Abstract
We investigate the existence of infinitely many weak solutions for a class of Neumann quasilinear elliptic systems driven by a (p_{1}, ..., p_{n})Laplacian operator. The technical approach is fully based on a recent three critical points theorem.
AMS subject classification: 35J65; 34A15.
Keywords:
infinitely many solutions; Neumann system; critical point theory; variational methods1 Introduction
The purpose of this article is to establish the existence of infinitely many weak solutions for the following Neumann quasilinear elliptic system
for i = 1, ..., n, where Ω ⊂ ℝ^{N }(N ≥ 1) is a nonempty bounded open set with a smooth boundary ∂Ω, p_{i }> N for i = 1, ..., n, is the p_{i}Laplacian operator, a_{i }∈ L^{∞ }(Ω) with ess inf_{Ω }a_{i }> 0 for i = 1, ..., n, λ > 0, and F: Ω × ℝ^{n }→ ℝ is a function such that the mapping (t_{1}, t_{2},..., t_{n}) → F (x, t_{1}, t_{2},..., t_{n}) is in C^{1 }in ℝ^{n }for all is continuous in Ω × ℝ^{n }for i = 1,..., n, and F (x, 0,..., 0) = 0 for all x ∈ Ω and ν is the outward unit normal to ∂Ω. Here, denotes the partial derivative of F with respect to t_{i}.
Precisely, under appropriate hypotheses on the behavior of the nonlinear term F at infinity, the existence of an interval Λ such that, for each λ ∈ Λ, the system (1) admits a sequence of pairwise distinct weak solutions is proved; (see Theorem 3.1). We use a variational argument due to Ricceri which provides certain alternatives in order to find sequences of distinct critical points of parameterdepending functionals. We emphasize that no symmetry assumption is required on the nonlinear term F (thus, the symmetry version of the Mountain Pass theorem cannot be applied). Instead of such a symmetry, we assume a suitable oscillatory behavior at infinity on the function F.
We recall that a weak solution of the system (1) is any such that
For a discussion about the existence of infinitely many solutions for differential equations, using Ricceri's variational principle [1]and its variants [2,3] we refer the reader to the articles [416].
For other basic definitions and notations we refer the reader to the articles [1722]. Here, our motivation comes from the recent article [8]. We point out that strategy of the proof of the main result and Example 3.1 are strictly related to the results and example contained in [8].
2 Preliminaries
Our main tool to ensure the existence of infinitely many classical solutions for Dirichlet quasilinear twopoint boundary value systems is the celebrated Ricceri's variational principle [[1], Theorem 2.5] that we now recall as follows:
Theorem 2.1. Let X be a reflexive real Banach space, let Φ, Ψ: X → ℝ be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf_{X }Φ, let us put
and
Then, one has
(a) for every r > inf_{X }Φ and every , the restriction of the functional I_{λ }= Φ  λΨ to Φ^{1}(]  ∞, r[) admits a global minimum, which is a critical point (local minimum) of I_{λ }in X.
(b) If γ < +∞ then, for each , the following alternative holds:
either
(b_{1}) I_{λ }possesses a global minimum,
or
(b_{2}) there is a sequence {u_{n}} of critical points (local minima) of I_{λ }such that
(c) If δ < +∞ then, for each , the following alternative holds:
either
(c_{1}) there is a global minimum of Φ which is a local minimum of I_{λ},
or
(c_{2}) there is a sequence {u_{n}} of pairwise distinct critical points (local minima) of I_{λ }that converges weakly to a global minimum of Φ.
We let X be the Cartesian product of n Sobolev spaces , ,... and , i.e., , equipped with the norm
where
Since p_{i }> N for 1 ≤ i ≤ n, one has C < +∞. In addition, if Ω is convex, it is known [23] that
for 1 ≤ i ≤ n, where ·_{1 }= ∫_{Ω}·(x) dx, ·_{∞ }= sup_{x∈Ω }·(x) and m(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball.
For all γ > 0 we define
3 Main results
We state our main result as follows:
Theorem 3.1. Assume that
(A1)
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 = (u_{1}, ..., 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 = (v_{1}, ..., 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 = (v_{1}, ..., v_{n}) ∈ X. Furthermore, (Φ')^{1}: X* → X exists and is continuous.
Put I_{λ}: = Φ  λΨ. 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
for each for 1 ≤ i ≤ n, we have
for each u = (u_{1}, u_{2}, ..., u_{n}) ∈ X. This, for each r > 0, together with (4), ensures that
Hence, an easy computation shows that whenever u = (u_{1}, ..., u_{n}) ∈ Φ^{1}(]  ∞, r_{m}]). Hence, one has
Therefore, since from Assumption (A1) one has
we deduce
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 }= (u_{1m}, ..., u_{nm}) of weak solutions of the system (1) such that lim_{m→∞ }(u_{1m}, ..., 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
For all m ∈ ℕ define w_{m}(x) = (d_{1, m}, ..., d_{n, 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 }= (u_{1m}, ..., 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
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)
Then, for each
the system
has an unbounded sequence of weak solutions in X.
Proof. Set F (x, u_{1}, ..., u_{n}) = F (u_{1}, ..., u_{n}) for all x ∈ Ω and (u_{1}, ..., u_{n}) ∈ ℝ^{n}. The conclusion follows from Theorem 3.1. □
Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t_{1}, ..., t_{n}) → (+∞, ..., +∞) with ξ → 0^{+ }(t_{1}, ..., 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 ⊂ ℝ^{2 }be a nonempty 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
where B((a_{n+1}, a_{n+1}), 1)) be the open unit ball of center (a_{n+1}, a_{n+1}). We observe that the function F is nonnegative, F (0, 0) = 0, and F ∈ C^{1}(ℝ^{2}). We will denote by f and g, respectively, the partial derivative of F respect to t_{1 }and t_{2}. For every n ∈ ℕ, the restriction F on B((a_{n+1}, a_{n+1}), 1) attains its maximum in (a_{n+1}, a_{n+1}) and F (a_{n+1}, a_{n+1}) = (a_{n+1})^{5},
then
So
On the other by setting y_{n }= a_{n+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, +∞[.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
DMS has presented the main purpose of the article and has used GAA contribution due to reaching to conclusions. All authors read and approved the final manuscript.
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