Semilinear Elliptic Problems in Unbounded Domains: Solutions in Weighted Sobolev Spaces

Klaus Pflüger Institut für Mathematik I Freie Universität Berlin Preprint Nr. 21 A 95

Abstract

The semilinear elliptic equation Lu = f(x, u) in an unbounded domain Ω ⊂ IR^{N} is studied. We use several embedding theorems of weighted Sobolev spaces to prove the solvability of this equation under natural growth conditions on f and on the coefficients of the elliptic operator L. This paper gives a unified approach to such problems and generalizes some results in [1], [9] and [13].

email: pflueger@math.fu-berlin.de

1 Introduction

We study the problem

Lu = f(x, u) in Ω,  (1)
u = 0 on ∂Ω, (2)
lim_{|x|→∞} u(x) = 0  (3)

where Ω is an unbounded domain in IR^{N} (or Ω = IR^{N}) and L is a uniformly elliptic operator of the form

Lu = − ∑_{i,j=1}^{N} ∂_{j}(a_{ij}(x)∂_{i}u) + b(x)u

with sufficiently smooth coefficients satisfying a_{ij} = a_{ji} (here ∂_{i} = ∂/∂x_{i}).

In the case, when this problem is radially symmetric (for example if the coefficients of L and the nonlinearity f do not depend on x and Ω = IR^{N}), the solvability in the subcritical case is well known (see [2]). If the coefficients are asymptotically constant, the well-known concentration-compactness priciple of P. L. Lions [6] can be applied to obtain nontrivial solutions. In the present paper we study the case where non of these conditions is satisfied. Our approach via weighted Sobolev spaces generalizes some previous results of [1], [9] and [13].

Corresponding to L we define the bilinear form

⟨u, v⟩_{L} = ∑_{i,j} ∫_{Ω} a_{ij}(x)∂_{i}u∂_{j}v + b(x)uv dx

and the energy functional

J(u) = \frac{1}{2}⟨u, u⟩_{L} − ∫_{Ω} F (x, u) dx ,

where F (x, u) = ∫_{0}^{u} f(x, t)dt is the primitive of f.

Now a weak solution of (1), (2) is obtained as a critical point of J in an appropriate function space. The main result of this paper is the solvability of this problem in certain weighted Sobolev spaces (Theorem 3.4). The exact asymptotic behaviour of these solutions will be studied in a forthcoming paper.

2 Weighted Sobolev Spaces

For x ∈ IR^{N}, ρ > 0 let B(x, ρ) be the open ball in IR^{N} of radius ρ, centered at x. For an (unbounded) domain Ω ⊂ IR^{N} denote Ω_{n} = Ω ∩ B(0, n), and Ω^{n} = int (Ω \ Ω_{n}). We always assume that there exists an ñ ≥ 2 such that

Ωñ = {x ∈ IR^{N} | |x| > ñ}.

For a positive function σ, we define the weighted p-norm (1 ≤ p < ∞)

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

and denote by L^{q}(Ω; w) the space of all measurable functions u such that ∥u∥_{q,Ω,w} is finite. The weighted Sobolev space W^{1,p}(Ω; v_{0}, v_{1}) is defined as the space of all functions u ∈ L^{p}(Ω, v_{0}) such that all derivatives ∂u/∂x_{i} belong to L^{p}(Ω, v_{1}). The corresponding norm is defined by

∥u∥_{1,p,Ω,v_{0},v_{1}} = (∫_{Ω} |∇u(x)|^{p}v_{1}(x) + |u(x)|^{p}v_{0}(x) dx)^{1/p} .

Let r be a positive function defined on Ωñ, satisfying

r(x) ≤ |x|/3 for a. e. x ∈ Ωñ
c_{r}^{−1} ≤ r(y)/r(x) ≤ c_{r}  for a. e. x ∈ Ωñ, y ∈ B(x, r(x)),

where c_{r} ≥ 1 is a constant.

Then the following holds (cf. Kufner/Opic [5], Theorem 18.12/20.5):

Theorem 2.1 Let 1 ≤ p ≤ q < ∞, \frac{N}{q} − \frac{N}{p} +1 ≥ 0 and assume that there are constants 0 ≤ k_{0} ≤ K_{0}, 0 ≤ c_{0} ≤ C_{0}, 0 ≤ c_{1} ≤ C_{1}, and positive measurable functions b_{0}, b_{1}, defined in Ωñ, such that the weight functions v_{0}, v_{1}, w satisfy

k_{0}v_{0}(x) ≤ v_{1}(x)r^{−p}(x) ≤ K_{0}v_{0}(x)  for a. e. x ∈ Ωñ,  (4)
c_{0}b_{0}(x) ≤ w(y) ≤ C_{0}b_{0}(x)  for a. e. x ∈ Ωñ, y ∈ B(x, r(x)), (5)
c_{1}b_{1}(x) ≤ v_{1}(y) ≤ C_{1}b_{1}(x)  for a. e. x ∈ Ωñ, y ∈ B(x, r(x)). (6)

Assume that the embedding W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Ω_{n}; w) is compact for every n ∈ IN. Then the embedding

W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Ω; w)

is compact if and only if

lim_{n→∞} B_{n} = 0 , where B_{n} = sup_{x∈Ω^{n}} \frac{b_{0}^{1/q}(x)}{b_{1}^{1/p}(x)} r^{( \frac{N}{q} − \frac{N}{p} +1)}(x) .

Furthermore this embedding is continuous if W^{1,p}(Ω_{n}; v_{0}, v_{1}) ↪ L^{q}(Ω_{n}; w) is continuous for every n ∈ IN and lim_{n→∞} B_{n} < ∞.

In this paper we are mainly concerned with the following examples.

Example 2.2 (1) Let β and α be real numbers and define

v_{0}(x) = (1 + |x|)^{β−p} , v_{1}(x) = (1 + |x|)^{β} , w(x) = (1 + |x|)^{α} .

If we choose b_{0} = w, b_{1} = v_{1} and r(x) = (1 + |x|)/6, we see that the assumptions of Theorem 2.1 are satisfied, if

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

therefore W^{1,p}(Ω; v_{0}, v_{1}) ↪ L^{q}(Ω; w) is compact.

(2) Define v_{0}(x) = v_{1}(x) = e^{β|x|} , w(x) = e^{α|x|}.

If we choose b_{0} = w, b_{1} = v_{1} and r(x) = 1, the assumptions of Theorem 2.1 are satisfied, if

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

Another useful tool for the study of partial differential equations in unbounded domains is the Hardy-inequality. For this we assume that Ω has the property, that x ∈ Ω implies tx ∈ Ω for every t ≥ 1. Furthermore the weight functions v_{0}, v_{1} are assumed to be radial, i. e. of the form v_{0}(|x|), v_{1}(|x|) and are bounded from below and above on each compact subinterval in (0, ∞). Let p′ be defined by (1/p) + (1/p′) = 1. Then the following holds ([5], Theorem 21.8):

Theorem 2.3 (Hardy-inequality) Assume that there are constants k, t_{0} > 0 such that

v_{0}(t) ≥ kv_{1}(t)t^{−p}  for a. e. t > t_{0},
B_{R}(v_{0}(t)t^{N−1}, v_{1}(t)t^{N−1}, p) < ∞ ,

where

B_{R}(v_{0}(t)t^{N−1}, v_{1}(t)t^{N−1}, p) = sup_{0<x<∞} ∥(v_{0}(t)t^{N−1})^{1/p}∥_{p,(0,x)} ∥(v_{1}(t)t^{N−1})^{−1/p}∥_{p′,(x,∞)} .

Then there is a constant C ≥ 0 such that

∥u∥_{p,Ω,v_{0}} ≤ C ∥∇u∥_{p,Ω,v_{1}}

holds for every u ∈ W^{1,p}(Ω; v_{0}, v_{1}).

Example 2.4 We set p = 2 and

v_{0}(|x|) = (1 + |x|)^{β−2} , v_{1}(|x|) = (1 + |x|)^{β} , β ∈ IR

and obtain v_{0}(t) ≥ \frac{1}{4}v_{1}(t)t^{−2} for t ≥ 1 and B_{R}(v_{0}(t)t^{N−1}, v_{1}(t)t^{N−1}, 2) < ∞ for 0 ≤ β ≤ 2. As a consequence of the Hardy-inequality we get the equivalence of the norms

∥u∥_{1,2,Ω,v_{0},v_{1}}  and  |∥u∥|_{1,2,Ω,v_{1}} = (∫_{Ω} |∇u(x)|^{2}v_{1}(x) dx)^{1/2}

in W^{1,2}(Ω; v_{0}, v_{1}).

In the sequel we write ∥ · ∥_{v_{0},v_{1}} = ∥ · ∥_{1,2,Ω,v_{0},v_{1}} and ∥ · ∥_{q,w} = ∥ · ∥_{q,Ω,w} for the norms in W^{1,2}(Ω; v_{0}, v_{1}) and L^{q}(Ω; w), respectively.

3 Existence results

In this section we always assume that the weight functions v_{0}, v_{1}, w are choosen such that the embedding W^{1,2}(Ω; v_{0}, v_{1}) ↪ L^{q}(Ω; w) is compact for 2 < q < 2N/(N − 2) and continuous for 2 ≤ q ≤ 2N/(N −2). By S_{q} we denote the corresponding embedding constant, ∥u∥_{v_{0},v_{1}} ≤ S_{q}∥u∥_{q,w}. We assume that L defines a continuous bilinear form ⟨·, ·⟩_{L} on W^{1,2}(Ω; v_{0}, v_{1}) and satisfies the ellipticity condition (with some constant K > 0)

⟨u, u⟩_{L} ≥ 2K∥u∥_{v_{0},v_{1}}^{2} for every u ∈ W^{1,2}(Ω; v_{0}, v_{1}) . (7)

We make the following assumptions on f.

f1 f is a Caratheodory function on Ω×IR, f(x, 0) = 0, lim_{t→0} f(x, t)/t = 0 uniformly in x 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.

f2 |f(x, t)| ≤ f_{0}(x) + f_{1}(x)|t|^{q−1} for x ∈ Ω, t ∈ IR, where 2 < q < 2N/(N − 2) and f_{0}, f_{1} are positive functions satisfying

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

with appropriate constant C.

f3 There exists a θ ∈ IR, 0 ≤ θ < 1/2 such that F (x, t) ≤ θf(x, t)t for a. e. x ∈ Ω.

Under these assumptions we obtain the following results.

Lemma 3.1 The Nemytski operator N_{f} associated to f is bounded and continuous as operator N_{f} : L^{q}(Ω; w) → L^{q/(q−1)}(Ω; w^{1/(1−q)}).

The Nemytski operator N_{F} associated to F is bounded and continuous, N_{F} : L^{q}(Ω; w) → L^{1}(Ω).

Proof. For u ∈ L^{q}(Ω; w) we get (q′ = q/(q − 1))

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

≤ C + C ∫_{Ω} |u|^{q}w^{q/(q−1)}w^{1/(1−q)}dx
= C + C∥u∥_{q,w}^{q} .

Therefore, N_{f} is bounded. The continuity now follows from standard properties of Nemytski operators.

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|^{q}dx

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

This completes the proof. □

Lemma 3.2 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⟩_{L} − ∫_{Ω} f(x, u)v dx.  (8)

Proof. Since (8) 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, ·⟩_{L} and u ↦ ∫_{Ω} u · dx in the dual norm. The first mapping is clearly continuous since we assumed that ⟨·, ·⟩_{L} 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^{q}(Ω; w) ⟶^{N_{f}} L^{q/(q−1)}(Ω; w^{1/(1−q)}) ⟶^{j} (W^{1,2}(Ω; v_{0}, v_{1}))′ ,

where ⟨j(φ), v⟩ = ∫_{Ω} φv 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} |φv|dx ≤ (∫_{Ω} |φ|^{q′}w^{1/(1−q)}dx)^{1/q′} (∫_{Ω} |v|^{q}w dx)^{1/q}
≤ C∥φ∥_{q/(q−1),w^{1/(1−q)}}∥v∥_{v_{0},v_{1}} ,

which proves the lemma. □

Lemma 3.3 J satisfies the Palais-Smale condition on W^{1,2}(Ω; v_{0}, v_{1}).

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}⟩_{L} + ∫_{Ω} f(x, u_{k})u_{k} dx ≤ ε∥u_{k}∥_{v_{0},v_{1}} .

Multiplying this equation with θ from f3 and adding this to the inequality \frac{1}{2}⟨u_{k}, u_{k}⟩_{L} − ∫_{Ω} F (x, u_{k}) dx ≤ M, we obtain

(\frac{1}{2} − θ)⟨u_{k}, u_{k}⟩_{L} + ∫_{Ω} θf(x, u_{k})u_{k} − F (x, u_{k}) dx − θε∥u_{k}∥_{v_{0},v_{1}} ≤ M .  (9)

From f3 we get

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

Now (7) implies that ⟨u, u⟩_{L}/∥u∥_{v_{0},v_{1}} → ∞ as ∥u∥_{v_{0},v_{1}} → ∞, which shows that ∥u_{k}∥_{v_{0},v_{1}} must be bounded.

Therefore, we get a subsequence (again denoted by u_{k}) which is convergent in L^{q}(Ω; w). For this subsequence we get

⟨u_{n} − u_{k}, u_{n} − u_{k}⟩_{L} ≤ |⟨u_{n}, u_{n} − u_{k}⟩_{L}| |⟨u_{k}, u_{n} − u_{k}⟩_{L}|
≤ 2ε∥u_{n} − u_{k}∥_{v_{0},v_{1}} +

|∫_{Ω} f(u_{n})(u_{n} − u_{k})dx| + |∫_{Ω} f(u_{k})(u_{n} − u_{k})dx|

By Hölder’s inequality we get

|∫_{Ω} f(u_{n})(u_{n} − u_{k})dx| ≤ ∫_{Ω} |f(u_{n})w^{−1/q}| |(u_{n} − u_{k})w^{1/q}| dx

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

Since ∥u_{n} − u_{k}∥_{q,w} tends to zero and ∫_{Ω} |f(u_{n})|^{q′}w^{1/(1−q)}dx is bounded (by Lemma 3.1), it follows that ∥u_{n} − u_{k}∥_{v_{0},v_{1}} tends to zero. □

By means of the Mountain Pass Lemma (cf. [10]) we can now prove

Theorem 3.4 If f satisfies conditions f1–f3, and L defines a continuous bilinear form on W^{1,2}(Ω; v_{0}, v_{1}), which satisfies the ellipticity condition (7), then there exists a nontrivial positive solution of (1), (2).

Proof. From f1–f2 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|^{q}, 2 < q < 2N/(N − 2). Consequently

J(u) ≥ \frac{1}{2}⟨u, u⟩_{L} − ∫_{Ω} (εf_{0}(x)u^{2} + C_{ε}f_{1}(x)|u|^{q}) dx
≥ K∥u∥_{v_{0},v_{1}}^{2} − εC∥u∥_{q,w}^{2} − C_{ε}C∥u∥_{q,w}^{q}
≥ (K − εC̃) ∥u∥_{v_{0},v_{1}}^{2} − 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 f1 and f3, we can choose a positive v ∈ W^{1,2}(Ω; v_{0}, v_{1}), v ≥ R > 0 in O ⊂ Ω, such that |F (x, λv)| ≥ h(x)|λv|^{q} , q > 2 for λ ≥ 1 with h > 0 in O and get

J(λv) ≤ \frac{1}{2}λ^{2}⟨v, v⟩_{L} − |λ|^{q} ∫_{O} h(x)|v|^{q} 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(v) ≤ 0 for some v, ∥v∥_{v_{0},v_{1}} > ρ. The Mountain Pass Lemma (cf. [10]) now implies the existence of a critical point of J.

To show the existence of a positve solution, we consider the truncated function

f̃(x, u) = \{ \begin{matrix}f(x, u) & , u ≥ 0 \\ 0 & , u < 0 \end{matrix} .

With the same arguments as above we get a solution ũ of the corresponding problem. Let ũ^{+}, ũ^{−} denote the positive and negative parts of ũ, i. e. ũ = ũ^{+} − ũ^{−}. Since f̃(x, ũ^{−}) = 0 we get

⟨ũ^{−}, ũ^{−}⟩_{L} = ⟨ũ, ũ^{−}⟩_{L} = ∫_{Ω} f̃(x, ũ)ũ^{−} dx = 0 ,

which shows ũ^{−} = 0. Since ũ is also a solution of the original problem (1), (2), the proof is complete. □

Remark 3.5 The condition lim_{t→0} f(x, t)/t = 0 in f1 can be ommitted, if we assume instead the following

f4 Let K be as in (7), C from f2 and S_{q} denote the embedding constant. Assume that f_{0} ∈ L^{2}(Ω, w^{−1}) and there exists a number ρ, 0 < ρ < (K/CS_{q}^{q})^{1/(q−2)} (q from f2), such that

∥f_{0}∥_{2,w^{−1}} ≤ \frac{Kρ − CS_{q}^{q}ρ^{q−1}}{2S_{2}} .

Then the statement of Theorem 3.4 remains true.

In fact, the only difference in the proof concerns the estimate J(u) ≥ δ > 0. For ∥u∥_{v_{0},v_{1}} = ρ (ρ from f4) we now obtain

J(u) ≥ \frac{1}{2}⟨u, u⟩_{L} − C∥u∥_{q,w}^{q} − ∫_{Ω} f_{0}(x)u dx
≥ K∥u∥_{v_{0},v_{1}}^{2} − CS_{q}^{q}∥u∥_{v_{0},v_{1}}^{q} − ∥f_{0}∥_{2,w^{−1}}S_{2}∥u∥_{v_{0},v_{1}}
≥ Kρ^{2} − CS_{q}^{q}ρ^{q} − S_{2}ρ\frac{Kρ − CS_{q}^{q}ρ^{q−1}}{2S_{2}}
= \frac{1}{2}(Kρ^{2} − CS_{q}^{q}ρ^{q})
= δ > 0 .

Condition f4 allows to consider also nonhomogeneous problems under some more restrictive assumptions on f. For example, if f(x, u) = f_{0}(x) + f_{1}(x)u^{q−1}, f satisfies f1 – f2 (except lim_{t→0} f(x, t)/t = 0), f_{0} satisfies f4 and additionally f_{0}, f_{1} ∈ L^{1}(Ω), then f3 can be replaced by the weaker condition f3′ There exists a θ ∈ IR, 0 ≤ θ < 1/2 and R ≥ 0 such that F (x, t) ≤ θf(x, t)t for

every x ∈ Ω, t ≥ R.

The only difference in the arguments above now concerns the proof of Lemma 3.3. Let u_{k} be a Palais-Smale sequence for J. From the estimates

∫_{Ω∩{u_{k}(x)<R}} |θf(x, u_{k})u_{k}|dx ≤ C ∫_{Ω} |f_{0}(x)| + |f_{1}(x)|dx ≤ C_{1} ,
∫_{Ω∩{u_{k}(x)<R}} |F (x, u_{k})|dx ≤ C_{2} ,

we obtain ∫_{Ω} θf(x, u_{k})u_{k} − F (x, u_{k}) dx ≥ −(C_{1} + C_{2}). Then (9) leads to

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

Again we conclude that ∥u_{k}∥_{v_{0},v_{1}} must be bounded.

We remark that f3′ can be satisfied for example if f_{0} ≤ f_{1}, with θ > 1/q and R ≥ [(1 − θ)/(θ − 1/q)]^{1/(q−1)}.

4 Examples

We consider the problem

−Δu + b(x)u = f(x, u) in Ω,  (10)
u = 0 on ∂Ω, (11)
lim_{|x|→∞} u(x) = 0 ,  (12)

where c(x) ≥ 0 is in L^{∞}(Ω).

For the first example we consider the weight functions

v_{0}(x) = (1 + |x|)^{−2} , v_{1}(x) = 1 , w(x) = (1 + |x|)^{−α} (α > 0)

and we assume that f satisfies f1–f3 from Section 3. To get compactness of the embedding W^{1,2}(Ω; (1 + |x|)^{−2}, 1) ↪ L^{q}(Ω; (1 + |x|)^{−α}), we have to choose α such that

− \frac{α}{q} + \frac{N}{q} − \frac{N}{p} + 1 < 0 ,  (13)

where q ∈ (2, 2N/(N − 2)) is defined by f2 (see Example 2.2(1) with β = 0). This inequality is satisfied for α > [2N − q(N − 2)]/2, in particular for α = 2. On the other hand, if 0 < α < 2, then (13) holds only for

q − 1 > \frac{N + 2 − 2α}{N − 2}

(this is exactly condition (3) in [9]). Furthermore, the Hardy-inequality (Example 2.4)

shows that the norms ∥u∥_{v_{0},v_{1}} and |∥u∥|_{1,2,Ω,v_{1}} = (∫_{Ω} |∇u(x)|^{2}dx)^{1/2} are equivalent.

Consequently, if 0 ≤ b(x) ≤ Cv_{0}(x), then the norm ⟨·, ·⟩_{L}^{1/2} defined by L = −Δ + b is equivalent to ∥ · ∥_{v_{0},v_{1}}. Therefore, ⟨·, ·⟩_{L} is a continuous bilinear form on W^{1,2}(Ω; (1 + |x|)^{−2}, 1), which satisfies the ellipticity condition (7). Theorem 3.4 now shows that there is a positive solution of problem (10)–(12). Note that this includes also the socalled zero mass case (b = 0). This result is essentially the same as in [9] and [1], although the assumptions on f are somewhat different.

For the next example we assume b(x) ≥ b_{0} > 0 and choose

v_{0} = v_{1} = 1 , w(x) = e^{−α|x|} (α > 0).

From Example 2.2(2) we obtain compactness of the embedding W^{1,2}(Ω) ↪ L^{q}(Ω; e^{−α|x|}) for every α > 0, 2 < q < 2N/(N − 2). Clearly, L = −Δ + b defines a continuous bilinear form and satisfies (7). If f satisfies f1–f3, then Theorem 3.4 again gives a positive solution to (10)–(12).

Remark that if f is given as a function of the form

f(x, t) = ∑_{i=1}^{m} g_{i}(x)t^{k_{i}} , 1 < k_{1} < … < k_{m} < (N + 2)/(N − 2) ,

then assumption f2 reduces to |g_{i}(x)| ≤ Ce^{−α|x|}, i = 1, …, m. In particular, if sup_{x∈Ω} |g_{i}(x)e^{α|x|}| < ∞, then f2 holds. Therefore, our assumptions are somewhat weaker that those in [13], where (10)–(12) was solved by a fixed point method in a weighted subspace of C^{0}(IR^{N}).

Of course, instead of −Δ + b we can take any other elliptic operator in (10), which defines a continuous bilinear form and satisfies (7).

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