Existence of Homoclinic Orbits for Unbounded Time-dependent P-laplacian Systems

In this paper, we consider the following ordinary p-Laplacian system d dt | ˙ u(t)| p−2 ˙ u(t) − ∇K(t, u(t)) + ∇W(t, u(t)) = f (t), (HS) where t ∈ R and p > 1. Using the Mountain Pass Theorem, we establish the existence of a nontrivial homoclinic solution for (HS) under new assumptions on the growth of the potential which allow W(t, x) to be either super p-linear or asymptotically p-linear at infinity. Also, contrary to previous works, W(t, x) will be neither periodic nor bounded with respect to the variable t. Recent results in the literature are generalized even if p = 2.


Introduction
Consider the ordinary p-Laplacian system where t ∈ R, p > 1, K, W : R × R N → R are C 1 -maps and f : R −→ R N is a continuous and bounded function.We will say that a solution u of (HS) is a nontrivial homoclinic (to 0) if u ≡ 0 and u(t) −→ 0 as t −→ ±∞.
(1.1) Homoclinic orbits were introduced by Poincaré more than a century ago, and since then, they became a fundamental tool in the study of chaos.Their existence has been extensively Email: daouas − adel@yahoo.frinvestigated in the last two decades in many papers via critical point theory.In particular, the following second-order systems were considered in many works (see [1, 3-8, 11, 13, 16, 17, 19, 23, 26]) ü(t) − L(t)u(t) + ∇W(t, u(t)) = 0 (1.2) where L(t) is a symmetric matrix valued function.Later, the authors of [7] introduced the more general system (1.1)where the quadratic function (L(t)x, x) is replaced by K(t, x).
Most of the previous works treat the superquadratic case under the global Ambrosetti-Rabinowitz condition, i.e., there exists µ > 2 such that 0 < µW(t, x) ≤ (∇W(t, x), x), for all (t, x) ∈ R × R N \{0}. (AR) Moreover, they suppose that L(t) and W(t, x) are either periodic in t or independent of t.
In the case where W(t, x) and L(t) or K(t, x) are neither autonomous nor periodic, ∇W(t, x) is usually bounded with respect to the first variable.Indeed, a variant of the following condition is used: In a recent paper the authors of [9] studied the problem (HS) under new superquadratic conditions which allow W to be neither periodic nor bounded in t.Particularly, they suppose where the functions W 1 , W 2 satisfy some (AR)-type conditions to be either increasing or decreasing (see [9], Lemma 2.5).
Motivated by the above mentioned works, in the present paper we study the existence of homoclinic solutions for (HS) under more general conditions which cover the case of unbounded potentials with respect to the variable t.Here, to overcome the difficulty due to the unboundedness of the domain, a homoclinic solution will be obtained as a limit of a sequence of solutions of some nil-boundary-value problem.The existence of such sequence of solutions is guaranteed through a standard version of the Mountain Pass Theorem.Furthermore, the forcing term f satisfies an easier condition compared to that given in [12,20] mainly.Our results complete and improve recent works in the literature even in the case p = 2.

Preliminary results
Consider for each T > 0 the following problem where f T is the function defined on R by Let equipped with the norm , and I T : E T −→ R, be defined by Then I T ∈ C 1 (E T , R) and it's easy to show that for all u, v ∈ E T , we have By (H 1 ), we obtain, for all u ∈ E T It is well known that critical points of I T are classical solutions of the problem (2.1), (see [2,14]).We will obtain a critical point of I T by using a standard version of the Mountain Pass Theorem.It provides the minimax characterization for the critical value which is important for what follows.For completeness, we give this theorem.where Next we need an extension to the p-case of the following proposition first proved by Rabinowitz in [16].
Lemma 2.2 ([12, 22]).Let u : R −→ R N be a continuous map such that u ∈ L p loc (R, R N ).Then, for all t ∈ R, we have Corollary 2.3.For all u ∈ E T the following inequality holds: Remark 2.4.Note that for T ≥ 1 2 we have 2 Subsequently, we may assume this condition fulfilled.
Lemma 2.5.Assume that (H 1 ) holds, then for all t ∈ R, we have The proof of Lemma 2.5 is a routine so we omit it.
3 Proof of Theorem 1.2 Lemma 3.1.Under the assumptions of Theorem 1.2, the problem (2.1) possesses a nontrivial solution u T ∈ E T .
Proof.It suffices to prove that the functional I T satisfies all the assumptions of the Mountain Pass Theorem.
Step 1.The functional I T satisfies the (PS)-condition, i.e., for every constant c and sequence 2) and (2.3) there exists M T > 0 such that (3.1)

A. Daouas
Using (H 4 ) and Hölder's inequality, from (3.1), we get Without loss of generality, we can assume that u n E T = 0. Then from (H 1 ) and (2.6) we get Combining (3.2) and (3.3) we obtain From (H 4 ), we know that µ − b − p > 0, then the sequence {u n } is bounded in E T .In a similar way to Lemma 2 in [19], we can prove that {u n } has a convergent subsequence in E T .Hence I T satisfies the (PS)-condition.
Step 2. The functional I T satisfies the condition (I 2 ) of the Mountain Pass Theorem.
Step 3. The functional I satisfies the condition (I 3 ) of the Mountain Pass Theorem. Let ), it is easy to see that Furthermore, by (H 3 ), there exists ε 0 > 0 and r > 0 such that For T ≥ T 0 , define where ω = π T 0 and e 1 = (1, 0, . . ., 0) ∈ R N .Then by (2.2) and (3.7)-(3.9),we obtain Thus, we can choose ξ large enough such that e E T > ρ and I T (e) < 0. For our setting, clearly I T (0) = 0, then, by application of the Mountain Pass Theorem, there exists a critical point u T ∈ E T of I T such that I T (u T ) ≥ α for all T ≥ T 0 .Lemma 3.2.u T is bounded uniformly for T ≥ T 0 .Proof.Define the set of paths By Lemma 3.1, we know that there is a solution u T of (2.1) at which Therefore, for all solution u T of (2.1), we get Using the fact that I T (u T ) = 0 and (3.10), the rest of the proof is identical to Step 1 in Lemma 3.1.Hence there exists a constant M 0 > 0, independent of T such that This ends the proof of Lemma 3.2.Now, take an increasing sequence T n −→ ∞ with T 1 > T 0 and consider the problem (2.1) on the interval [−T n , T n ].By the conclusion of Lemma 3.1 and Lemma 3.2, there exists a nontrivial solution u n := u T n of (2.1) satisfying u n E Tn ≤ M 0 , for all n ∈ N.
(3.11) Lemma 3.3.Let (u n ) n∈N be the sequence given above.Then there exists a subsequence (u n j ) j∈N convergent to a certain function u 0 in C 1 loc (R, R N ).Proof.First of all from (2.6) and (3.11), we have and for all n ∈ N. By Hölder's inequality and (3.11), for t 1 ,   From (2.4) and (3.24), we receive our claim.