USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 343–357. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Nontrivial solutions of semilinear elliptic systems in ℝ^{N} ^{∗}

Jianfu Yang

Abstract We establish an existence result for strongly indefinite semilinear el-

liptic systems in ℝ^{N}.

1 Introduction

The main objective of this paper is to establish existence results for the semilinear elliptic system

−Δu + u = g(x, v), −Δv + v = f(x, u) inℝ^{N},  (1.1)
u(x) → 0 and v(x) → 0 as |x| → ∞.  (1.2)

The existence of solutions of (1.1)-(1.2) is usually investigated by finding critical points of a related functional. Typical features of the problem are that firstly, the related functional is strongly indefinite; secondly, the growths of f in u and g in v at infinity may not be ‘symmetric’; and lastly, Sobolev embeddings in general are not compact, then the Palais - Smale condition may not be satisfied. Existence results were recently obtained in [12] and [15] in bounded domains. The arguments lie in the decomposition of Sobolev spaces by eigenfunctions of Laplacian operator and a use of linking theorems. Using spectral family theory of non-compact operator, the author and Figueiredo [13] find a suitable linking structure for the functional associate to (1.1)-(1.2) and prove that problem (1.1)- (1.2) possesses at least a positive solution if f and g depend on the variable x radially. Furthermore, it is also shown in [13] that all positive solutions of problem (1.1)-(1.2) are exponentially decaying. In this paper, we establish existence results for general cases. Assume that

H1) f, g : ℝ^{N} × ℝ → ℝ are measurable in the first variable, continuous in the second variable. Both F (x, t) = ∫_{0}^{t} f(x, s)ds and G(x, t) = ∫_{0}^{t} g(x, s)ds are increasing and strictly convex in t.

H2) lim_{t→0} f(x, t)/t = 0, lim_{t→0} g(x, t)/t = 0 uniformly in x ∈ ℝ^{N}.

∗Mathematics Subject Classifications: 35J50, 35J55.

Key words: indefinite, semilinear, elliptic system.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

H3) There is a constant c > 0 such that |f(x, t)| ≤ c(|t|^{p} + 1) and |g(x, t)| ≤ c(|t|^{q} + 1), where 0 < p, q < (N + 2)/(N − 2), N ≥ 3.

H4) There are constants α, β > 2 such that 0 < αF (x, t) ≤ tf(x, t) and 0 < βG(x, t) ≤ tg(x, t), for t ≠ 0.

H5) f(x, t) → f̄(t) and g(x, t) → ḡ(t) uniformly for t bounded as |x| → ∞. |f(x, t) − f̄(t)| ≤ ϵ(R)|t| and |g(x, t) − ḡ(t)| ≤ ϵ(R)|t| whenever |x| ≥ R, |t| ≤ δ, where ϵ(R) → ∞ as R → ∞.

H6) F (x, t) ≥ F̄(t) and G(x, t) ≥ Ḡ(t), meas{x ∈ R^{N} : f(x, t) ≢ f̄(t)} > 0 or meas{x ∈ R^{N} : g(x, t) ≢ ḡ(t)} > 0.

H7) Both f̄(t)/t and ḡ(t)/t are increasing in t.

Our main result is as follows.

Theorem 1.1 Assume (H1)-(H7). Then problem (1.1)-(1.2) possesses at least one nontrivial exponentially decaying solution.

The restriction of exponents in (H3) is due to the fact that we only know the decaying law in the case.

We analyze the convergence of Palais-Smale sequence of associate functional to (1.1)-(1.2) in Section 3. It is shown that the energy levels of solutions of the related autonomous system

−Δu + u = ḡ(v), −Δv + v = f̄(u) in ℝ^{N},  (1.3)
u(x) → 0, v(x) → 0 as |x| → 0 .  (1.4)

are obstacle levels preventing strong convergence of Palais-Smale sequences of (1.1)-(1.2). The possible critical values to be found are between obstacle levels. To retain the compactness, we have to get control of critical values. It is harder to handle critical values described by linking structure than that by the Mountain Pass Theorem. We use dual variational method as [3], [4] and [11]. The method is of the advantage avoiding the indefinite character of original functional. We start with problem (1.1)-(1.2) in bounded domains. Although existence result in the case is known, it has no control of critical values. We establish in Section 2 an existence result by the Mountain Pass Theorem and bound uniformly corresponding critical values by the energy level of ground state of problem (1.3)-(1.4). Then we construct a Palais - Smale sequence for the functional associated to problem (1.1)-(1.2). Theorem 1.1 is proved in Section 4.

2 Existence results in bounded domains

Let Ω be a bounded domain. We consider the problem

−Δu + u = g(x, v), −Δv + v = f(x, u) in Ω,  (2.1)
u = 0, v = 0 on ∂Ω .  (2.2)

The solutions of (2.1)-(2.2) will be found by looking for critical points of associate functional. The main result in this section is as follows.

Theorem 2.1 Assume (H1) − (H4). Then problem (2.1)-(2.2) possesses at least a nontrivial solution.

To prove Theorem 2.1 we will need the lemmas below. First we define the dual functional associate to (2.1)-(2.2). It is well known that the inclusions

i_{r} : W_{o}^{1,r}(Ω) → L^{p+1}(Ω), i_{s} : W_{o}^{1,s}(Ω) → L^{q+1}(Ω)

are compact if 2 < p + 1 < \frac{rN}{N−r} , N > r and 2 < q + 1 < \frac{sN}{N−s}, N > s. The operator −Δ + id : W_{o}^{1,r}(Ω) → W_{o}^{−1,r′}(Ω) is an isomorphism, where r′ = \frac{r}{r−1}. Hence

T = i_{2}(−Δ + id)^{−1}i_{2}^{∗} : L^{1+1/q}(Ω) → L^{p+1}(Ω).

is continuous. Denote by X = L^{p+1}(Ω)×L^{q+1}(Ω), X^{∗} = L^{1+1/p}(Ω)×L^{1+1/q}(Ω) and let

A = ( \begin{matrix}0 & T \\ T & 0 \end{matrix} ) , K = A^{−1} = ( \begin{matrix}0 & T^{−1} \\ T^{−1} & 0 \end{matrix} ) .

For each x, the Legendre-Fenchel transformations F^{∗}(x, ·) of F (x, ·), and G^{∗}(x, ·) of G(x, ·) are defined by

F^{∗}(x, s) = sup_{t∈R}{st − F (x, t)}, G^{∗}(x, s) = sup_{t∈R}{st − G(x, t)}  (2.3)

respectively. Equivalently, we have

F^{∗}(x, s) = st − F (x, t) with s = f(x, t), t = F_{s}^{∗′}(x, s)  (2.4)

and

G^{∗}(x, s) = st − G(x, t) with s = g(x, t), t = G_{s}^{∗′}(x, s).  (2.5)

In the same way, we define F̄^{∗}, Ḡ^{∗} for F̄, Ḡ. By (H6) and properties of Legendre-Fenchel transformations, we have

F^{∗}(x, s) ≤ F̄^{∗}(s), G^{∗}(x, s) ≤ Ḡ^{∗}(s).  (2.6)

We may verify following properties of F^{∗}, G^{∗} in Lemmas 2.2 and 2.3 as [3], [10] and [16].

Lemma 2.2 F^{∗}, G^{∗} ∈ C^{1} and

F^{∗}(x, s) ≥ (1 − \frac{1}{α})sF^{∗′}(x, s), G^{∗}(x, s) ≥ (1 − \frac{1}{β})sG^{∗′}(x, s),  (2.7)
F^{∗}(s, x) ≥ C|s|^{1+1/p} − C, G^{∗}(x, s) ≥ C|s|^{1+1/q} − C.  (2.8)

Lemma 2.3 There exist δ > 0, C_{δ} and C′_{δ} > 0 such that

F^{∗}(x, s) ≥ \{ \begin{matrix}C_{δ}|s|^{2}, & if |s| ≤ δ \\ C′_{δ}|s|^{1+ \frac{1}{p}} , & if |s| ≥ δ \end{matrix} , G^{∗}(x, s) ≥ \{ \begin{matrix}C_{δ}|s|^{2}, & if |s| ≤ δ \\ C′_{δ}|s|^{1+ \frac{1}{q}} , & if |s| ≥ δ \end{matrix} .

We may verify that the dual functional

Ψ(w) = Ψ_{Ω}(w) = ∫_{Ω}(F^{∗}(x, w_{1}) + G^{∗}(x, w_{2})) dx − \frac{1}{2} ∫_{Ω}⟨w, Kw⟩ dx,

is well defined and C^{1} on X^{∗}. A critical point w of Ψ satisfies

(−Δ + id)^{−1}w_{2} = F_{s}^{∗′}(x, w_{1}) and (−Δ + id)^{−1}w_{1} = G_{s}^{∗′}(x, w_{2}).

Let

u = (−Δ + id)^{−1}w_{2}, v = (−Δ + id)^{−1}w_{1}.

Then (u, v) satisfies (2.1)-(2.2). Furthermore, denoting by

Φ(z) = ∫_{Ω}(∇u∇v + uv) dx − ∫_{Ω} F (x, u) dx − ∫_{Ω} G(x, v) dx

the functional of (2.1) -(2.2) defined on H_{o}^{1}(Ω)×H_{o}^{1}(Ω), we deduce by (2.4) and (2.5) that Φ(z) = Ψ(w). Such a result is also valid for solutions of (1.1)-(1.2). Now we use the Mountain Pass Theorem to find critical points of Ψ.

Following arguments of [6], we know that (H2) implies F^{∗}(x, t)/t^{2} → ∞ and G^{∗}(x, t)/t^{2} → ∞. Thus 0 is a local minmum of Ψ. Precisely,

Lemma 2.4 There exist constants α, ρ > 0, independent of Ω, such that

Ψ(w) ≥ α > 0 if ∥w∥_{X^{∗}} = ρ.

Lemma 2.5 There exist T > 0 and w ∈ E such that Ψ(tw) ≤ 0 whenever t ≥ T.

Proof. Taking w ∈ X^{∗}, w ≢ 0 such that

∫_{Ω}⟨w, Kw⟩ dx > 0,

whence by (H4), for t > 0

Ψ(tw) ≤ t^{\frac{α}{α−1}} ∫_{Ω} |w_{1}|^{\frac{α}{α−1}} dx + t^{\frac{β}{β−1}} ∫_{Ω} |w_{2}|^{\frac{β}{β−1}} dx − \frac{1}{2}t^{2} ∫_{Ω}⟨w, Kw⟩ dx.

Since \frac{α}{α−1}, \frac{β}{β−1} < 2, the assertion follows for t > 0 large. □

Let

Γ = {g ∈ C([0, 1], X^{∗}) : g(0) = 0, g(1) = e},

where e = T w. We define

c = c_{Ω} = inf_{g∈Γ} sup_{t∈[0,1]} Ψ(g(t)).  (2.9)

The Mountain Pass Theorem will imlpy that c is a critical value of Ψ if the Palais-Smale ((PS) for short) condition holds. It is known from Lemma 2.4 that corresponding critical points are nontrivial. Then the proof of Theorem 2.1 is completed.

Now we verify the (PS) condition. By a (PS) condition for Ψ we mean that any sequence {w_{n}} ⊂ X^{∗} such that |Ψ(w_{n})| is uniformly bounded in n and Ψ′(w_{n}) → 0 as n → ∞ possesses a convergent subsequence.

Lemma 2.6 The (PS) condition holds for Ψ.

Proof. Let {w_{n}} be a (PS) sequence of Ψ, that is

|Ψ(w_{n})| ≤ C Ψ′(w_{n}) → 0 as n → ∞

for some constant C > 0. This inequality and lemma 2.2 yield

∫_{Ω}[F^{∗}(x, w_{n}^{1}) + G^{∗}(x, w_{n}^{2})] dx

≤ \frac{1}{2} ∫_{Ω}⟨w_{n}, Kw_{n}⟩ dx + C
≤ \frac{1}{2} ∫_{Ω}(F_{s}^{∗′}(x, w_{n}^{1})w_{n}^{1} + G_{s}^{∗′}(x, w_{n}^{2})w_{n}^{2}) dx + o(1)∥w_{n}∥_{X^{∗}} + C
≤ \frac{1}{2} \frac{α}{α − 1} ∫_{Ω} F^{∗}(x, w_{n}^{1}) dx + \frac{1}{2} \frac{β}{β − 1} ∫_{Ω} G^{∗}(x, w_{n}^{2}) dx + o(1)∥w_{n}∥_{X^{∗}}.

That is

∫_{Ω}[F^{∗}(x, w_{n}^{1}) + G^{∗}(x, w_{n}^{2})] dx ≤ C + o(1)∥w_{n}∥_{X^{∗}}.

By Lemma 2.3, we obtain

∥w_{n}^{1}∥_{L^{1+1/p}}^{1+1/p} + ∥w_{n}^{2}∥_{L^{1+1/q}}^{1+1/q} ≤ C + o(1)∥w_{n}∥_{X^{∗}}.

It implies that ∥w_{n}∥_{X^{∗}} is bounded. We may assume w_{n} → w weakly in X^{∗} as n → ∞. Since the operator (−Δ + id)^{−1} is compact, it follows

u_{n} := (−Δ + id)^{−1}w_{n}^{2} → (−Δ + id)^{−1}w^{2} in X^{∗} as n → ∞,
v_{n} := (−Δ + id)^{−1}w_{n}^{1} → (−Δ + id)^{−1}w^{1} in X^{∗} as n → ∞.

As a result, w_{n} = (f(x, u_{n}), g(x, v_{n})) → w in X^{∗} as n → ∞ which completes the present proof.

3 Palais-Smale sequence

In this section, we prove a global compact result for problem (1.1)-(1.2). Let E = H^{1}(ℝ^{N}) × H^{1}(ℝ^{N}). By our assumptions, the functional

Φ(z) = ∫_{ℝ^{N}} (∇u∇v + uv) dx − ∫_{ℝ^{N}} (F (x, u) + G(x, v)) dx

is C^{1} on E. The functional Φ^{∞} is defined with F̄ and Ḡ replacing F and G in Φ respectively.

Proposition 3.1 Asumme (H1)-(H6). Let {z_{n}} be a (P S)_{c} sequence of Φ, i.e.

Φ(z_{n}) → c and Φ′(z_{n}) → 0 as n → 0.  (3.1)

Then there exists a subsequence (still denoted by {z_{n}}) for which the following holds: there exist an integer k ≥ 0, sequences {x_{n}^{i}} ⊂ ℝ^{N}, |x_{n}^{i}| → ∞ as n → ∞ for 1 ≤ i ≤ k, a solution z of (1.1)-(1.2) and solutions z^{i}(1 ≤ i ≤ k) of (1.3)-(1.4) such that

z_{n} → z weakly in E,  (3.2)
Φ(z_{n}) → Φ(z) + ∑_{i=1}^{k} Φ^{∞}(z^{i}),  (3.3)
z_{n} − (z + ∑_{i=1}^{k} z^{i}(x − x_{n}^{i})) → 0 in E  (3.4)

as n → ∞, where we agree that in the case k = 0 the above holds without z^{i}, x_{n}^{i}.

Proof. The result can be derived from the arguments for one equation [5]. First we remark that the boundedness of {z_{n}} in E can be deduced as [13] by (3.1). Therefore we may assume

z_{n} → z weakly in E,
z_{n} → z strongly in L_{loc}^{p+1}(ℝ^{N}) × L_{loc}^{q+1}(ℝ^{N}),
z_{n} → z a.e. in ℝ^{N}

as n → ∞. Denote Q(z) = ∫_{ℝ^{N}} (∇u∇v + uv) dx, we have

Q(z_{n}) = Q(z_{n} − z) + Q(z) + o(1).  (3.5)

It follows from Brezis & Lieb’s lemma [8] that

∫_{ℝ^{N}} F (x, u_{n}) dx = ∫_{ℝ^{N}} F (x, u_{n} − u) dx + ∫_{ℝ^{N}} F (x, u) dx + o(1)  (3.6)

and

∫_{ℝ^{N}} G(x, v_{n}) dx = ∫_{ℝ^{N}} G(x, v_{n} − v) dx + ∫_{ℝ^{N}} G(x, v) dx + o(1).  (3.7)

Hence we obtain

Φ(z_{n}) = Φ(z_{n} − z) + Φ(z) + o(1), Φ′(z_{n}) = Φ′(z_{n} − z) + Φ′(z) + o(1) (3.8)

as n → ∞. Let z_{n}^{1} = z_{n} − z. We may deduce from (H5) as [17] and [19] that

∫_{ℝ^{N}} u_{n}^{1}[f(x, u_{n}^{1}) − f̄(u_{n}^{1})] dx → 0 and ∫_{ℝ^{N}} v_{n}^{1}[g(x, v_{n}^{1}) − ḡ(v_{n}^{1})] dx → 0

as well as

∫_{ℝ^{N}} [F (x, u_{n}^{1}) − F̄(u_{n}^{1})] dx → 0 and ∫_{ℝ^{N}} [G(x, v_{n}^{1}) − Ḡ(v_{n}^{1})] dx → 0

as n → ∞. Therefore

Φ^{∞}(z_{n}^{1}) = Φ(z_{n}^{1}) + o(1) = Φ(z_{n}) − Φ(z) + o(1)  (3.9)
Φ^{∞′}(z_{n}^{1}) = Φ′(z_{n}^{1}) + o(1) = Φ′(z_{n}) − Φ′(z) + o(1).  (3.10)

Suppose z_{n}^{1} = z_{n} − z ↛ 0 strongly in E(otherwise we shall have finished). We want to show that there exists x_{n}^{1} ⊂ ℝ^{N} such that |x_{n}^{1}| → +∞ and z_{n}^{1}(x+x_{n}^{1}) → z^{1} ≢ 0 weakly in E. We note that

Φ^{∞}(z_{n}^{1}) ≥ α > 0

because ∥z_{n}^{1}∥_{E} ↛ 0. In fact, were it not true, we would have

Φ^{∞}(z_{n}^{1}) → 0, < Φ^{∞′}(z_{n}^{1}), η >= o(1)∥η∥_{E} as n → ∞.  (3.11)

Taking η = (\frac{β}{α+β} u_{n}^{1}, \frac{α}{α+β} v_{n}^{1}) =: η_{n} in (3.11), it follows

o(1)∥η_{n}∥_{E} = \frac{β}{α + β} ∫_{ℝ^{N}} u_{n}^{1}f̄(u_{n}^{1}) dx + \frac{α}{α + β} ∫_{ℝ^{N}} v_{n}^{1}ḡ(v_{n}^{1}) dx
− ∫_{ℝ^{N}} F̄(u_{n}^{1}) dx − ∫_{ℝ^{N}} Ḡ(v_{n}^{1}) dx.  (3.12)

Using hypothesis (H4) we obtain

∫_{ℝ^{N}} (F̄(u_{n}^{1}) + Ḡ(v_{n}^{1})) dx = o(1)∥η∥_{E}.

This and (3.12) yield

∫_{ℝ^{N}} u_{n}^{1}f̄(u_{n}^{1}) dx = o(1)∥η_{n}∥_{E}, ∫_{ℝ^{N}} v_{n}^{1}ḡ(v_{n}^{1}) dx = o(1)∥η_{n}∥_{E}.  (3.13)

It follows from assumptions (H2)-(H4) that

|f̄(t)|^{2} ≤ Ctf̄(t) if |t| ≤ 1, |f̄(t)|^{(p+1)′} ≤ Ctf̄(t) if |t| > 1. (3.14)

Taking η = (ϕ, 0) in (3.11) and using (3.14) and Hölder’s inequality, we obtain

| ∫_{ℝ^{N}} (∇ϕ∇v_{n}^{1} + ϕv_{n}^{1}) dx|

≤ | ∫_{{|u_{n}^{1}|≤1}} + ∫_{{|u_{n}^{1}|>1}} ϕf̄(u_{n}^{1}) dx|  (3.15)
≤ C(∫_{ℝ^{N}} |f(u_{n}^{1})|^{2} dx)^{\frac{1}{2}} ∥ϕ∥_{L^{2}} + C(∫_{ℝ^{N}} |f̄(u_{n}^{1})|^{(p+1)′} dx)^{1/(p+1)′}∥ϕ∥_{L^{p+1}}
≤ C∥ϕ∥_{H^{s}}[(∫_{ℝ^{N}} u_{n}^{1}f̄(u_{n}^{1}) dx)^{\frac{1}{2}} + C(∫_{ℝ^{N}} u_{n}^{1}f̄(u_{n}^{1}) dx)^{1/(p+1)′}].

which with (3.13) imply that

∥v_{n}^{1}∥_{H^{1}} = o(1).  (3.16)

Similarly, we show that

∥u_{n}^{1}∥_{H^{1}} = o(1).  (3.17)

(3.16) and (3.17) yield ∥z_{n}^{1}∥_{E} → 0 , we get a contradiction.

We decompose ℝ^{N} into N-dimensional unit hypercubes Q_{j} with vertices having integer coordinates and put

d_{n} = max_{j}(∥u_{n}^{1}∥_{L^{p+1}(Q_{j})} + ∥v_{n}^{1}∥_{L^{q+1}(Q_{j})}).

We claim that there is a β > 0 such that

d_{n} ≥ β > 0 ∀n ∈ ℕ.  (3.18)

Suppose, by contradiction, that d_{n} → 0 as n → ∞. Since

Φ^{∞′}(z_{n}^{1}) → 0 as n → ∞,  (3.19)

noting that ∥z_{n}^{1}∥_{E} is bounded, we have by (H2) and (H3) that

0 ≤ Φ^{∞}(z_{n}^{1}) ≤ ∫_{ℝ^{N}} u_{n}^{1}f̄(u_{n}^{1}) dx + ∫_{ℝ^{N}} v_{n}^{1}ḡ(v_{n}^{1}) dx + o(1)
≤ C_{ϵ}(∥u_{n}^{1}∥_{L^{p+1}(R^{N})}^{p+1} + ∥v_{n}^{1}∥_{L^{q+1}(R^{N})}^{q+1}) + ϵ(∥u_{n}^{1}∥_{L^{2}(R^{N})}^{2} + ∥v_{n}^{1}∥_{L^{2}(R^{N})}^{2})
≤ C_{ϵ} ∑_{j}(∥u_{n}^{1}∥_{L^{p+1}(Q_{j})}^{p+1} + ∥v_{n}^{1}∥_{L^{q+1}(Q_{j})}^{q+1}) + ϵ(∥u_{n}^{1}∥_{L^{2}(R^{N})}^{2} + ∥v_{n}^{1}∥_{L^{2}(R^{N})}^{2})
≤ C_{ϵ}d_{n}^{p−1} ∑_{j} ∥u_{n}^{1}∥_{L^{p+1}(Q_{j})}^{2} + C_{ϵ}d_{n}^{q−1} ∑_{j} ∥v_{n}^{1}∥_{L^{q+1}(Q_{j})}^{2} + ϵC
≤ C_{ϵ}d_{n}^{p−1} ∑_{j} ∥u_{n}^{1}∥_{H^{1}(Q_{j})}^{2} + C_{ϵ}d_{n}^{q−1} ∑_{j} ∥v_{n}^{1}∥_{H^{1}(Q_{j})}^{2} + ϵC
≤ C_{ϵ}d_{n}^{p−1}∥u_{n}^{1}∥_{H^{1}}^{2} + C_{ϵ}d_{n}^{q−1}∥v_{n}^{1}∥_{H^{1}}^{2} + ϵC.

Let n → ∞ and then ϵ → 0, we obtain Φ^{∞}(z_{n}^{1}) → 0 as n → ∞. This and (3.19) imply as above that ∥z_{n}^{1}∥_{E} → 0 as n → ∞, a contradiction, hence we have (3.18).

Let {x_{n}^{1}} be the center of a hypercube Q_{j} in which

d_{n} = ∥u_{n}^{1}∥_{L^{p+1}(Q_{j})} + ∥v_{n}^{1}∥_{L^{q+1}(Q_{j})}.

Now we show that

|x_{n}^{1}| → ∞ as n → ∞.  (3.20)

If {x_{n}^{1}} were bounded, by passing to a subsequence if necessary we should find that x_{n}^{1} would be in the same Q_{j} and so they should coincide. Therefore in that Q_{j}, for every n > n_{o}, n_{o} fixed and large enough, we should have

Φ^{∞}|_{E(Q_{j})}(z̄_{n}^{1}) = ∫_{Q_{j}}(∇ū_{n}^{1}∇v̄_{n}^{1} + ū_{n}^{1}v̄_{n}^{1}) dx − ∫_{Q_{j}}(F̄(ū_{n}^{1}) + Ḡ(v̄_{n}^{1})) dx + o(1)
≥ (α − 1) ∫_{R^{N}} F̄(ū_{n}^{1}) dx + (β − 1) ∫_{R^{N}} Ḡ(v̄_{n}^{1}) dx + o(1)
≥ C(∥ū_{n}^{1}∥_{L^{α}(Q_{j})}^{α} + ∥v̄_{n}^{1}∥_{L^{β}(Q_{j})}^{β}) + o(1)
≥ C(∥ū_{n}^{1}∥_{L^{p+1}(Q_{j})}^{α} + ∥v̄_{n}^{1}∥_{L^{q+1}(Q_{j})}^{β}) + o(1),

and

Φ^{∞′}(z̄_{n}^{1}) → 0 as n → 0,

where

z̄_{n}^{1}(x) = \{ \begin{matrix}z_{n}^{1}(x) & z ∈ Q_{j} \\ 0 & x ∈ ℝ^{N}\Q_{j}. \end{matrix}

Hence z̄_{n}^{1} should converge strongly in E(Q_{j}) to a nonzero function, contradicting to z_{n}^{1} → 0 weakly in E, so we have (3.20). Let

z_{n}^{1}(· + x_{n}^{1}) → z^{1} weakly in E.

Denote by Q̄ the unit hypercube centered at the origin, we have ∥z_{n}^{1}∥_{E(Q̄)} ≥ β > 0, thus z^{1} ≢ 0 and

⟨Φ^{∞′}(z^{1}), η⟩ = 0, ∀η ∈ E.  (3.21)

Iterating the procedure, we obtain sequences x_{n}^{l}, |x_{n}^{l}| → ∞ and

z_{n}^{l}(x) = z_{n}^{l−1}(x + x_{m}) − z^{l−1}(x), j ≥ 2
z_{n}^{l}(x + x_{n}^{l}) → z^{l}(x) weakly in E

as n → 0, where each z^{l} satisfying (3.21) and by induction

∥z_{n}^{l}∥_{E} = ∥z_{n}^{l−1}∥_{E}^{2} − ∥z^{l−1}∥_{E}^{2} = ∥z_{n}∥_{E}^{2} − ∥z∥_{E}^{2} − ∑_{i=1}^{l−1} ∥z^{i}∥_{E}^{2} + o(1).

Φ^{∞}(z_{n}^{l}) = Φ^{∞}(z_{n}^{l−1}) − Φ^{∞}(z^{l−1}) + o(1) = Φ(z_{n}) − Φ(z) − ∑_{i=1}^{l−1} Φ(z^{i}) + o(1).

Since z^{l} is a solution of (1.3)-(1.4) and z^{l} ≢ 0, we may prove as Lemma 4.1 below that ∥z^{l}∥_{E} ≥ C > 0. Thus the iteration will terminate at some index k ≥ 0. The assertion follows.

4 Uniform bounds and proof of Theorem 1.1

We shall bound critical values defined in (2.9) by the energy of the ground state of problem (1.3)-(1.4). By a ground state of problem (1.3)-(1.4) we mean a minimizer of the variational problem

Φ^{∞} = inf{Φ^{∞}(u, v) : (u, v) ∈ E is a solution of (1.3)-(1.4), (u, v) ≢ (0, 0)}.

(4.1)

It is shown in [13] that problem (1.3)-(1.4) has a positive radially decaying solution, so the variational problem (4.1) is well defined.

Lemma 4.1 Variational problem (4.1) is assumed by a nontrivial solution of (1.3)-(1.4).

Proof. Let z_{n} = (u_{n}, v_{n}) be a minimizing sequence of Φ^{∞}. It is obvious that {z_{n}} is a (PS) sequence of Φ^{∞}. We deduce by Proposition 3.1 that

Φ^{∞} = Φ(z_{n}) + o(1) = ∑_{j=1}^{k} Φ^{∞}(z_{j}) + o(1),

where z_{j} is a solution of (1.3)-(1.4). By the definition of Φ^{∞}, k = 1. The proof will be completed if we show z_{1} ≠ 0. To this end, we bound solutions of (1.3)-(1.4) in H^{1} norm below by a positive constant.

Suppose z = (u, v) is a solution of (1.3)-(1.4), we have

∥u∥_{H^{1}}^{2} = ∫_{ℝ^{N}} uḡ(v) dx, ∥v∥_{H^{1}}^{2} = ∫_{ℝ^{N}} vf̄(u) dx,  (4.2)

and

∫_{ℝ^{N}} (∇u∇v + uv) dx = ∫_{ℝ^{N}} vḡ(v) dx = ∫_{ℝ^{N}} uf̄(u) dx.  (4.3)

By assumptions (H2), (H3) and (H5), we obtain

f̄(u) ≤ C_{ϵ}|u|^{\frac{N+2}{N−2}} + ϵu, ḡ(v) ≤ C_{ϵ}|v|^{\frac{N+2}{N−2}} + ϵv.  (4.4)

We deduce by (4.2)-(4.4) and Hölder’s inequality that

∥u∥_{H^{1}}^{2} ≤ C_{ϵ}∥v∥_{L^{2^{∗}}}^{2^{∗}−1}∥u∥_{L^{2^{∗}}} + ϵ∥u∥_{L^{2}}∥v∥_{L^{2}},

where 2^{∗} = \frac{2N}{N−2}. Using Young’s inequality and Sobolev embedding, we obtain

∥u∥_{H^{1}}^{2} ≤ C_{ϵ}(∥u∥_{H^{1}}^{2^{∗}} + ∥v∥_{H^{1}}^{2^{∗}}) + ϵ∥v∥_{H^{1}}^{2}.

Similarly,

∥v∥_{H^{1}}^{2} ≤ C_{ϵ}(∥u∥_{H^{1}}^{2^{∗}} + ∥v∥_{H^{1}}^{2^{∗}}) + ϵ∥u∥_{H^{1}}^{2}.

So for ϵ small, we have

∥u∥_{H^{1}}^{2} + ∥v∥_{H^{1}}^{2} ≤ C(∥u∥_{H^{1}}^{2^{∗}} + ∥v∥_{H^{1}}^{2^{∗}}).

It yields that ∥u∥_{H^{1}} or ∥v∥_{H^{1}} ≥ C > 0, uniformly for solutions of (1.3)-(1.4), and where C > 0 is independent of z = (u, v). Consequently, z_{1} = (u_{1}, v_{1}) ≢ 0. □

Let R_{n} → ∞, B_{n} = B_{R_{n}}(0). Taking Ω = B_{n} in problem (2.1)-(2.2), we infer from Theorem 2.1 that there exists a solution z_{n} of problem (2.1)- (2.2) defined on B_{n} for each n. Moreover,

Φ(z_{n}) = Ψ(w_{n}) = c_{n} ≥ α > 0,  (4.5)

where z_{n} = Kw_{n} , Φ = Φ_{R^{N}} and Ψ = Ψ_{R^{N}}. We have extended z_{n} to ℝ^{N} by letting z_{n} = 0 outside B_{n}.

Proposition 4.1 c_{n} < Φ^{∞} for n large.

Proof. Since each element w in X_{n}^{∗} = L^{1+1/p}(B_{n}) × L^{1+1/q}(B_{n}) can be extended to an element of X^{∗} by letting w = 0 outside B_{n}, we shall denote Ψ_{B_{n}} as Ψ in brief. By Lemma 4.1, Φ^{∞} is assumed. Let z_{o} = (u_{o}, v_{o}) be a minimizer of Φ^{∞}. Choosing

w_{1}^{o} = f̄(u_{o}), w_{2}^{o} = ḡ(v_{o})

and using (H4)-(H5) and equations (1.3)-(1.4), one has ∫_{ℝ^{N}} < w_{o}, Kw_{o} > dx > 0, where w_{o} = (w_{1}^{o}, w_{2}^{o}). Moreover, we know as Lemma 2.5 that there are t_{1}, t_{2} > 0 such that

max_{t≥0}Ψ(tw_{o}) = max_{t_{1}≤t≤t_{2}}Ψ(tw_{o}).

Suppose that t_{o} ∈ (t_{1}, t_{2}) and

Ψ(t_{o}w_{o}) = max_{t_{1}≤t≤t_{2}}Ψ(tw_{o}).

Because F (x, t) ≥ F̄(t) and G(x, t) ≥ Ḡ(t), one has F^{∗}(x, s) ≤ F̄^{∗}(s) and G^{∗}(x, s) ≤ Ḡ(s). By the assumption (H6),

Ψ(t_{o}w_{o}) < Ψ^{∞}(t_{o}w_{o}),

it follows

sup_{t≥0} Ψ(tw_{o}) < sup_{t≥0} Ψ^{∞}(tw_{o}).  (4.6)

The density of real number field implies that there exists ϵ > 0 such that

sup_{t≥0} Ψ(tw_{o}) + 2ϵ < sup_{t≥0} Ψ^{∞}(tw_{o}).  (4.7)

Let ϕ ∈ C_{o}^{∞}(ℝ^{N}), 0 ≤ ϕ ≤ 1 and ϕ ≡ 1 if |x| ≤ \frac{1}{2}; ϕ ≡ 0 if |x| > 1; ϕ_{n}(x) = ϕ(\frac{x}{R_{n}}). Then z_{n} := (ϕ_{n}u_{o}, ϕ_{n}v_{o}) converges to (u_{o}, v_{o}) in E. Let

w_{1}^{n} = f̄(ϕ_{n}u_{o}), w_{2}^{n} = ḡ(ϕ_{n}v_{o}).

We also have w_{n} → w_{o} in X^{∗}. Suppose

Ψ(t_{n}w_{n}) = sup_{t≥0} Ψ(tw_{n}),

then {t_{n}} is bounded. Indeed, if t_{n} → ∞, arguments in Lemma 2.5 would yield sup_{t≥0} Ψ(tw_{n}) → −∞. It is not possible because the value is not negative. Suppose t_{n} → t̄_{o}, the continuity of the functional Ψ gives

Ψ(t_{n}w_{n}) → Ψ(t̄_{o}w_{o}).

We claim that Ψ(t̄_{o}w_{o}) = sup_{t≥0} Ψ(tw_{o}). In fact, for every ϵ > 0 there exists δ > 0 such that

Ψ(t_{o}w_{o}) − ϵ ≤ Ψ(tw_{o})

whenever |t − t_{o}| < δ. By the continuity of Ψ, we may find n_{o} > 0 such that if n ≥ n_{o}

Ψ(tw_{o}) ≤ Ψ(tw_{n}) + ϵ, Ψ(t_{n}w_{n}) ≤ Ψ(t̄_{o}w_{o}) + ϵ.

Therefore if n ≥ n_{o} we have

Ψ(t_{o}w_{o}) − ϵ ≤ Ψ(t_{n}w_{n}) + ϵ ≤ Ψ(t̄_{o}w_{o}) + 2ϵ ≤ Ψ(t_{o}w_{o}) + 2ϵ.

Since ϵ is arbitrary, the conclusion holds. By the same arguments, we find that there exists s_{n} such that s_{n} → s̄_{o} and

Ψ^{∞}(s_{n}w_{n}) = sup_{t≥0} Ψ^{∞}(tw_{n}) → Ψ^{∞}(s̄_{o}w_{o}) = sup_{t≥0} Ψ^{∞}(tw_{o})

as n → ∞. By (4.7), we obtain Ψ(t_{n}w_{n}) + ϵ < Ψ^{∞}(s_{n}w_{n}) for n large enough. We may assume s_{n} > 0, and then

\frac{dΨ^{∞}(tw_{n})}{dt} |_{t=s_{n}}= 0,

that is

∫_{ℝ^{N}} [F̄_{s}^{∗′}(s_{n}w_{1}^{n})w_{1}^{n} + Ḡ_{s}^{∗′}(s_{n}w_{2}^{n})w_{2}^{n}] dx − s_{n} ∫_{ℝ^{N}} < w_{n}, Kw_{n} > dx = 0. (4.8)

By the definition of Legendre - Fenchel transformation, we obtain

∫_{ℝ^{N}} [F̄^{∗}(s_{n}w_{1}^{n}) + Ḡ^{∗}(s_{n}w_{2}^{n})] dx

= ∫_{ℝ^{N}} [F̄_{s}^{∗′}(s_{n}w_{1}^{n})s_{n}w_{1}^{n} + Ḡ_{s}^{∗′}(s_{n}w_{2}^{n})s_{n}w_{2}^{n}] dx
− ∫_{ℝ^{N}} [F̄(f̄^{−1}(s_{n}w_{1}^{n})) + Ḡ(ḡ^{−1}(s_{n}w_{2}^{n}))] dx  (4.9)
= s_{n}^{2} ∫_{ℝ^{N}} < w_{n}, Kw_{n} > dx − ∫_{ℝ^{N}} [F̄(f̄^{−1}(s_{n}w_{1}^{n})) + Ḡ(ḡ^{−1}(s_{n}w_{2}^{n}))] dx.

Consider

(−Δ + id)^{−1}w_{2}^{n} = u_{o} + σ_{n}, (−Δ + id)^{−1}w_{1}^{n} = v_{o} + μ_{n} in ℝ^{N},

we obtain

(−Δ + id)σ_{n} = ḡ(ϕ_{n}v_{o}) − ḡ(v_{o}), (−Δ + id)μ_{n} = f̄(ϕ_{n}u_{o}) − f̄(u_{o}).

In terms of L^{p}−estimates, σ_{n} → 0 and μ_{n} → 0 in H^{2,2} as n → ∞. Furthermore, we infer from (4.8) that

∫_{ℝ^{N}} s_{n}(w_{1}^{n})^{2}[\frac{f̄^{−1}(s_{n}w_{1}^{n})}{s_{n}w_{1}^{n}} − \frac{f̄^{−1}(w_{1}^{n})}{w_{1}^{n}}] dx

+ ∫_{ℝ^{N}} s_{n}(w_{2}^{n})^{2}[\frac{ḡ^{−1}(s_{n}w_{2}^{n})}{s_{n}w_{2}^{n}} − \frac{ḡ^{−1}(w_{2}^{n})}{w_{2}^{n}}] dx

= ∫_{ℝ^{N}} [w_{1}^{n}σ_{n} + w_{2}^{n}μ_{n} + (1 − ϕ_{n})(w_{1}^{n} + w_{2}^{n})] dx = o(1)

as n → ∞. The equality and assumption (H7) imply s_{n} → 1 as n → ∞. hence we deduce by (4.8) and (4.9) that

sup_{t≥0} Ψ^{∞}(tw_{n})

≤ \frac{1}{2} ∫_{ℝ^{N}} (u_{o}f̄(u_{o}) + v_{o}ḡ(v_{o})) dx − ∫_{ℝ^{N}} (F̄(u_{o}) + Ḡ(v_{o})) dx + ϵ_{n}
= Ψ^{∞} + ϵ_{n},

where

ϵ_{n} = \frac{1}{2}(s_{n}^{2} − 1) ∫_{ℝ^{N}} (u_{o}f̄(u_{o}) + v_{o}ḡ(v_{o})) dx
− ∫_{ℝ^{N}} [(F̄(ϕ_{n}u_{o}) − F̄(u_{o})) + (Ḡ(ϕ_{n}v_{o}) − Ḡ(v_{o}))] dx
+ ∫_{ℝ^{N}} [(F̄(ϕ_{n}u_{o}) − F̄(f̄^{−1}(s_{n}w_{1}^{n})) + (Ḡ(ϕ_{n}v_{o}) − Ḡ(ḡ^{−1}(s_{n}w_{2}^{n}))] dx.

The above estimates imply ϵ_{n} = o(1) as n → ∞. Therefore

sup_{t≥0} Ψ(tw_{n}) < sup_{t≥0} Ψ(tw_{n})^{∞} − ϵ ≤ Ψ^{∞} − ϵ + o(1),

the assertion follows for n large. □

Lemma 4.2 z_{n} is a (PS) sequence of Φ in E.

Proof. It is readily to verify that c_{n} = Φ(z_{n}) ≤ c_{n−1} = Φ(z_{n−1}), thus by Proposition 4.2

α ≤ c_{n} ≤ c_{1} < Φ^{∞},  (4.10)

we obtain

c_{n} = Φ(z_{n}) → c, α ≤ c ≤ c_{1} < Φ^{∞}.  (4.11)

Now we show that

Φ′(z_{n}) → 0, as n → ∞.  (4.12)

In fact, ∀(ϕ, ψ) ∈ C_{o}^{∞}(ℝ^{N})×C_{o}^{∞}(ℝ^{N}), there is n_{o} > 0 such that suppϕ, suppψ ⊂ B_{n} whenever n ≥ n_{o} and

Φ′(z_{n})(ϕ, ψ) = 0, if n ≥ n_{o}.

This implies that Φ′(z_{n})z → 0 as n → ∞ for all z ∈ C_{o}^{∞}(ℝ^{N})×C_{o}^{∞}(ℝ^{N}). Hence (4.12) follows because C_{o}^{∞}(ℝ^{N}) × C_{o}^{∞}(ℝ^{N}) is dense in H^{1}(ℝ^{N}) × H^{1}(ℝ^{N}). □

Completion of the proof of Theorem 1.1 We may prove that the (PS) sequence z_{n} of Φ is bounded in E as [13], and assume z_{n} → z_{o} weakly in E. Obviously, z_{o} solves (1.1)-(1.2). We claim that z_{o} is nontrivial. In fact, Lemma 2.4, Proposition 3.1 and Proposition 4.2 give that

α ≤ Φ(z_{n}) = Φ(z_{o}) + ∑_{j} Φ^{∞}(z_{j}) + o(1) < Φ^{∞}.

If j = 0, Φ(z_{o}) ≥ α > 0, z_{o} is a nontrivial solution; if j ≥ 1, then Φ(z_{o}) < 0, also implying z_{o} ≢ 0. The decaying law of z_{o} at infinity was proved in [13]. □

ACKNOWLEDGEMENTS

The work was partially supported by 21CSPJ, NSFJ and NSFC in China.

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Jianfu Yang Department of Mathematics, IMECC-Unicamp Caixa Postal 6065 Campinas 13081-970, SP, Brazil e-mail: jfyang@ime.unicamp.br

and Department of Mathematics, Nanchang University Nanchang, Jiangxi 330047 P. R. of China