ω} |f(x, u_{n})w^{−1/p}| |(u_{n} − u_{k})w^{1/p}| dx
≤ (∫_{Ω} |f(x, u_{n})w^{−1/p}|^{p′}dx)^{1/p′} (∫_{Ω} |u_{n} − u_{k}|^{p}w dx)^{1/p} .

and

|∫_{Γ} φ(ξ, u_{n})(u_{n} − u_{k})dξ| ≤ (∫_{Γ} |φ(ξ, u_{n})w^{−1/q}|^{q′}dξ)^{1/q′} (∫_{Γ} |u_{n} − u_{k}|^{q}w dξ)^{1/q} .

Since ∥u_{n} − u_{k}∥_{p,Ω,w} and ∥u_{n} − u_{k}∥_{q,Γ,w} tend to zero and ∫_{Ω} |f(x, u_{n})|^{p′}w^{1/(1−p)}dx and ∫_{Γ} |φ(ξ, u_{n})|^{q′}w^{1/(1−q)}dξ are bounded (by Lemma 8), it follows that ∥u_{n} − u_{k}∥_{v_{0},v_{1}} tends to zero. □

By means of the Mountain Pass Lemma (cf. [9]) we can now prove

\underline{THEOREM 11.} If f and φ satisfy conditions 2^{o}–4^{o}, and A defines a continuous bilinear form on W^{1,2}(Ω; v_{0}, v_{1}), which satisfies the ellipticity condition 1^{o}, then there exists a nontrivial solution of (15), (16).

\underline{Proof.} From 2^{o}–3^{o} we obtain to every ε > 0 a constant C_{ε} ≥ 0 such that F (x, t) ≤ εf_{0}(x)t^{2} + C_{ε}f_{1}(x)|t|^{p}, 2 < p < 2N/(N − 2) and Φ(ξ, t) ≤ εφ_{0}(ξ)t^{2} + C_{ε}φ_{1}(ξ)|t|^{q}, 2 < q < 2(N − 1)/(N − 2). Consequently

J(u) ≥ \frac{1}{2}⟨u, u⟩_{A} − ∫_{Ω} (εf_{0}(x)u^{2} + C_{ε}f_{1}(x)|u|^{p}) dx − ∫_{Γ} (εφ_{0}(ξ)u^{2} + C_{ε}φ_{1}(ξ)|u|^{q}) dξ
≥ K∥u∥_{v_{0},v_{1}}^{2} − εC∥u∥_{p,Ω,w}^{2} − C_{ε}C∥u∥_{p,Ω,w}^{p} − εC∥u∥_{q,Γ,w}^{2} − C_{ε}C∥u∥_{q,Γ,w}^{q}
≥ (K − εC̃) ∥u∥_{v_{0},v_{1}}^{2} − C̃_{ε}∥u∥_{v_{0},v_{1}}^{p} − C̃_{ε}∥u∥_{v_{0},v_{1}}^{q} .

For ε small we can now choose a ρ > 0 such that J(u) ≥ δ > 0 if ∥u∥_{v_{0},v_{1}} = ρ. On the other hand, by 2^{o} and 4^{o}, we can choose a positive g ∈ W^{1,2}(Ω; v_{0}, v_{1}) with compact support satisfying g ≥ R > 0 in O ⊂ Ω, such that |F (x, λg)| ≥ h(x)|λg|^{p} , p > 2 for λ ≥ 1 with h > 0 in O and get

J(λg) ≤ \frac{1}{2}λ^{2}⟨g, g⟩_{A} − |λ|^{p} ∫_{O} h(x)|g|^{p} dx + c .

For λ → ∞ the right hand side of this inequality tends to −∞. Therefore, we have shown, that the functional J satisfies the Palais-Smale condition, J(0) = 0, J(u) ≥ δ > 0 if ∥u∥_{v_{0},v_{1}} = ρ (small enough) and J(g) ≤ 0 for some g, ∥g∥_{v_{0},v_{1}} > ρ. The Mountain Pass Lemma now implies the existence of a critical point of J. □

As an example we consider the weight functions from Section 5 with β = 0 in W^{1,2}:

v_{0}(x) = |x|^{−2} , v_{1}(x) = 1 , w(x) = |x|^{α} for x ∈ IR^{N} \ B_{1} , α ∈ IR

and v_{0} = v_{1} = w ≡ 1 in B_{1}. They are in A_{2} for −N < α < N. Then A defines a continuous bilinear form and satisfies 1^{o} if there are constants c ≤ C such that

c v_{0}(x) ≤ b(x) ≤ C v_{0}(x) for a. e. x ∈ Ω .

We assume that f and φ satisfy assumptions 2^{o}–4^{o}. To get compactness of the embedding W^{1,2}(Ω; v_{0}, 1) ↪ L^{p}(Ω; w) and of the trace operator W^{1,2}(Ω; v_{0}, 1) ↪ L^{p}(Γ; w) , we have to choose α such that

\frac{α}{p} + \frac{N}{p} − \frac{N}{2} + 1 < 0  and  \frac{α}{q} + \frac{N − 1}{q} − \frac{N}{2} + 1 < 0

where p ∈ (2, 2N/(N − 2)) and q ∈ (2, 2(N − 1)/(N − 2)) are defined by 3^{0}. This inequality is for example satisfied for α = −2. Thus Theorem 11 shows that there exists a nontrivial solution of (15)–(16) in W^{1,2}(Ω; v_{0}, 1).

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