Infinitely Many Homoclinic Solutions for Fourth Order p-Laplacian Differential Equations

The existence of infinitely many homoclinic solutions for the fourth-order differential equation ( φp (u′′ (t)) )′′ + w ( φp (u′ (t)) )′ + V(t)φp (u (t)) = a(t) f (t, u(t)), t ∈ R is studied in the paper. Here φp(t) = |t|p−2 t, p ≥ 2, w is a constant, V and a are positive functions, f satisfies some extended growth conditions. Homoclinic solutions u are such that u(t)→ 0, |t| → ∞, u 6= 0, known in physical models as ground states or pulses. The variational approach is applied based on multiple critical point theorem due to Liu and Wang.


Introduction
In this paper, we study the existence of infinitely many nonzero solutions homoclinic solutions for the fourth-order p-Laplacian differential equation ϕ p u (t) + w ϕ p u (t) + V(t)ϕ p (u (t)) = a(t) f (t, u(t)), where t ∈ R, w is a constant, ϕ p (t) = |t| p−2 t, for p ≥ 2, V is a positive bounded function, a is a positive continuous function and f ∈ C 1 (R, R) satisfies some growth conditions with respect to p. As usual, we say that a solution u of (1) is a nontrivial homoclinic solution to zero solution of (1) if They are known in phase transitions models as ground states or pulses (see [1]). The existence of homoclinic and heteroclinic solutions of fourth-order equations is studied by various authors (see [2][3][4][5][6][7][8][9][10][11][12] and references therein). Sun and Wu [4] obtained existence of two homoclinic solutions for a class of fourth-order differential equations: where w is a constant, λ > 0, 1 ≤ p < 2, a ∈ C (R, R + ) and h ∈ L 2 2−p (R) by using mountain pass theorem.
Yang [8] studies the existence of infinitely many homoclinic solutions for a the fourth-order differential equation: where w is a constant, a ∈ C (R) and f ∈ C (R × R, R). A critical point theorem, formulated in the terms of Krasnoselskii's genus (see [13], Remark 7.3), is applied, which ensures the existence of infinitely many homoclinic solutions.
We suppose the following conditions on the functions a, f and V. (A) a ∈ C(R, R + ) and a(t) → 0 as |t| → +∞. (F 1 ) There are numbers p and q s.t. 1 < q < 2 ≤ p and for f ∈ C 1 (R, R) Denote by X the Sobolev's space The functional I : X → R is defined as follows where Φ(t) = |t| p p for p ≥ 2. Under conditions (A), (F 1 ) − (F 3 ) and V the functional I is differentiable and for all u, v ∈ X we have where ., . means the duality pairing between X and it's dual space X * . The homoclinic solutions of the Equation (1) are the critical points of the functional I, i.e., u 0 is a homoclinic solution of the problem if I (u 0 ), v = 0 for every v ∈ X (see [6,11,12]).
Let v 0 = min{1, v 1 }, where v 1 is the positive constant from condition (V). Our main result is: Let p ≥ 2, w < v 0 w * and the functions a, f and V satisfy the assumptions (A), (F 1 ) − (F 3 ) and (V) . Then the Equation (1) has at least one nonzero homoclinic solution u 0 ∈ X. Additionally if (F 4 ) holds, the Equation (1) has infinitely many nonzero solutions u j such that ||u j || ∞ → 0 as j → ∞.

Remark 1.
An example of a function f (t, u), which satisfies the assumptions (F 1 ) − (F 4 ) is as follows. Let p = 3, q = 3 2 and f (t, u) = α(t)|u| 1/2 u, where As an open problem we state the existence of weak solutions of the problem when 1 < q < p < 2. This paper is organized as follows. In Section 2 we present the variational formulation of the problem and critical point theorems used in the proof of the main result. In Section 3, we give the proof of Theorem 1.

Preliminaries
In this section we give the variational formulation of the problem and present two critical point theorems.
Let X 1 be the Sobolev's space equipped by the norm and v 0 = min{1, v 1 }. The next lemma shows that under condition (V) for w < v 0 w * the norms ||.|| and ||.|| X are equivalent and X = X 1 .
Proof of Lemma 1. In view of Lemma 4.10 in [14], there exists a positive constant which completes the proof.
By Brezis [15], Theorem 8.8 and Corollary 8.9 for u ∈ X and s > p We consider the functional I : X → R where Φ(t) = |t| p p for p ≥ 2. One can show that under conditions (A), (F 1 ) − (F 3 ) and V the functional I is differentiable and for all u, v ∈ X we have Proof of Lemma 2. The embedding X ⊂ L p a (R) is continuous by the boundedness of the function a by (A). We show that the inclusion is compact. Let u j ⊂ X be a sequence such that ||u j || ≤ M and u j u weakly in X. We'll show that u j → u strongly in L p a (R). Without loss of generality we can assume that u = 0, considering the sequence u j − u . By (A) for any ε > 0, there exists R > 0, such that for |t| ≥ R Then |t|≥R a(t)|u j (t)| p dt ≤ εM p 2(1 + M p ) .
By Sobolev's imbedding theorem u j → 0 strongly in C([−R, R]) and there exists j 0 such that for j > j 0 : Then, for j > j 0 we have R a(t)|u j (t)| p dt < ε, which shows that u j → 0 strongly in L p a (R).
By Lemma 2, u j u weakly in X implies that there exists a subsequence {u j }, such that u j → u strongly in L p a (R). By analogous way as above we have that there exists B > 0, such that Let 0 < a R < a(t) ≤ A for |t| ≤ R. By u j → u strongly in L p a (R) it follows that |t|≤R a(t)|u j (t) − u(t)| p dt ≥ a R |t|≤R |u j (t) − u(t)| p dt → 0 and u j (t) − u(t) → 0 a.e. in |t| ≤ R. Then, by Lebesque's dominated convergence theorem Let j 0 is sufficiently large, such that for j > j 0 , 0 ≤ I R < ε 2 . Then by (7) for j > j 0 we have which completes the proof.
Next we have: , (V) the functional I ∈ C 1 (X, R) and the identity (6) holds for all u, v ∈ X. holds.
By (F 2 ) we have Then, {u j } is a bounded sequence in X and up to a subsequence, still denoted by {u j }, u j u weakly in X. There exists M 2 > 0, such that ||u j || ≤ M 2 , ||u|| ≤ M 2 . By Lemma 2, u m → u in L 2 a (R) and by Lemma 3, f (t, u m (t)) → f (t, u(t)) in L 2 a (R) . By Hölder inequality we have: As in the proof of Lemma 3, by assumption (F 2 ), b ∈ L p p−q (R) and Hölder inequality we have for Then, by u j → u in L 2 a (R) it follows that I j → 0 as j → ∞. Next, we have which shows that u j → u in X.
Next, we recall a minimization theorem which will be used in the proof of Theorem 1. (see [16], Theorem 2.7 of [13]).

Theorem 2.
(Minimization theorem) Let E be a real Banach space and J ∈ C 1 (E, R) satisfying (PS) condition. If J is bounded below, then c = inf E I is a critical value of J.
We will use also the following generalization of Clark's theorem (see Rabinowitz [13], p. 53) due to Z. Liu and Z. Wang [17]: Theorem 3. (Generalized Clark's theorem, [17]) Let E be a Banach spa ce, J ∈ C 1 (E, R). Assume that J satisfies the (PS) condition, it is even, bounded from below and J(0) = 0. If for any k ∈ N, there exists a k−dimensional subspace E k of E and ρ k > 0 such that sup E k ∩S ρ k J < 0, where S ρ = {u ∈ E , u E = ρ}, then at least one of the following conclusions holds: 1. There exists a sequence of critical points {u k } satisfying J(u k ) < 0 for all k and lim k→∞ u k E = 0. 2. There exists r > 0 such that for any 0 < α < r there exists a critical point u such that u E = α and J(u) = 0.
Note that Theorem 3 implies the existence of infinitely many pairs of critical points (u k , −u k ), u k = 0 of J, s.t. J(u k ) ≤ 0, lim k→+∞ J(u k ) = 0 and lim k→+∞ u k E = 0. Lemma 6. Assume that assumptions (A), (F 2 ) and (V) hold. Then the functional I is bounded from below.
Proof of Lemma 6. By (F 2 ) and the proof of Lemma 3 we have and By p > q it follows that I is bounded from below functional.

Proof of the Main Result
In this section we prove Theorem 1. The proof is based on the minimization Theorem 2 and multiplicity result Theorem 3. Their conditions are satisfied according to Lemmas 1-6. By 1 < q < p and the last inequality it follows that I(v) < 0 for v ∈ S ρ n−1 := {u ∈ X n : ||u|| = ρ}. Finally, all assumptions of Theorem 3 are satisfied and by Remark 1 there exist infinitely many weak solutions {u j } of the problem (1), such that I({u j }) ≤ 0 and ||u j || → 0. By imbedding X ⊂ L ∞ (R) it follows that ||u j || ∞ → 0 as j → ∞ which completes the proof.

Conlusions
In this paper, we obtained the existence of infinitely many homoclinic solutions of Equation (1) under conditions (A), (F 1 ) − (F 4 ), (V) in the case 1 < q < 2 ≤ p. The equation is an extension of the stationary Fisher-Kolmogorov equation which appears in the phase transition models. The variational approach is applied based on the multiple critical point theorem due to Liu and Wang. It will be interesting to extend the result to the case 1 < q < p < 2.