Infinitely many weak solutions for a mixed boundary value system with ( p 1 , . . . , p m )-Laplacian

The aim of this paper is to prove the existence of infinitely many weak solutions for a mixed boundary value system with (p1, . . . , pm)-Laplacian. The approach is based on variational methods.


Introduction
The aim of this paper is to establish the existence of infinitely many weak solutions for the following mixed boundary value system with (p 1 , . . ., p m )-Laplacian.
Among the papers which have dealt with the nonlinear mixed boundary value problems we cite [1,3,10,13].
We investigate the existence of infinitely many weak solutions for system (1.1) by using Theorem 1.1.This theorem is a refinement, due to Bonanno and Molica Bisci, of the variational principle of Ricceri [12,Theorem 2.5] and represents a smooth version of an infinitely many critical point theorem obtained in [5,Theorem 2.1].
Theorem 1.1.Let X be a reflexive Banach space, Φ : X → R is a continuously Gâteaux differentiable, coercive and sequentially weakly lower semicontinuous functional, Ψ : X → R is sequentially weakly upper semicontinuous and continuously Gâteaux differentiable functional, λ is a positive real parameter.
Put, for each r One has (a) For every r > inf X Φ and every λ ∈ 0, 1 ϕ(r) , the restriction of the functional Φ − λΨ to γ [, the following alternatives hold: either (b 1 ) Φ − λΨ possesses a global minimum, or (b 2 ) there is a sequence {u n } of critical points (local minima) of Φ − λΨ such that lim n→+∞ Φ(u n ) = +∞.
(c) If δ < +∞, then for each λ ∈]0, 1 δ [, the following alternatives hold: either (c 1 ) there is a global minimum of Ψ which is a local minimum of Φ − λΨ, or (c 2 ) there is a sequence {u n } of pairwise distinct critical points (local minima) of Φ − λΨ, with lim n→+∞ Φ(u n ) = inf X Φ which weakly converges to a global minimum of Φ.
The paper is arranged as follows.At first we prove the existence of an unbounded sequence of weak solutions of system (1.1) under some hypotheses on the behaviour of potential F at infinity (see Theorem 3.1).And as a consequence, we obtain the existence of infinitely many weak solutions for autonomous case (see Corollary 3.4).

Preliminaries
Let us introduce notation that will be used in the paper.Let be the Sobolev space with the norm defined by Infinitely many weak solutions for a mixed boundary value system 3 Now, let X be the Cartesian product of m Sobolev spaces X p i , i.e.X = ∏ m i=1 X p i endowed with the norm In order to study system (1.1), we will use the functionals Φ, Ψ : X → R defined by putting for every u = (u 1 , . . ., u m ) ∈ X.
Clearly, Φ is coercive, weakly sequentially lower semicontinuous and continuously Gâteaux differentiable and the Gâteaux derivative at point u = (u 1 , . . ., u m ) ∈ X is defined by On the other hand Ψ is well defined, weakly upper sequentially semicontinuous, continuously Gâteaux differentiable and the Gâteaux derivative at point u = (u 1 , . . ., u m ) ∈ X is defined by A critical point for the functional Hence, the critical points for functional I λ := Φ − λΨ are exactly the weak solutions to system (1.1).
A function u Standard methods show that solutions to system (1.1) coincide with weak ones when F is a C 1 function.Now, put where ) we suppose λ 1 = 0 if B = +∞, and

Main results
Our main result is the following theorem.
Let {c n } be a real sequence such that lim n→+∞ c n = +∞ and lim Infinitely many weak solutions for a mixed boundary value system 5 Hence, for all n ∈ N, one has , therefore, since from (i 2 ) one has A < ∞, we obtain For all n ∈ N define clearly, ω n = (ω 1n , . . ., ω mn ) ∈ X and where k is given by (2.6).
Taking into account (i 1 ), we have Then, by using (3.2) and (3.3) for all n ∈ N we have Now, if B < ∞, we fix ∈ k λB , 1 , from (3.1) there exists ν ∈ N such that On the other hand, if B = +∞, we fix Hence, our claim is proved.Since all assumptions of Theorem 1.1 (b) are verified, the functional I λ = Φ − λΨ admits a sequence {u n } of critical points such that lim n→∞ u n = +∞ and the conclusion is achieved.Remark 3.2.In Theorem 3.1 we can replace r → +∞ by r → 0 + , applying in the proof part (c) of Theorem 1.1 instead of (b).In this case a sequence of pairwise distinct weak solutions to the system (1.1) which converges uniformly to zero is obtained.Remark 3.3.We consider the system by using the usual norm in X p i , and the constant k = max 1≤i≤m we can prove in a very similar way to that used to prove Theorem 3.1, that for each λ ∈]λ 1 , λ 2 [, with λ 1 and λ 2 given by (2.5), the system (3.5) has a sequence of weak solutions which is unbounded in X .Now, we point out a special case of Theorem 3.1.
Then, the system possesses a sequence of pairwise distinct solutions which is unbounded in X.