High perturbations of a new Kirchhoff problem involving the p-Laplace operator

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                </mml:math></jats:alternatives></jats:inline-formula>) is a bounded smooth domain. Under suitable conditions on <jats:italic>g</jats:italic>, we show the existence and multiplicity of solutions in the case of high perturbations (<jats:italic>λ</jats:italic> large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.</jats:p>

In what follows, we suppose that the continuous function g satisfies the following conditions: (g 1 ) g ∈ C 1 ( × R, R) and g(x, s) ≤ C 1 + |s| r-1 ∀(x, s) ∈ × R, where C > 0 and p < r < p * ; (g 2 ) g(x, s) = o(|s| p-1 ) as s → 0; (g 3 ) (AR-condition) there exist θ ∈ (p, p * ) and T > 0 such that where G(x, s) = s 0 g(x, t) dt; |t| p = ∞ uniformly in x ∈ ; (g * * 3 ) G(x, 0) ∈ L 1 ( ) and there exists σ ∈ L 1 ( ) such that for all 0 ≤ t ≤ s or s ≤ t ≤ 0, where G(x, t) := g(x, t)t -pG(x, t); (g 4 ) g(x, -s) = -g(x, s) for all (x, s) ∈ × R. The novelty of our work is the fact that we combine several different phenomena in one problem. The features of this paper are the following: (1) The continuous function g may satisfy the Ambrosetti-Rabinowitz condition or not. (2) The presence of the new nonlocal term (ab |∇u| 2 dx). To the best of our knowledge, there few papers proving the existence and multiplicity of solutions with the combined effects generated by the above features.
Recently, nonlocal problems and operators have been widely studied in the literature and have attracted the attention of a lot of mathematicians coming from different research areas. A typical model proposed by Kirchhoff [4] serves as a generalization of the classic D' Alembert wave equation by taking into account the effects of the changes in the length of the strings during the vibrations. Thanks to the pioneering work of Lions [13], a lot of attention has been drawn to these nonlocal problems during the last decade. After that, some studies on this kind of problems have been performed by using different approaches, see [2, 6-12, 14, 15, 18, 19, 23] and the references therein.
In this paper, we mainly consider a new Kirchhoff problem involving the p-Laplace operator, that is, the form with a nonlocal coefficient (ab |∇u| p dx). Its background is derived from negative Young's modulus, when the atoms are pulled apart rather than compressed together and the strain is negative. Recently, the authors in [22] first studied this kind of problem where 2 < p < 2 * := (2N)/(N -2), and they obtained the existence of solutions by using the mountain pass lemma. Furthermore, some interesting results have been obtained for this kind of Kirchhoff-type problem. We refer the readers to [1,3,5,16,[20][21][22] and the references therein. From the above-mentioned papers, it is a natural question to see what results can be recovered when this new Kirchhoff problem involves the p-Laplace operator. Compared to the above papers, some difficulties arise in our paper when dealing with problem (1.1), because of the appearance of the nonlocal coefficient (ab |∇u| p dx) which provokes some mathematical difficulties, and these make the study of problem (1.1) particularly interesting. In addition, we need more delicate estimates which are not trivial, the method of this paper is obviously different from the literature works mentioned above.
If the nonlinear term g satisfies not the AR-condition, our next main result in this paper is the following. Theorem 1.3 Suppose that assumptions (g 1 )-(g 2 ), (g * 3 ), (g * * 3 ), and (g 5 ) are fulfilled. Then there exists λ 3 > 0 such that, for any λ ≥ λ 3 , problem (1.1) admits a sequence of solutions with unbounded energy.
there is no doubt that we encounter serious difficulties because of the lack of compactness. To overcome the challenge, we must estimate precisely the value of c and give a threshold value. So the variational technique for problem (1.1) becomes more delicate. To the best of our knowledge, the present paper results have not been covered yet in the literature. This paper is organized as follows. In Sect. 2, we give some necessary preliminary knowledge on the functional setting and prove the Palais-Smale compactness condition. In Sect. 3, we prove Theorem 1.1 by using the mountain pass theorem. In Sect. 4, we prove Theorems 1.2 and 1.3 via the symmetric mountain pass theorem where the nonlinear term g satisfies the AR-condition or not, respectively.

Preliminaries and compactness results
We seek weak solutions to problem (1.1) in W 1,p 0 ( ) which is the usual Sobolev space with respect to the norm u = |∇u| p dx. We then have that W 1,p 0 ( ) is continuously and compactly embedded into the Lebesgue space L τ ( ) endowed the norm |u| τ = ( |u| τ dx) 1 τ , p < τ < p * . Denote by S τ the best constant for this embedding, that is, In particular, if S is the best constant for the embedding W 1,p 0 ( ) → L p * ( ), then it is defined by We consider the energy functional J λ : W It is well known that a critical point of J λ is a weak solution of problem (1.1) and the func- [17]). Denote by J λ the derivative operator of J λ in the weak sense. Then Proof First, let {u n } n ⊂ W 1,p 0 ( ) be a (PS) c sequence associated with the functional J λ , that is, Then, {u n } n is bounded in W 1,p 0 ( ). In fact, arguing by contradiction, we assume that, passing eventually to a subsequence, still denoting by {u n } n , we have u n → +∞ as n → +∞. By (g 3 ) we have Thus, (2.4) leads to contradiction since 1 < p. Therefore, there exists u ∈ W 1,p 0 ( ) such that up to a subsequence u n u weakly in W 1,p 0 ( ) and u n → u strongly in L s ( ) with p ≤ s < p * .
In the following, we will prove u n → u strongly in W 1,p 0 ( ).
In fact, by the Hölder inequality, one has and thus On the other hand, by conditions (g 1 ) and (g 2 ), we have that for every ε > 0 there exists C ε > 0 such that This fact implies that Therefore, we can deduce from (2.5) and (2.6) that ab |∇u n | p dx |∇u n | p-2 ∇u n ∇(u nu) dx → 0 as n → ∞. (2.7) Since {u n } n is bounded in W 1,p 0 ( ), passing to a subsequence, if necessary, we may assume that |∇u n | p dx → d ≥ 0 as n → ∞.
Next, we distinguish the following two steps.
Step I: d = a b . In this step, we have ab |∇u n | p dx → 0 as n → ∞.
This means that {ab |∇u n | p dx} is bounded.
Step II: d = a b . In this step, we have Then ψ (u), v = λ |u| q-2 uv dx + g(x, u)v dx.
By the fundamental lemma of the variational method (see [17]), it follows that u = 0. So Hence, we see that This is a contradiction since Then (2.8) is not true and d = a b , this means that any subsequence of {ab u n p } does not converge to zero. Therefore there exists δ > 0 such that |ab u n p | > δ when n is large enough. It is clear that {ab u n } is bounded. Therefore, it follows from (2.7) that |∇u n | p-2 ∇u n ∇(uu n ) dx → 0 as n → ∞.
Thus by the (S + ) property, u n → u strongly in W 1,p 0 ( ). The proof is complete.

Proof of Theorem 1.1
In this section, we first begin by giving the following general mountain pass theorem (see [17]).

Theorem 3.1 Let E be a Banach space and I
there exists e ∈ E satisfying e E > ρ such that I λ (e) < 0.
and there exists a (PS) c sequence {u n } n ⊂ E. Now, we begin proving that J λ satisfies the assumptions of the mountain pass geometry.

Lemma 3.1 Under conditions (g 1 )-(g 3 ), the functional J λ satisfies the mountain pass geometry, that is,
Proof First, by conditions (g 1 ) and (g 2 ), we have that for every ε > 0 there exists C ε > 0 such that

By (2.1) and (2.3), we have
Taking ε > 0 satisfies a 2 -εS -p p > 0. Then, we can choose ρ, α > 0 such that J λ (u) ≥ α for u = ρ, since p < q and p < r. Hence (A 1 ) in Theorem 3.1 holds. Next, we verify condition (A 2 ) of Theorem 3.1. By (g 3 ) we know that, for all T > 0, there exists C T > 0 such that Set v ∈ C ∞ 0 ( ) and v = 0. It follows from (3.2) that From the fact that q, θ > p, we deduce that J λ (t 0 v) < 0 and t 0 v > ρ for t 0 large enough. Set e = t 0 v. Hence e is the required function and (A 2 ) in Theorem 3.1 is valid. This completes the proof.
Proof of Theorem 1.1 Now, we claim that for all sufficiently large λ.
In the following, we prove that Arguing by contradiction, we can assume that there exist t 0 > 0 and a sequence {λ n } n with λ n → ∞ as n → ∞ such that t λ n → t 0 as n → ∞. By the Lebesgue dominated convergence theorem, we deduce that |t λ n ν| q dx → |t 0 ν| q dx as n → ∞, from which it follows that λ n |t λ n ν| q dx → +∞ as n → ∞.
Clearly, Lemmas 2.1, 3.1, and Theorem 3.1 give the existence of nontrivial critical points of J λ , and this concludes the proof.

Proof of Theorems 1.2 and 1.3
To prove Theorems 1.2 and 1.3, we shall use the following symmetric mountain pass theorem in [17]. (I 3 ) There exist constants ρ, α > 0 such that J(u) ≥ α for all u ∈ ∂B ρ Z; Then J possesses an unbounded sequence of critical values.
Proof of Theorem 1.2 We shall apply Theorem 4.1 to J λ . On the one hand, we know that On the other hand, using a similar discussion as in (3.4), there exists λ 2 > 0 such that for all λ > λ 2 .
By Lemma 2.1, one knows that J λ satisfies the (PS) c condition. Then (I 1 ) of Theorem 4.1 is satisfied.
In the following, we will prove that assumptions (I 3 ) and (I 4 ) of Theorem 4.1 are satisfied.
In fact, the proof of (I 3 ) is similar to the proof of (i) in Lemma 3.1. So, we omit the proof here.
In order to prove (I 4 ) of Theorem 4.1, we take X is the finite dimensional subspace of Banach space X. By (3.2), we deduce that Because all the norms on the finite dimensional subspace X are equivalent, there exists C > 0 such that |u| q dx ≥ C X u q and |u| θ dx ≥ C X u θ .
From these and (4.1), we have Thus, there exists R = R( X) such that we can derive J λ (u) < 0 for all u ∈ X with u ≥ R.
Since all the assumptions of Theorem 4.1 are satisfied, problem (1.1) possesses infinitely many nontrivial solutions with unbounded energy.
Indeed, we take {u n } ⊂ W 1,p 0 ( ) to be a (PS) c sequence associated with the functional J λ , that is, J λ (u n ) → c and J λ (u n ) → 0 as n → ∞. By using (g * 3 ) and (g * * 3 ), we have This fact implies that {u n } is bounded in W 1,p 0 ( ) since 1 < p, σ ∈ L 1 ( ) and G(x, 0) ∈ L 1 ( ). Arguing as in the proof of Lemma 2.1, we can also prove u n → u strongly in W 1,p 0 ( ), we omit the details here. Next, we claim that assumptions (I 3 ) and (I 4 ) of Theorem 4.1 are also satisfied. In fact, it is easy to prove that (I 3 ) of Theorem 4.1 is satisfied. In the following, we prove that (I 4 ) of Theorem 4.1 is satisfied. On the one hand, by (g * 3 ), we have Hence, we deduce that Similarly, arguing as in the proof of Theorem 1.2, we know that (I 4 ) of Theorem 4.1 is satisfied. Thus, problem (1.1) possesses infinitely many nontrivial solutions with unbounded energy.