Nontrivial Solutions for a Class of p-Kirchhoff Dirichlet Problem

<jats:p>This paper is devoted to the following<jats:italic>p</jats:italic>-Kirchhoff type of problems<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mfenced open="{" close="" separators="|"><mml:mrow><mml:mtable class="cases"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">∫</mml:mo></mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:msub><mml:mrow><mml:msup><mml:mrow><mml:mfenced open="|" close="|" separators="|"><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mtext>d</mml:mtext><mml:mi>x</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math>with the Dirichlet boundary value. We show that the<jats:italic>p</jats:italic>-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.</jats:p>


Introduction and Main Results
Consider the following p-Kirchhoff type of problems with the Dirichlet boundary value: where Ω is a smooth bounded domain in R N , a > 0 and b ≥ 0 are real constants, Δ p denotes the p-Laplacian operator by △ p u � div(|∇u| p− 2 ∇u)(1 < p < N), and f(x, u) is continuous on Ω × R.
We look for the weak solutions of (1) which are the same as the critical points of the functional I: W 1,p 0 (Ω) ⟶ R defined by where F(x, u) � I is of W 1 (Ω) and u, v ∈ W 1,p 0 (Ω) with derivatives given by Problem (1) began to attract the attention of researchers after the work of Lions [1], where a functional analysis approach was proposed to attack it. Since then, much attention has been paid to the existence of nontrivial solutions, sign-changing solutions, ground state solutions, multiplicity of solutions, and concentration of solutions.
Many researchers studied the existence of weak solution of (1) in N-dimensional whole spaces (see [2][3][4][5][6][7] and references therein). For example, Wu [2] showed that problem has a nontrivial solution and a sequence of high-energy solutions by using the mountain-pass theorem and the symmetric mountain-pass theorem. Similar consideration can be found in Nie and Wu [3], where radial potentials were considered. Jia and Li [6] studied multiplicity and concentration behaviour of positive solutions for Schr€ odinger-Kirchhoff type equations involving the p-Laplacian. Recently, Li and Niu [7] have obtained the existence of nontrivial solutions for the p-Kirchhoff type equations with critical exponents by the Ekeland variational principle and mountain-pass lemma.
Kirchhoff-type problem setting on a bounded domain also attracts a lot of attention, see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein. For example, we refer to some recent works [8,9] in which some interesting results on the problems with Dirichlet or Neumann boundary conditions have been obtained. Chung [10] studied Kirchhoff type problems with Robin boundary conditions and indefinite weights. Chen et al. [11] treated problem (1) when f(x, t) � λa(x)|u| q− 2u + b(x)|u| r− 2 u(1 < q < p � 2 < r < 2 * ), using the Nehari manifold and fibering maps, and they established the existence of multiple positive solutions for (1). In [14][15][16][17][18][19], the authors studied the existence of solutions of the p-Kirchhoff problem in the following form − [M(‖u‖ p )] p − 1Δ p u � f(x, u), x ∈ Ω or p-Kirchhoff problem such as (1). In [16,17], the authors obtained the existence and multiplicity of solutions for the p-Kirchhoff equation by using the genus theory. In [18,19], using variational methods, the researchers studied fractional p-Kirchhoff equations and received the multiplicity and centration of solutions. In [20,21], the authors used the fountain theorem and concentration-compactness principle to consider multiplicity of solutions for p-Kirchhoff equations. By applying a variant of the mountain-pass theorem and the Ekeland variational principle, Cheng et al. [22] obtained the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity. In [23,24], the authors studied the equations with nonlinearities which are superlinear in one direction and linear in the other. ey use the Ambrosetti--Rabinowitz (AR) condition to express the superlinear growth at +∞. e AR condition has been used in a technical but crucial way not only in establishing the mountain-pass geometry of the functional but also in obtaining the boundedness of Palais-Smale (PS) sequences. Since the publication of [25], the AR condition has been used extensively throughout the literature (see [26,27]). However, this condition is very restrictive, eliminating many nonlinearities.
ere are always many functions that satisfy the AR condition. For example, Under the motivation of Pei and Zhang [28], the aim of this paper is to consider existence of nontrivial solutions for problem (1) with 1 < p < N. We shall assume that the nonlinearity term f(x, t) does not satisfy the AR condition and it is asymmetric as t ∈ R approaches +∞ and − ∞.
Nevertheless, to ensure the global compactness, one needs to impose the subcritical growth condition on the nonlinearity f(x, t): there exists a position constant C 0 such that where p < q < p * � pN/(N − p). However, under the motivation of Lan and Tang [29], we consider a class of elliptic partial differential equations with a more general growth condition. We assume that the following condition holds: It is weaker than subcritical growth condition.
We next state the other hypotheses on f(x, t). Suppose f(x, t) ∈ C(Ω × tR), and this satisfies: x ∈ Ω, then problem (1) has at least one nontrivial solution.
Here, λ 1 is the first eigenvalue of (− Δ p , W 1,p 0 (Ω)) and there is a corresponding eigenfunction e paper is organized as follows. In Section 1, we introduced the main purpose of this paper and get some conclusions. e proofs of main results will be given in Section 2. We denote various positive constants as C or C i (i � 0, 1, 2, 3, . . .) for convenience.

Proofs of the Main Results
Proof of eorem 1.
Step 1 (the (PS) c condition). Let u n ⊂ W 1,p 0 (Ω) be a (PS) c sequence, then we have First, we prove that u n is bounded. Assume for contradiction that, up to a sequence, ‖u n ‖ ⟶ +∞ as n ⟶ ∞. Since I ′ (u n ) ⟶ 0, for every w ∈ W 2 Discrete Dynamics in Nature and Society Define z n � u n /‖u n ‖. Obviously, z n ∈ W 1,p 0 (Ω) with ‖z n ‖ � 1.
en, there exists a subsequence (denoted also by z n ) such that where z 0 ∈ W 1,p 0 (Ω) and q ∈ W 1,p 0 (Ω). Dividing both sides of (10) by ‖u n ‖ 2p− 1 , we obtain Passing to the limit, we get Let us now prove that z 0 (x) ≤ 0 for a.e. x ∈ Ω. To verify this, we chose w � z + 0 � max z 0 , 0 in (13), and we have where Ω + � x ∈ Ω | z 0 (x) > 0 . Moreover, using lim n⟶∞ u n (x) � +∞ a. e. x ∈ Ω + and by (A 3 ), we have erefore, if |Ω + | > 0, by Fatou's lemma, we will obtain that which contradicts (10). us, |Ω + | � 0 and the claim (z 0 (x) ≤ 0) is proved. Clearly, z 0 (x) ≡ 0. By contradiction z 0 � 0, we know that z n ⇀ 0 in W 1,p 0 (Ω) and z n ⟶ 0 in L r (Ω), where p ≤ r ≤ p * . erefore, we obtain ‖z n ‖ ⟶ 0, which is a contradiction. When z 0 < 0, we have u n � z n ‖u n ‖ ⟶ − ∞ as n ⟶ ∞. If this is the case, by 〈I ′ (u n ),z n 〉 � I ′ (u n )z n � o(1), we first obtain Dividing both sides of (17) by ‖u n ‖ 2p− 1 , we obtain Passing to the limit, we get en, Using Lebesgue's dominated convergence theorem and (A 2 ) for the left hand side of (20), we have en, (20) can be written as follows: bv is means that v is an eigenvalue, which contradicts our assumption. So, u n is bounded. Now, we prove that u n has a convergent subsequence. By the continuity of embedding, we have ‖u n ‖ p * p * ≤ C 1 < ∞ for all n. Going if necessary to a subsequence, we have where p ≤ q < p * .

(23)
By (F), for every ε > 0, we can find a constant C(ε) > 0 such that Discrete Dynamics in Nature and Society Hence, Ω (x, u n )u n dx, n ∈ N is equiabsolutely continuous. Moreover, f(x, u n )u n , n ∈ N is equi-integrable and bounded in L 1 (Ω), and f(x, u n )u n ⟶ f(x, u)u in measure. From the Vitali convergence theorem, it follows that By (F), for every ε > 0, we can find a constant C(ε) > 0 such that where c 1 ≥ ( Ω |u n | p * dx) p * − 1/p * for all n and c 2 � ( Ω |u| p * dx) 1/p * are positive constants with the assumption of u ≡ 0. Form H€ older's inequality, for every E ⊆ Ω, we have Hence, Ω f(x, u n )u dx, n ∈ N is equiabsolutely continuous. Moreover, f(x, u n )u dx, n ∈ N is equi-integrable and bounded in L 1 (Ω), and f(x, u n )u ⟶ f(x, u)u in measure. From the Vitali convergence theorem, it follows that it follows that us, we have u n ⟶ u in W 1,p 0 (Ω), which means that I satisfies the (PS) condition.
Step 3 (a critical value of I ). For e in Step 2, we define It turns out that the mountain-pass theorem holds. en, c is a critical value of I.
Discrete Dynamics in Nature and Society It contradicts (43). Hence, u n is bounded. According to Step 1, the proof of eorem 1, we have u n ⟶ uinW 1,p 0 (Ω), which means that I satisfies (C c ).