We state our main result as follows:
Theorem 3.1. Assume that
(A1)
where (see (3)).
Then, for each
the system (1) has an unbounded sequence of weak solutions in X.
Proof. Define the functionals Φ, Ψ: X → ℝ for each u = (u1, ..., u
n
) ∈ X, as follows
and
It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point u ∈ X is the functional Ψ'(u) ∈ X*, given by
for every v = (v1, ..., v
n
) ∈ X, and Ψ': X → X* is a compact operator. Moreover, Φ is a sequentially weakly lower semicontinuous and Gâteaux differentiable functional whose Gâteaux derivative at the point u ∈ X is the functional Φ' (u) ∈ X*, given by
for every v = (v1, ..., v
n
) ∈ X. Furthermore, (Φ')-1: X* → X exists and is continuous.
Put I
λ
: = Φ - λ Ψ. Clearly, the weak solutions of the system (1) are exactly the solutions of the equation . Now, we want to show that
Let {ξ
m
} be a real sequence such that ξ
m
→ +∞ as m → ∞ and
Put for all m ∈ ℕ. Since
for each for 1 ≤ i ≤ n, we have
(4)
for each u = (u1, u2, ..., u
n
) ∈ X. This, for each r > 0, together with (4), ensures that
Hence, an easy computation shows that whenever u = (u1, ..., u
n
) ∈ Φ-1(] - ∞, r
m
]). Hence, one has
Therefore, since from Assumption (A1) one has
we deduce
(5)
Assumption (A1) along with (5), implies
Fix λ ∈ Λ. The inequality (5) concludes that the condition (b) of Theorem 2.1 can be applied and either I
λ
has a global minimum or there exists a sequence {u
m
} where u
m
= (u1m, ..., u
nm
) of weak solutions of the system (1) such that limm→∞||(u1m, ..., u
nm
)|| = +∞.
Now fix λ ∈ Λ and let us verify that the functional I
λ
is unbounded from below. Arguing as in [8], consider n positive real sequences such that as m → ∞
and
(6)
For all m ∈ ℕ define w
m
(x) = (d1, m, ..., dn, m). For any fixed m ∈ ℕ, w
m
∈ X and, in particular, one has
Then, for all m ∈ ℕ,
Now, if
we fix . From (6) there exists τ
ε
such that
therefore
and by the choice of ε, one has
If
let us consider . From (6) there exists τ
K
such that
therefore
and by the choice of K, one has
Hence, our claim is proved. Since all assumptions of Theorem 2.1 are satisfied, the functional I
λ
admits a sequence {u
m
= (u1m, ..., u
nm
)} ⊂ X of critical points such that
and we have the conclusion. □
Here, we give a consequence of Theorem 3.1.
Corollary 3.2. Assume that
(A2)
(A3)
Then, the system
for 1 ≤ i ≤ n, has an unbounded sequence of classical solutions in X.
Now, we want to present the analogous version of the main result (Theorem 3.1) in the autonomous case.
Theorem 3.3. Assume that
(A4)
where (see (3)).
Then, for each
the system
has an unbounded sequence of weak solutions in X.
Proof. Set F (x, u1, ..., u
n
) = F (u1, ..., u
n
) for all x ∈ Ω and (u1, ..., u
n
) ∈ ℝn. The conclusion follows from Theorem 3.1. □
Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t1, ..., t
n
) → (+∞, ..., +∞) with ξ → 0+ (t1, ..., t
n
) → (0+, ..., 0+), respectively, that by the same way as in the proof of Theorem 3.1 but using conclusion (c) of Theorem 2.1 instead of (b), the system (1) has a sequence of weak solutions, which strongly converges to 0 in X.
Finally, we give an example to illustrate the result.
Example 3.1. Let Ω ⊂ ℝ2 be a non-empty bounded open set with a smooth boundary ϑΩ and consider the increasing sequence of positive real numbers given by
for every n ≥ 1. Define the function
(7)
where B((an+1, an+1), 1)) be the open unit ball of center (an+1, an+1). We observe that the function F is non-negative, F (0, 0) = 0, and F ∈ C1(ℝ2). We will denote by f and g, respectively, the partial derivative of F respect to t1 and t2. For every n ∈ ℕ, the restriction F on B((an+1, an+1), 1) attains its maximum in (an+1, an+1) and F (an+1, an+1) = (an+1)5,
then
So
On the other by setting y
n
= an+1- 1 for every n ∈ ℕ, one has
Then
and hence
Finally
So, since all assumptions of Theorem 3.3 is applicable to the system
for every λ ∈ [0, +∞[.