A SEMILINEAR ELLIPTIC PROBLEM

IN AN UNBOUNDED DOMAIN

Klaus Pflüger

Let Ω = D ×ℝ^{m} (m ≥ 1) be an unbounded domain in ℝ^{n} (n ≥ 2), where D ⊂ ℝ^{n−m} is bounded, with smooth boundary Γ = ∂D ×ℝ^{m}. In this paper we study the problem

(1)  Au ≡ − ∑_{i,j=1}^{n} ∂_{j}(a_{ij}(x)∂_{i}u) + b(x)u = f(x, u)  in Ω,

(2)  Bu ≡ ∑_{i,j=1}^{n} n_{j}(ξ)a_{ij}(ξ)∂_{i}u = φ(ξ, u)  on Γ,
(3)  u(x) ⟶ 0  as |x| → ∞,

where A is a uniformly elliptic operator and n_{j} (j = 1, …, n) are the components of the outer normal vector on Γ. It is assumed that f and φ are of subcritical growth in the second variable. Problems of this kind arise for example in electromagnetic wave propagation in fibres. In [4] the existence of a nontrivial solution to this problem was shown in the special case

f(x, u) = P (x)|u|^{p−1}u, φ(ξ, u) = Q(ξ)|u|^{p−1}u, 1 < p < n/(n − 2),

(n ≥ 3), where a Mountain Pass Lemma approach on a sequence of bounded subdomains of Ω and a comparison argument were used. In this paper we make the following assumptions.

1^{∘} The coefficients a_{ij} are in C^{1}(\overline{Ω}), b is an L^{∞}-function satisfying b(x) ≥ b_{0} > 0, and A satisfies the ellipticity condition ∑_{i,j=1}^{n} a_{ij}(x)ζ_{i}ζ_{j} ≥ α|ζ|^{2} for every ζ ∈ ℝ^{n}. 2^{∘} The functions f(x, t), φ(ξ, t) are continuously differentiable with respect to t, ∂f/∂t and ∂φ/∂t are bounded and uniformly continuous on Ω×[−T, T] (resp. Γ × [−T, T]) for each T > 0, and there exists a real θ ∈ (0, 1) such that for every t ≠ 0

0 ≤ t^{−1}f(x, t) ≤ θ \frac{∂f}{∂t} (x, t), 0 ≤ t^{−1}φ(ξ, t) ≤ θ \frac{∂φ}{∂t}(ξ, t).

3^{∘} We have

lim_{|t|→∞} f(x, t)/|t|^{(n+2)/(n−2)} = lim_{|t|→∞} φ(ξ, t)/|t|^{n/(n−2)} = 0,

lim sup_{|t|→0} f(x, t)/t ≤ 0, lim sup_{|t|→0} φ(ξ, t)/t ≤ 0, and lim sup_{|t|→∞} |f(x, t)| > 0

uniformly in x and ξ. (If n = 2, (n + 2)/(n − 2) and n/(n − 2) may be replaced by any finite power p.)

Assumption 1^{∘} implies that the norm

∥u∥_{A}^{2} = ∫_{Ω} (∑ a_{ij}(x)∂_{i}u∂_{j}u + b(x)u^{2}) dx

is equivalent to the usual W^{1,2}(Ω)-norm. We define

E(u) = \frac{1}{2} ∫_{Ω} (∑ a_{ij}(x)∂_{i}u∂_{j}u + b(x)u^{2}) dx

− ∫_{Ω} F (x, u) dx − ∫_{Γ} Φ(ξ, u) dΓ,

J(u) = ∫_{Ω} (∑ a_{ij}(x)∂_{i}u∂_{j}u + b(x)u^{2}) dx

− ∫_{Ω} f(x, u)u dx − ∫_{Γ} φ(ξ, u)u dΓ,

where F and Φ are the indefinite integrals of f and φ, respectively. We consider the following minimization problem which goes back to Nehari [2]:

(4)  I = inf{E(u): u ∈ W^{1,2}(Ω), u ≠ 0, J(u) = 0}.

It follows from our assumptions that every u ≠ 0 with J(u) = 0 satisfies E(u) ≥ δ > 0 and ⟨J′(u), u⟩ < 0. Therefore 0 is an isolated point in the set {u ∈ W^{1,2}(Ω) | J(u) = 0}, and the Lagrange multiplier appearing in (4) is zero. Thus a minimum is a weak solution of (1)–(3). The main result of this paper is the existence of a weak solution in the case of asymptotically constant coefficients. In addition to 1^{∘}–3^{∘} we assume that

4^{∘} a_{ij}(x) → â_{ij}, b(x) → b̂ > 0 as |x| → ∞ and f(x, t) → f̂(t), φ(ξ, t) → φ̂(t) as |x| → ∞, |ξ| → ∞, uniformly in t ∈ [−T, T] for every T.

We also consider the problem at infinity: Define

E^{∞}(u) = \frac{1}{2} ∫_{Ω} (∑ â_{ij}∂_{i}u∂_{j}u + b̂u^{2}) dx − ∫_{Ω} F̂(u) dx − ∫_{Γ} Φ̂(u) dΓ,

J^{∞}(u) = ∫_{Ω} (∑ â_{ij}∂_{i}u∂_{j}u + b̂u^{2}) dx − ∫_{Ω} f̂(u)u dx − ∫_{Γ} φ̂(u)u dΓ,

and

(5)  I^{∞} = inf{E^{∞}(u): u ∈ W^{1,2}(Ω), u ≠ 0, J^{∞}(u) = 0}.

We remark that the inequalities 0 < I ≤ I^{∞} always hold, since E^{∞} and J^{∞} are invariant with respect to translations of the form (0, …, 0, η_{1}, …, η_{m}).

Theorem 1. Under the conditions 1^{∘}–4^{∘}, every minimizing sequence for (4) is relatively compact up to translations of the form (0, …, 0, η_{1}, …, η_{m}).

Every minimizing sequence for (4) is relatively compact if and only if I < I^{∞}. In this case there exists a minimum of (4) which is a nontrivial solution of (1)–(3).

Proof. The proof is based on the concentration–compactness principle of P. L. Lions [1]. For a minimizing sequence u_{k} for (4) we define ρ_{k} = |∇u_{k}|^{2}+ |u_{k}|^{2}. We see from 3^{∘} that for every ε > 0 we can find a constant C_{ε} > 0 such that the estimates

f(x, u)u ≤ ε|u|^{2} + C_{ε}|u|^{p}, φ(ξ, u)u ≤ ε|u|^{2} + C_{ε}|u|^{q}

hold for some 2 < p < 2n/(n − 2), 2 < q < 2(n − 1)/(n − 2). From the constraint J(u_{k}) = 0 we deduce that

∥u_{k}∥_{A}^{2} = ∫_{Ω} f(x, u_{k})u_{k} dx + ∫_{Γ} φ(ξ, u_{k})u_{k} dΓ
≤ ε(∥u_{k}∥_{L^{2}}^{2} + ∥u_{k}∥_{L^{2}(Γ)}^{2}) + C_{ε}(∥u_{k}∥_{L^{p}}^{p} + ∥u_{k}∥_{L^{q}(Γ)}^{q})
≤ εC∥u_{k}∥_{A}^{2} + C_{ε}C(∥u_{k}∥_{A}^{p} + ∥u_{k}∥_{A}^{q}),

which shows that

(6)  ∥u_{k}∥_{A} ≥ const. > 0.

Therefore we can assume that (at least for a subsequence of ρ_{k}) λ_{k} = ∫_{Ω} ρ_{k}(x) dx → λ > 0 as k → ∞. Now Lemma I.1 from [1, I] states that there is a subsequence (denoted again by ρ_{k}) satisfying one of the following conditions:

(i) Vanishing. lim_{k→∞} sup_{y∈Ω} ∫_{y+B_{R}∩Ω} ρ_{k}(x) dx = 0 for every R < ∞.

(ii) Dichotomy. There exists an α ∈ (0, λ) such that for every ε > 0 there is a k_{0} ≥ 1 and positive functions ρ_{k}^{1}, ρ_{k}^{2} ∈ L^{1}(Ω) which satisfy for k ≥ k_{0}

∥ρ_{k} − (ρ_{k}^{1} + ρ_{k}^{2})∥_{L^{1}} ≤ ε, | ∫_{Ω} ρ_{k}^{1}(x) dx − α| ≤ ε, | ∫_{Ω} ρ_{k}^{2}(x) dx − (λ − α)| ≤ ε

and dist(supp ρ_{k}^{1}, supp ρ_{k}^{2}) → ∞ as k → ∞. (iii) Compactness. There exists a sequence (y_{k})_{k} in Ω such that for every

ε > 0 there is an R < ∞ satisfying ∫_{y_{k}+B_{R}∩Ω} ρ_{k}(x) dx ≥ λ − ε.

First we show that the case (i) cannot occur. By integration we get from 2^{∘} the estimates

(7) 0 ≤ F (x, t) ≤ \frac{θ}{1 + θ} f(x, t)t, 0 ≤ Φ(ξ, t) ≤ \frac{θ}{1 + θ} φ(ξ, t)t, t ∈ ℝ

and from the constraint J(u_{k}) = 0 we deduce that

C ≥ E(u_{k}) = ∫_{Ω} \frac{1}{2}f(x, u_{k})u_{k} − F (x, u_{k}) dx + ∫_{Γ} \frac{1}{2}φ(ξ, u_{k})u_{k} − Φ(ξ, u_{k}) dΓ

≥ (\frac{1}{2} − \frac{θ}{1 + θ} ) (∫_{Ω} f(x, u_{k})u_{k} dx + ∫_{Γ} φ(ξ, u_{k})u_{k} dΓ)

= (\frac{1}{2} − \frac{θ}{1 + θ} ) ∥u_{k}∥_{A}^{2},

which shows that u_{k} is bounded in W^{1,2}(Ω).

Now if lim_{k→∞} sup_{y∈Ω} ∫_{y+B_{R}∩Ω} ρ_{k}(x) dx = 0, we conclude from Lemma I.1 in

[1, II] that u_{k} → 0 in L^{p}(Ω), 2 < p < 2n/(n − 2). The same argument then shows that u_{k} → 0 in L^{q}(Γ), 2 < q < 2(n − 1)/(n − 2) (in the sense of traces). Then we obtain ∫_{Ω} f(x, u_{k})u_{k} dx → 0 and ∫_{Γ} φ(ξ, u_{k})u_{k} dΓ → 0, and from J(u_{k}) = 0 we finally get ∥u_{k}∥_{A} → 0, which contradicts (6).

Next we show that the case (ii) is also impossible. For α ∈ ℝ define

I_{α} = inf\{E(u) − \frac{1}{2}J(u): J(u) = α\}

(and I_{α}^{∞} correspondingly). From (7) we infer that I_{α}, I_{α}^{∞} ≥ 0 for every α ∈ ℝ and I < I_{α}, I^{∞} < I_{α}^{∞} if α < 0 (see [1, II]). From (ii) we obtain sequences u_{k}^{1}, u_{k}^{2} satisfying ∥u_{k} − (u_{k}^{1} + u_{k}^{2})∥_{W^{1,2}} ≤ ε for large k and dist(supp u_{k}^{1}, supp u_{k}^{2}) → ∞. Since u ↦ f(x, u)u and u ↦ φ(ξ, u)u define continuous Nemytskii operators from L^{p} to L^{1} (cf. [5]), we obtain

∫_{Ω} |f(x, u_{k})u_{k} − (f(x, u_{k}^{1})u_{k}^{1} + f(x, u_{k}^{2})u_{k}^{2})| dx

+ ∫_{Γ} |φ(ξ, u_{k})u_{k} − (φ(ξ, u_{k}^{1})u_{k}^{1} + φ(ξ, u_{k}^{2})u_{k}^{2})| dΓ → 0.

Consequently, at least for a subsequence, we get J(u_{k}^{1}) → β_{1}, J(u_{k}^{2}) → β_{2} with β_{1} = −β_{2} and

I = lim (E(u_{k}) − \frac{1}{2}J(u_{k}))
≥ lim inf (E(u_{k}^{1}) − \frac{1}{2}J(u_{k}^{1})) + lim inf (E(u_{k}^{2}) − \frac{1}{2}J(u_{k}^{2})).

Without loss of generality we may assume that supp u_{k}^{2} tends to ∞, while supp u_{k}^{1} remains bounded. Then we conclude that I ≥ I_{β_{1}} + I_{β_{2}}^{∞}. The following three cases should be considered:

if β_{1} = β_{2} = 0, then I ≥ I + I^{∞} > I, since I^{∞} > 0;

if β_{1} < 0, then I < I_{β_{1}} and I ≥ I_{β_{1}} + I_{β_{2}}^{∞} ≥ I_{β_{1}};

if β_{1} > 0, then I ≥ I_{β_{1}} + I_{β_{2}}^{∞} ≥ I_{β_{2}}^{∞} > I^{∞}.

In all three cases we arrive at a contradiction.

This shows that the case (iii) holds. If the sequence (y_{k})_{k} is bounded, we may assume that y_{k} = 0 and then, from the embedding and trace theorem, there is a subsequence satisfying u_{k} → u in L^{p}(Ω), 2 < p < 2n/(n − 2), and in L^{q}(Γ), 2 < q < 2(n − 1)/(n − 2), respectively. Therefore we have ∫_{Ω} f(x, u_{k})u_{k} dx → ∫_{Ω} f(x, u)u dx, ∫_{Ω} F (x, u_{k}) dx → ∫_{Ω} F (x, u) dx, ∫_{Γ} φ(ξ, u_{k})u_{k} dΓ → ∫_{Γ} φ(ξ, u)u dΓ, and ∫_{Γ} Φ(ξ, u_{k}) dΓ → ∫_{Γ} Φ(ξ, u) dΓ. Since J(u_{k}) = 0, we get J(u) ≤ 0, E(u) ≤ lim E(u_{k}) = I and E(u) = I + \frac{1}{2}J(u). If J(u) = α < 0, then E(u) − \frac{1}{2}J(u) ≥ I_{α} > I, which is a contradiction. Therefore J(u) = 0, E(u) = I and u_{k} → u strongly in W^{1,2}(Ω).

Now suppose that y_{k} → ∞. We may assume that y_{k} = (0, …, 0, η_{k,1}, …, η_{k,m}). Then for the sequence v_{k} = u_{k}(· + y_{k}) we infer as above that v_{k} → v in L^{p}(Ω), L^{q}(Γ), J^{∞}(v) = 0, E^{∞}(v) = I^{∞} = I, and v_{k} → v strongly in W^{1,2}(Ω). If I < I^{∞}, then the case y_{k} → ∞ is not possible. On the other hand, if I = I^{∞}, then we can construct a minimizing sequence which is not compact. □

From Theorem 1 we get immediately the following

Corollary 2. In the case of constant coefficients a_{ij}(x) = â_{ij}, b(x) = b̂, f(x, t) = f̂(t), φ(ξ, t) = φ̂(t), there exists a minimum of (5) which is a solution of (1)–(3).

If the coefficients and nonlinearities satisfy the inequalities

(8)  a_{ij}(x) ≤ â_{ij}, b(x) ≤ b̂, f(x, t)t ≥ f̂(t)t, φ(ξ, t)t ≥ φ̂(t)t

then we have the following corollary. Corollary 3. Let (8) be satisfied, and assume that at least one of the first three inequalities in (8) is strict. If in addition

F̂(t)−F (x, t) ≤ \frac{1}{2}(f̂(t)t−f(x, t)t) and Φ̂(t)−Φ(ξ, t) ≤ \frac{1}{2}(φ̂(t)t−φ(ξ, t)t),

then I < I^{∞}, i.e., every minimizing sequence of (4) is relatively compact and there exists a nontrivial solution of (1)–(3). Proof. Let u_{0} be a minimizer for I^{∞}, J^{∞}(u_{0}) = 0, E^{∞}(u_{0}) = I^{∞}. From the strictness of one of the inequalities in (8) we may assume that J(u_{0}) = α < 0. Then

α = ∫_{Ω} (∑ a_{ij}(x)∂_{i}u_{0}∂_{j}u_{0} + b(x)u_{0}^{2}) dx − ∫_{Ω} (∑ â_{ij}∂_{i}u_{0}∂_{j}u_{0} + b̂u_{0}^{2}) dx

+ ∫_{Ω} (f̂(u_{0})u_{0} − f(x, u_{0})u_{0}) dx + ∫_{Γ} (φ̂(u_{0})u_{0} − φ(ξ, u_{0})u_{0})dΓ

≥ ∫_{Ω} (∑ a_{ij}(x)∂_{i}u_{0}∂_{j}u_{0} + b(x)u_{0}^{2}) dx − ∫_{Ω} (∑ â_{ij}∂_{i}u_{0}∂_{j}u_{0} + b̂u_{0}^{2}) dx

+ 2 ∫_{Ω} (F̂(u_{0}) − F (x, u_{0})) dx + 2 ∫_{Γ} (Φ̂(u_{0}) − Φ(ξ, u_{0})) dΓ,

and we obtain

I_{α} ≤ E(u_{0}) − \frac{1}{2}J(u_{0}) = E(u_{0}) − \frac{α}{2}

≤ \frac{1}{2} ∫_{Ω} (∑ â_{ij}∂_{i}u_{0}∂_{j}u_{0} + b̂u_{0}^{2}) dx − ∫_{Ω} F̂(u_{0}) dx − ∫_{Γ} Φ̂(u_{0}) dΓ = I^{∞}.

Since I < I_{α}, the proof is complete. □

References

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