Infinitely many solutions for a class of quasilinear two-point boundary value systems

The existence of infinitely many solutions for a class of Dirichlet quasilinear elliptic systems is established. The approach is based on variational methods.


Introduction
The aim of this paper is to investigate the existence of infinitely many weak solutions for the following doubly eigenvalue quasilinear two-point boundary value system −(p i − 1)|u i (x)| p i −2 u i (x) = (λF u i (x, u 1 , . . ., u n )+µG u i (x, u 1 , . . ., u n ))h i (x, u i ) in (a, b) where ) and G(x, 0, . . ., 0) = 0 for all x ∈ [a, b] and h i : [a, b] × R →]0, +∞[ is a bounded and continuous function with m i := inf (x,t)∈[a,b]×R h i (x, t) > 0. Here, F u i and G u i denote respectively the partial derivatives of F and G with respect to u i for 1 ≤ i ≤ n.
On the existence of multiple solutions for two-point boundary value problems of the type (D λ,µ ), several results are known when n = 1, see for example [2,3,18,23] and the references cited therein.Existence results for nonlinear elliptic systems with Dirichlet boundary conditions have also received a great deal of interest in recent years; see, for instance, the papers [11,13,19,20,22].
In the present paper, employing a smooth version of [5, Theorem 2.1], under some hypotheses on the behavior of the nonlinear terms at infinity, under conditions on the potential of h i for 1 ≤ i ≤ n, we determine the exact collections of the parameters λ and µ in which the system (D λ,µ ) admits infinitely many weak solutions (Theorem 3.1).We also list some consequences of Theorem 3.1 and one example.Here, due to the facts, no symmetric assumptions are requested on the nonlinearities, the infinitely many solutions are local minima of the energy functionals associated to the problem, and the nonlinearities depend on the term h i (x, u i ) being h i a continuous bounded function and u i is the weak derivative of the component u i of the weak solution u = (u 1 , u 2 , . . ., u n ) of the system (D λ,µ ), the application of variational methods to investigate the system (D λ,µ ) is not standard.
A special case of our main result is the following theorem.
dν is integrable and let F be a primitive of w such that F(0, 0) = 0. Fix two integers p, q > 2, with p ≤ q, and assume that Then, for every nonnegative arbitrary and for every µ ∈ [0, µ G [ where in (0, 1), admits a sequence of pairwise distinct positive weak solutions.

Preliminaries
Our main tool to investigate the existence of infinitely many weak solutions for the system (D λ,µ ) is a smooth version of Theorem 2.1 of [5] that we recall here.
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 λ ∈]0, 1 ϕ(r) [, the restriction of the functional Let X be the Cartesian product of n Sobolev spaces W where In order to apply Theorem 2.1 we set for 1 ≤ i ≤ n and for all (x, t) ∈ [a, b] × R, and consider the functionals Φ, Ψ : X → R for each u = (u 1 , . . ., u n ) ∈ X, as follows 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.Furthermore, Φ is sequentially weakly lower semicontinuous.By a classical solution of the system (D λ,µ ), we mean a function u = (u 1 , . . ., u n ) such that, and u satisfies (D λ,µ ).We say that a function

Main results
In this section, we present our main results.To be precise, we establish an existence result of infinitely many solutions to problem (D λ,µ ).For all ξ > 0 we denote by K(ξ) the set Put , where for each t i ∈ R, for all i = 1, . . ., n, Theorem 3.1.Assume that .
Proof.Our aim is to apply Theorem 2.1.To this end, fix λ, µ and G satisfying our assumptions.Let X be the Sobolev space and It is well known that they satisfy all regularity assumptions requested in Theorem 2.1 and that the critical points in X of the functional I λ = Φ − λΨ are precisely the weak solutions of problem (D λ,µ ).
Let {ξ k } be a real sequence of positive numbers such that lim k→+∞ ξ k = +∞, and for each u = (u 1 , u 2 , . . ., u n ) ∈ X.This, for each r > 0, along with (3.6), ensures that Therefore, one has for all k ∈ N. Therefore, from assumption (A2) and the condition (3.3) one has Now, let {(η i,k )} ⊆ R n be positive real sequences such that η i,k > 0 for all i = 1, . . ., n and for all k ∈ N, and = +∞.Let {w k = (w 1,k (x), ..., w n,k (x))} be a sequence in X defined by for each i = 1, . . ., n. Clearly (3.12) On the other hand, since G is nonnegative and bearing assumption (A1) in mind, from (3.5) one has and so Now, consider the following cases.
and so and moreover, Since 1 − λM < 0, and arguing as before, we have and that I λ does not possess a global minimum, from part (b) of Theorem 2.1, there exists an unbounded sequence {u k } of critical points which are the weak solutions of (D λ,µ ).So, our conclusion is achieved.
Proof of Theorem 1.1.Since f 1 and f 2 are positive, then F is nonnegative in R 2 + .Moreover, one has that the functions t 1 → F(t 1 , t 2 ), t 2 ∈ R, and ξ p = 0. On the other hand, one has By simple calculations, we see that Moreover, and Since p ≤ q, one has . Now, arguing as before we obtain Therefore, since one has also that where B((a k+1 , a k+1 ), 1) denotes the open unit ball of center (a k+1 , a k+1 ) and radius 1.Now, put for each y ∈ R. By simple calculations, we see that for each y ∈ R, and By definition, F is nonnegative and F(0, 0) = 0. Further it is a simple matter to verify that F ∈ C 1 (R 2 ).We will denote by f 1 and f 2 respectively the partial derivative of F respect to t 1 and t 2 .Now, for every k ∈ N, the restriction F(t 1 , t 2 )| B((a k+1 ,a k+1 ),1) attains its maximum in (a k+1 , a k+1 ) and one has F(a k+1 , a k+1 ) = (a k+1 ) 4 .Clearly lim sup On the other hand, by setting ξ k = a k+1 − 1 for every k ∈ N, one has max Hence, condition (A2) is provided.Now, let G : R 2 → R be a function defined by By definition G ∈ C 1 (R 2 ) and G(0, 0) = 0.For any sequence {ρ k } k∈N such that lim All hypotheses of Theorem 3.1 are satisfied.Then for all (λ, µ) ∈]0, +∞[×[0, +∞[, the system admits a sequence of weak solutions which is unbounded in W 1,2 0 ([0, 1]) × W 1,2 0 ([0, 1]).Remark 3.3.Under the conditions A = 0 and B = +∞, Theorem 3.1 concludes that for every λ > 0 and for each , the system (D λ,µ ) admits infinitely many weak solutions in X.Moreover, if G ∞ = 0, the result holds for every λ > 0 and µ ≥ 0.
We explicitly observe that assumption (A2) in Theorem 3.1 could be replaced by the following more general condition .
We have the same conclusion as in Theorem 3.1 with Λ replaced by Λ := λ 1 , λ 2 .
r[) admits a global minimum, which is a critical point (local minimum) of I λ in X.(b) If γ < +∞ then, for each λ ∈]0, 1 γ [, 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 lim n→+∞ Φ(u n ) = +∞.(c) If δ < +∞ then, for each λ ∈]0, 1 δ [, 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 of pairwise distinct critical points (local minima) of I λ which weakly converges to a global minimum of Φ.

and λ 2 Example 3 . 2 .
= +∞.Theorem 3.1, taking into account the positivity of f and g, ensures the conclusion.We now exhibit an example in which the hypotheses of Theorem 3.1 are satisfied.Put p 1 = p 2 = 2, [a, b] = [0, 1] and consider the increasing sequence of positive real numbers given by a 1 = 2, a k+1 = ka 2 k + 2, for every k ∈ N. Let F : R 2 → R be a function such that