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Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations
Boundary Value Problems volume 2019, Article number: 125 (2019)
Abstract
We are concerned with the following p-biharmonic equations:
where \(2< 2p<N\), \(\Delta _{p}^{2}u=\Delta (|\Delta u|^{p-2} \Delta u)\), the function \(\varphi (x,v)\) is of type \(\lvert v \rvert ^{p-2}v\), \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\), the potential function \(V:\mathbb{R}^{N}\to (0,\infty )\) is continuous, and \(f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems.
1 Introduction
We are concerned with the following p-biharmonic equations:
where \(2< 2p<N\), \(1< p< p_{*}:=\frac{Np}{N-2p}\), \(\Delta _{p} ^{2}u=\Delta (|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator, the function \(\varphi (x,v)\) is of type \(\lvert v \rvert ^{p-2}v\), \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\), the potential function \(V:\mathbb{R}^{N}\to (0,\infty )\) is continuous, and \(f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition.
The fourth-order differential equations arise in the study of deflections of elastic beams on nonlinear elastic foundations. Thus, they become very significant in engineering and physics. Many authors considered this type of equation in recent years, and we refer to [9, 13, 27] and the references therein. For this reason, the existence of solutions of p-biharmonic equations has been studied by several authors; see [6, 8, 12, 15, 21, 24, 30, 31, 34]. To obtain the existence and multiplicity results for the p-Laplace type operators, which generalize the usual p-Laplacian, the authors in [10, 28] considered the following condition:
for all \(x\in \mathbb{R}^{N}\) and \(v\in \mathbb{R}^{N}\), and for some positive constant d; see also [14]. On the other hand, Kirchhoff in [20] initially proposed the following equation:
which is a generalization of the classical D’Alembert’s wave equation. Also, Woinowsky and Krieger [33] in the 1950s considered a stationary analogue of the evolution equation of Kirchhoff type, namely
as a model for the deflection of an extensible beam on nonlinear foundations. Here, u denotes the displacement, f is the force that the foundations exert on the beam, and M models the effects of the small changes in the length of the beam (see, e.g., [3,4,5, 7] for the physics viewpoint model). In view of mathematics, many researchers have extensively studied the existence of weak solutions for the elliptic problem of Kirchhoff type in recent years (see, e.g., [11, 16, 18]). Based on these references, we consider the generalized elliptic equation (P) involving the p-biharmonic and generalized p-Laplacian of Kirchhoff type.
Since the seminal paper of Ambrosetti and Rabinowitz in [2], the existence of solutions for the elliptic problem has been studied by many researchers. A common feature of these works is that the following condition, which is originally due to Ambrosetti and Rabinowitz, is imposed on the nonlinearity f:
-
(AR)
There exist positive constants m and ζ such that \(\zeta >p\) and
$$ 0< \zeta F(x,t)\le f(x,t)t \quad \text{for } x\in \varOmega \text{ and } \lvert t \rvert \ge m, $$where \(F(x,t)=\int _{0}^{t}f(x,s) \,ds\), and Ω is a bounded domain in \(\mathbb{R}^{N}\).
The (AR) condition above is somewhat natural and important to guarantee the boundedness of Palais–Smale sequence of Euler–Lagrange functional for an elliptic equation, however, this condition is very restrictive and eliminates many nonlinearities. Thus, many researchers have tried to drop the (AR) condition for elliptic equations associated with the p-Laplacian; see, e.g., [1, 23, 25, 26, 29].
The purpose of this paper is to study the existence of weak solutions for problem (P) without assuming the (AR) condition, but imposing various assumptions for the divergence part φ and nonlinear term f. In particular, as observed by Remark 1.8 in [23], there are many examples which do not fulfill the condition of the nonlinear term f given in [1, 25, 26]. On the other hand, in case of the whole space \(\mathbb{R}^{N}\), the main difficulty of this problem is the lack of compactness for the Sobolev theorem. In that sense, our study is to pursue two goals. First, we show the existence of nontrivial weak solutions for the problem above using the mountain pass theorem. To be precise, we prove the existence of weak solutions for problem (P) under Cerami condition, as a weak version of the Palais–Smale condition. Also, we try to do analysis using the properties of Kirchhoff function M and function φ. Second, we show the multiplicity of weak solutions to problem (P) via the fountain theorem. To the best of our knowledge, there were no such existence results for our problem in this situation.
2 Preliminaries
In this section, we briefly describe the framework for our problem. We assume that the potential \(V\in C(\mathbb{R}^{N})\) is a continuous function with
-
(V)
\(\inf_{x\in \mathbb{R}^{N}}V(x)>0\), and \(\operatorname{meas} \{x\in \mathbb{R}^{N}:V(x)\le K \}<+\infty \) for all \(K\in \mathbb{R}\).
Also, we set \(D^{p}(\mathbb{R}^{N}) = \{u\in L^{p_{*}}(\mathbb{R}^{N})| \Delta u \in L^{p}(\mathbb{R}^{N})\}\). Thus, we define the function space as follows:
equipped with the norm
For our problem, we first assume that \(M:\mathbb{R}^{+} \to \mathbb{R}^{+}\) satisfies the following conditions:
-
(M1)
\(M\in C(\mathbb{R}^{+})\) satisfies \(\inf_{t\in \mathbb{R}^{+}} M(t) \geq m_{0} > 0\), where \(m_{0}\) is a constant.
-
(M2)
There exists \(\theta \in [1,\frac{N}{N-p})\) such that \(\theta \mathcal{M}(t)=\theta \int _{0}^{t} M(\tau )\,d \tau \geq M(t)t \) for any \(t\geq 0\).
A typical example for M is given by \(M(t)=b_{0} +b_{1}t^{n}\) with \(n>0\), \(b_{0}>0\), and \(b_{1}\geq 0\).
Next, we assume that \(\varphi : \mathbb{R}^{N}\times \mathbb{R}^{N} \to \mathbb{R}^{N}\) is a continuous function with the continuous derivative with respect to v of the mapping \(\varPhi _{0}:\mathbb{R}^{N}\times \mathbb{R}^{N} \to \mathbb{R}\), \(\varPhi _{0}=\varPhi _{0}(x,v)\), that is, \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\). Suppose that φ and \(\varPhi _{0}\) satisfy the following assumptions:
-
(J1)
The equality
$$ \varPhi _{0}(x,\mathbf{0})=0 $$holds for almost all \(x\in \mathbb{R}^{N}\).
-
(J2)
There are a nonnegative function \(a\in L^{p'}(\mathbb{R} ^{N})\) and a nonnegative constant b such that
$$ \bigl\vert \varphi (x,v) \bigr\vert \le a(x)+b \vert v \vert ^{p-1} $$holds for almost all \(x\in \mathbb{R}^{N}\) and for all \(v\in \mathbb{R}^{N}\). Here, \(p'\) is a conjugate number of p.
-
(J3)
The relations
$$ d \vert v \vert ^{p}\le \varphi (x,v)\cdot v\quad \text{and} \quad d \vert v \vert ^{p}\le p \varPhi _{0}(x,v) $$hold for all \(x\in \mathbb{R}^{N}\) and \(v\in \mathbb{R}^{N}\), where d is a positive constant.
-
(J4)
\(\varPhi _{0}(x,\cdot )\) is strictly convex in \(\mathbb{R} ^{N}\) for all \(x\in \mathbb{R}^{N}\).
-
(J5)
The relation
$$ p\varPhi _{0}(x,v)-\varphi (x,v)\cdot v \ge 0 $$holds for all \(x\in \mathbb{R}^{N}\) and all \(v\in \mathbb{R}^{N}\).
Let us define the functional \(\varPhi : X \to \mathbb{R}\) by
It is not difficult to prove that the functional \(\varPhi \in C^{1}(X, \mathbb{R})\), and its Fréchet derivative is given by
We give some examples satisfying assumptions (J1)–(J5).
Example 2.1
-
(1)
Let us consider the following functions:
$$ \varphi (x,v)= \lvert v \rvert ^{p-2}v \quad \text{and} \quad \varPhi _{0}(x,v)=\frac{ \lvert v \rvert ^{p}}{p} $$for \(v\in \mathbb{R}^{N}\) and \(x\in \mathbb{R}^{N}\). Then it is obvious that assumptions (J1)–(J5) hold.
-
(2)
Suppose that \(a\in L^{2p'}(\mathbb{R}^{N})\), and there is a positive constant \(a_{0}\) such that \(a(x)\ge a_{0}\) for almost all \(x\in \mathbb{R}^{N}\). We consider
$$ \varphi (x,t)= \bigl(a(x)+t^{2} \bigr)^{\frac{p-2}{2}}t \quad \text{and} \quad \varPhi _{0}(x,t)=\frac{1}{p} \bigl[ \bigl(a(x)+t^{2} \bigr)^{\frac{p}{2}}-a(x)^{\frac{p}{2}} \bigr] $$for \(t\in \mathbb{R}\), where \(p\geq 2\) for all \(x\in \mathbb{R}^{N}\). Then assumptions (J1)–(J5) hold.
By analogous arguments as in [19, 22], the following lemma is easily checked, and thus we omit the proof. That is, the operator \(\varPhi ^{\prime }\) is a mapping of type \((S_{+})\).
Lemma 2.2
Assume that (V), (M1), (M2), and (J1)–(J4) hold. Then the functional \(\varPhi :X\to \mathbb{R}\) is convex and weakly lower semicontinuous on X. Moreover, the operator \(\varPhi ^{\prime }\) is a mapping of type \((S_{+})\), i.e., if \(u_{n}\rightharpoonup u\) in X and \(\limsup_{n\to \infty } \langle \varPhi ^{\prime }(u_{n})-\varPhi ^{ \prime }(u), u_{n}-u \rangle \le 0\), then \(u_{n}\to u\) in X as \(n\to \infty \).
Denoting \(F(x,t)=\int _{0}^{t}f(x,s) \,ds\), for the number θ given in (M2), we assume that
-
(F1)
\(f: \mathbb{R}^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Carathéodory condition in the sense that \(f(\cdot ,t)\) is measurable for all \(t\in \mathbb{R}\) and \(f(x,\cdot )\) is continuous for almost all \(x\in \mathbb{R}^{N}\).
-
(F2)
There exist nonnegative functions \(\rho \in L^{q^{\prime }}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) and \(\sigma \in L^{ \infty }(\mathbb{R}^{N})\) such that
$$ \bigl\lvert f(x,t) \bigr\rvert \le \rho (x)+\sigma (x) \lvert t \rvert ^{q-1},\quad q\in (\theta p,p_{*}) $$for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\).
-
(F3)
There exists \(\delta >0\) such that
$$ F(x,t)\leq 0 \quad \text{for } x\in \mathbb{R}^{N} \text{ and } \lvert t \rvert < \delta . $$ -
(F4)
\(\lim_{ \lvert t \rvert \to \infty }{\frac{F(x,t)}{ \lvert t \rvert ^{\theta p}}}=\infty \) uniformly for almost all \(x\in \mathbb{R}^{N}\).
-
(F5)
There exist \(c_{0}\ge 0\), \(r_{0}\ge 0\), and \(\kappa > \frac{N}{p}\) such that
$$ \bigl\lvert F(x,t) \bigr\rvert ^{\kappa }\le c_{0} \lvert t \rvert ^{\kappa p}{\mathfrak{F}}(x,t) $$for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\) and \(\lvert t \rvert \ge r_{0}\), where \({\mathfrak{F}}(x,t)=\frac{1}{\theta p}f(x,t)t-F(x,t) \ge 0\).
-
(F6)
There exist \(\mu >\theta p\) and \(\varrho >0\) such that
$$ \mu F(x,t)\leq tf(x,t)+\varrho t^{p} $$for all \((x,t)\in \mathbb{R}^{N} \times \mathbb{R}\).
Next, we give some examples with respect to assumptions (F1)–(F6).
Since assumption (F5) is weaker than the following assumption, namely that
for any \(x\in \mathbb{R}^{N}\), we check that the following example satisfies assumption (F5) by applying condition (2.1).
Example 2.3
Let us consider
for all \(t\in \mathbb{R}\). It is clear that function f satisfies assumptions (F1)–(F4). Since the following ratio, namely
is increasing for \(t> 0\) and decreasing for \(t< 0\) if \(q>p=\theta \), it follows that assumption (F5) holds.
The following example can be found in [23] for the case of p-Laplace operator.
Example 2.4
Consider the following function:
Then this function satisfies conditions (F2), (F6), but not the (AR) condition.
Define the functional \(\varPsi :X\to \mathbb{R}\) by
Then it is easy to check that \(\varPsi \in C^{1}(X,\mathbb{R})\) and its Fréchet derivative is
for any \(u,v \in X\). Next we define the functional \(I_{\lambda }:X \to \mathbb{R}\) by
Then it follows that the functional \(I_{\lambda }\in C^{1}(X,\mathbb{R})\) and its Fréchet derivative is
for any \(u,v\in X\).
In our setting, first of all, we need the following lemma. Using a similar argument as in [17, Lemma 3.2], we can see that the functionals Ψ and \(\varPsi ^{\prime }\) are weakly strongly continuous on X. We give a detailed proof for the convenience of the reader.
Lemma 2.5
Assume that (V) and (F1)–(F2) hold. Then Ψ and \(\varPsi ^{\prime }\) are weakly strongly continuous on X.
Proof
See Appendix. □
3 Existence of weak solutions
Definition 3.1
We say that \(u\in X\) is a weak solution of problem (P) if
for any \(v\in X\).
The following result is used to show that the energy functional \(I_{\lambda }\) satisfies the geometric conditions of the mountain pass theorem.
Lemma 3.2
Assume that (V), (M1), (M2), (J1)–(J3), and (F1)–(F4) hold. Then the geometric conditions in the mountain pass theorem hold, i.e.,
-
(1)
\(u=0\) is a strict local minimum for \(I_{\lambda }(u)\),
-
(2)
\(I_{\lambda }(u)\) is unbounded from below on X.
Proof
By assumption (F3), \(u=0\) is a strict local minimum for \(I_{\lambda }(u)\). Next we claim that condition (2) holds. Assumption (F4) implies that for any \(K_{0}>0\), there exists a constant \(\delta >0\) such that
for \(\lvert t \rvert >\delta \) and for almost all \(x \in \mathbb{R}^{N}\). Note that for \(t>1\), we can easily check that \(\mathcal{M}(t)\leq \mathcal{M}(1)t\). For any \(v\in X\setminus \{0 \}\), from assumptions (J2), (J3) and relation (3.1), we have
for sufficiently large \(t>1\). If \(K_{0}\) is large enough, then we assert that \(I_{\lambda }(tv)\to -\infty \) as \(t \to \infty \). Hence we conclude that the functional \(I_{\lambda }\) is unbounded from below. This completes the proof. □
With the aid of Lemmas 2.2 and 2.5, we prove that the energy functional \(I_{\lambda }\) satisfies the Cerami condition \((C)_{c}\) condition, for short, i.e., for \(c\in \mathbb{R}\), any sequence \(\{u_{n} \}\subset X\) such that
has a convergent subsequence. This plays a key role in obtaining the existence of a nontrivial weak solution for the given problem.
Lemma 3.3
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. Then the functional \(I_{\lambda }\) satisfies the \((C)_{c}\) condition for any \(\lambda >0\).
Proof
For \(c\in \mathbb{R}\), let \(\{u_{n}\}\) be a \((C)_{c}\)-sequence in X, that is,
This says that
where \(o(1)\to 0\) as \(n\to \infty \). It follows from Lemmas 2.2 and 2.5 that \(\varPhi ^{\prime }\) and \(\varPsi ^{\prime }\) are mappings of type \((S_{+})\). Since \(I_{\lambda }^{\prime }\) is of type \((S_{+})\) and X is reflexive, it suffices to prove that the sequence \(\{u_{n}\}\) is bounded in X. We argue by contradiction. Suppose that the sequence \(\{u_{n} \}\) is unbounded in X. Then we may assume that \(\|u_{n}\|_{X}>1\) and \(\|u_{n}\|_{X}\to \infty \) as \(n\to \infty \). Define a sequence \(\{w_{n} \}\) by \(w_{n}={u_{n}}/{\|u_{n}\|_{X}}\). It is clear that \(\{w_{n} \} \subset X\) and \(\|w_{n}\|_{X}=1\). Hence, up to a subsequence still denoted by \(\{w_{n} \}\), we obtain \(w_{n}\rightharpoonup w\) in X as \(n\to \infty \) and note that
for \(1< s<p_{*}\). According to assumptions (M1), (M2), (J3), and relation (3.3), we obtain that
Since \(\|u_{n}\|_{X}\to \infty \) as \(n\to \infty \), we have
In addition, we assert that
Combining this with relation (3.3), we obtain that
for sufficiently large n. Assumption (F4) implies that there exists \(t_{0}>1\) such that \({F(x,t)}>{ \lvert t \rvert ^{\theta p}}\) for all \(x\in \mathbb{R}^{N}\) and \(\lvert t \rvert >t _{0}\). From assumptions (F1) and (F2), there exists \(\mathcal{C}>0\) such that \(\lvert F(x,t) \rvert \leq \mathcal{C}\) for all \((x,t)\in \mathbb{R}^{N} \times [-t_{0},t_{0}]\). Therefore we can choose a real number \(\mathcal{C}_{0}\) such that \(F(x,t)\geq \mathcal{C}_{0}\) for all \((x,t)\in \mathbb{R}^{N}\times \mathbb{R}\), and thus
for all \(x\in \mathbb{R}^{N}\) and for all \(n\in \mathbb{N}\). Set \(\varOmega _{1}= \{ x\in \mathbb{R}^{N} : w(x)\neq0 \}\). By the convergence in (3.4), we know that
for all \(x\in \varOmega _{1}\). So then, it follows from assumptions (M2), (J2), (F4), and Hölder’s inequality that, for all \(x\in \varOmega _{1}\), we have
where we have used the inequality \(\mathcal{M} (t)\leq \mathcal{M}(1) (1+{t}^{\theta } )\) for all \(t\in \mathbb{R}^{+}\), since if \(0\le t<1\), then \(\mathcal{M} (t)=\int _{0}^{t} M(\tau ) \,d\tau \leq \mathcal{M}(1)\) and if \(t>1\), then \(\mathcal{M} (t)\leq \mathcal{M}(1)t ^{\theta }\). Hence we get that \(\operatorname{meas}(\varOmega _{1})=0\). Indeed, if \(\operatorname{meas}(\varOmega _{1})\neq0\), according to (3.6)–(3.8) and Fatou’s lemma, we would obtain
which is a contradiction. Thus \(w(x)=0\) for almost all \(x\in \mathbb{R} ^{N}\). Using assumptions (M1)–(M2) and (J5), we get
for n large enough. Let us define \(\varOmega _{n}(a,b):=\{ x\in \mathbb{R} ^{N} : a \le \lvert u_{n}(x) \rvert < b \}\) for \(a \geq 0\). The convergence in (3.4) means that
for \(1 < r < p_{*}\). Hence by using (3.5) we get
On the other hand, from assumption (F2) and relation (3.11), it follows that
for some positive constants \(C_{i}\) \((i=1,2,3)\). Set \(\kappa ^{\prime }=\kappa / (\kappa -1)\). Since \(\kappa > N / p\), we get \(1 < \kappa ^{\prime } p < p_{*}\). Hence, it follows from (F5), (3.10), and (3.11) that
Combining the estimates in (3.13) with (3.14), we have
which contradicts (3.12). This completes the proof. □
Using Lemma 3.3, we prove the existence of a nontrivial weak solution for our problem under the considered assumptions.
Theorem 3.4
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. Then problem (P) has a nontrivial weak solution for all \(\lambda >0\).
Proof
Note that \(I_{\lambda }(0)=0\). In view of Lemma 3.2, the geometric conditions in the mountain pass theorem are fulfilled. And also \(I_{\lambda }\) satisfies the \((C)_{c}\) condition for any \(\lambda >0\) by Lemma 3.3. Hence, problem (P) has a nontrivial weak solution for all \(\lambda >0\). This completes the proof. □
Next, under assumption (F6) instead of (F5), we show that \(I_{\lambda }\) satisfies the Cerami condition.
Lemma 3.5
Assume that (V), (M1), (M2), (J1)–(J5), (F1)–(F4), and (F6) hold. Then the functional \(I_{\lambda }\) satisfies the \((C)_{c}\) condition for any \(\lambda >0\).
Proof
For \(c\in \mathbb{R}\), let \(\{u_{n}\}\) be a \((C)_{c}\)-sequence in X satisfying (3.2). Following the proof of Lemma 3.3, we only prove that \(\{u_{n}\}\) is bounded in X. To this end, arguing by contradiction, suppose that \(\|u_{n}\|_{X}\rightarrow \infty \) as \(n\to \infty \). Let \(v_{n}=u_{n}/\|u_{n}\|_{X}\). Then \(\|v_{n}\|_{X}=1\). Passing to a subsequence, we may assume that \(v_{n}\rightharpoonup v\) as \(n\to \infty \) in X. Thus by an embedding theorem, for \(1< s< p_{*}\), we have
From (M1), (M2), (J5), and (F6), it follows that
This implies
Hence, due to (3.15), we see that \(v\neq 0\). From the same argument as in Lemma 3.3, we can show that the relations (3.6), (3.7), and (3.8) hold, and hence we conclude that relation (3.9) is true. Therefore we get a contradiction. Thus \(\{u_{n}\}\) is bounded in X. This completes the proof. □
Next, applying the fountain theorem in [32, Theorem 3.6] with the oddity of f, we demonstrate infinitely many weak solutions for problem (P). To do this, let W be a reflexive and separable Banach space. Then there are \(\{e_{n}\}\subseteq W\) and \(\{f_{n}^{*} \}\subseteq W^{*}\) such that
and
Let us denote \(W_{n}=\operatorname{span}\{e_{n}\}\), \(Y_{k}=\bigoplus_{n=1} ^{k}W_{n}\), and \(Z_{k}= \overline{\bigoplus_{n=k}^{\infty }W_{n}}\). In order to establish the existence result, we use the following Fountain theorem.
Lemma 3.6
Let X be a real reflexive Banach space, \(I \in C^{1}(X,\mathbb{R})\) satisfies the \({(C)_{c}}\) condition for any \(c>0\) and I is even. If for each sufficiently large \(k \in \mathbb{N}\), there exist \(\rho _{k}> \delta _{k}>0\) such that the following conditions hold:
-
(1)
\(b_{k}:=\inf \{ I(u):u\in Z_{k}, \|u\|_{X}= \delta _{k}\}\to \infty\) as \(k\to \infty \);
-
(2)
\(a_{k}:=\max \{ I(u):u\in Y_{k}, \|u\|_{X}=\rho _{k}\} \le 0\).
Then the functional I has an unbounded sequence of critical values, i.e., there exists a sequence \(\{u_{n}\}\subset X\) such that \(I^{\prime }(u_{n})=0\) and \(I(u_{n})\to +\infty \) as \(n\to +\infty \).
Theorem 3.7
Assume that (V), (M1), (M2), (J1)–(J5), and (F1)–(F5) hold. If \(\varPhi _{0}(x,-v)=\varPhi _{0}(x,v)\) holds for all \((x,v)\in \mathbb{R}^{N}\times \mathbb{R}^{N}\) and \(f(x,-t)=-f(x,t)\) holds for all \((x,t)\in \mathbb{R}^{N} \times \mathbb{R}\), then for any \(\lambda >0\), problem (P) possesses an unbounded sequence of nontrivial weak solutions \(\{u_{n}\}\) in X such that \(I_{\lambda }( u_{n})\to \infty \) as \(n\to \infty \).
Proof
It is obvious that \(I_{\lambda }\) is an even functional and satisfies the \((C)_{c}\) condition. It suffices to show that there exist \(\rho _{k}> \delta _{k}>0\) such that
-
(1)
\(b_{k}:=\inf \{I_{\lambda }(u):u\in Z_{k}, \|u\|_{X}= \delta _{k}\}\to \infty \quad \text{as } n\to \infty \);
-
(2)
\(a_{k}:=\max \{I_{\lambda }(u):u\in Y_{k}, \|u\|_{X}= \rho _{k}\}\le 0\),
for k large enough. Denote
Then we have \(\alpha _{k} \to 0\) as \(k \to \infty \). In fact, assume to the contrary that there exist \(\varepsilon _{0}>0\) and a sequence \(\{u_{k}\}\) in \(Z_{k}\) such that
for all \(k \ge k_{0}\). By the boundedness of the sequence \(\{u_{k}\}\) in X, we can find an element \(u \in X\) such that \(u_{k} \rightharpoonup u\) in X as \(n\to \infty \) and
for \(j=1,2,\dots \). Thus we deduce \(u=0\). However, we see that
which is a contradiction.
For any \(u \in Z_{k}\), we may suppose that \(\|u\|_{X}>1\). According to assumptions (M1), (M2), (J3), and (F2), we obtain that
where \(C_{4}\) and \(C_{5}\) are positive constants. If we take
then \(\delta _{k}\to \infty \) as \(k\to \infty \) because \(\theta p< q\) and \(\alpha _{k}\to 0\) as \(k\to \infty \). Hence, if \(u \in Z_{k}\) and \(\|u\|_{X}=\delta _{k}\), then we conclude that
This implies that condition (1) holds.
The proof of condition (2) proceeds analogously as in the proof of [1, Theorem 1.3]. For the reader’s convenience, we give the proof. Assume that condition (2) is not true. Then for some k there exists a sequence \(\{u_{n}\}\) in \(Y_{k}\) such that
Set \(w_{n}=u_{n}/\|u_{n}\|_{X}\). Note that \(\|w_{n}\|_{X}=1\). Since \(\dim {Y_{k}}<\infty \), there exists \(w\in Y_{k}\setminus \{0\}\) such that, up to a subsequence,
for almost all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). If \(w(x)\neq 0\), then \(\lvert u_{n}(x) \rvert \to \infty \) for all \(x\in \mathbb{R}^{N}\) as \(n\to \infty \). Hence we obtain by assumption (F4) that
for all \(x\in \varOmega _{2}:= \{ x\in \mathbb{R}^{N} : w(x)\neq0 \}\). As in the proof of Lemma 3.3, we have
Therefore, we conclude that
which contradicts (3.16). This completes the proof. □
Remark 3.8
Although we replace (F5) with (F6) in the assumptions of Theorem 3.7, we can show that problem (P) possesses an unbounded sequence of nontrivial weak solutions \(\{u_{n}\}\) in X such that \(I_{\lambda }( u_{n})\to \infty \) as \(n\to \infty \).
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Acknowledgements
The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments, leading to the improved version of this paper.
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The first author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2017R1D1A1B03031104). J.-M. Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1D1A1B03930422). J. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620) and (2019R1A6A1A11051177). K. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070225).
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Appendix: Proof of Lemma 2.5
Appendix: Proof of Lemma 2.5
In this section, we give a proof of Lemma 2.5 for the reader’s convenience. In fact, we consider that it is a well-known fact to researchers in this area.
Proof
Let \(\{u_{n}\}\) be a sequence in X such that \(u_{n}\rightharpoonup u\) in X as \(n\to \infty \). Then \(\{u_{n}\}\) is bounded in X, and we know the embeddings \(X\hookrightarrow L^{p}(\mathbb{R}^{N})\) and \(X\hookrightarrow L^{q}(\mathbb{R}^{N})\) are compact for \(p< q< p_{*}\). So we know that
First we prove that Ψ is weakly strongly continuous in X. Let \(u_{n}\to u\) in \(L^{p}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})\) as \(n\to \infty \). By the convergence principle, there exist a subsequence \(\{u_{n_{k}}\}\) such that \(u_{n_{k}}(x)\to u(x)\) as \(k\to \infty \) for almost all \(x\in \mathbb{R}^{N}\) and a function \(u_{0}\in L^{p}(\mathbb{R} ^{N})\cap L^{q}(\mathbb{R}^{N})\) such that \(|u_{n_{k}}(x)|\leq u_{0}(x)\) for all \(k\in \mathbb{N}\) and for almost all \(x\in \mathbb{R}^{N}\). Therefore from (F2), we deduce
Since function f satisfies the Carathéodory condition by (F1), we obtain that \(F(x,u_{n_{k}})\to F(x,u)\) as \(k\to \infty \) for almost all \(x\in \mathbb{R}^{N}\). Therefore, the Lebesgue convergence theorem tells us that
as \(k\to \infty \), which says \(\varPsi (u_{n_{k}}) \to \varPsi (u)\) as \(k\to \infty \). Thus Ψ is weakly strongly continuous in X.
Next, we show that \(\varPsi '\) is weakly strongly continuous on X. By (F2) and Hölder’s inequality, we obtain
for some positive constants \(C_{6}\), \(C_{7}\), which implies that \(\lvert f(x,u_{n})-f(x,u) \rvert ^{q'}\leq g(x)\) for almost all \(x\in \mathbb{R}^{N}\) and for some \(g\in L^{1}(\mathbb{R}^{N})\). Since \(u_{n}\to u\) in \(L^{p}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N})\) and almost all in \(\mathbb{R}^{N}\), it follows from (A.1) and the convergence principle that \(f(x,u_{n})\to f(x,u)\) for almost all \(x\in \mathbb{R} ^{N}\). Combining this with the Lebesgue convergence theorem, we have
Therefore, we derive that \(\varPsi '(u_{n})\to \varPsi '(u)\) in X. This completes the proof. □
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Bae, JH., Kim, JM., Lee, J. et al. Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations. Bound Value Probl 2019, 125 (2019). https://doi.org/10.1186/s13661-019-1237-6
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DOI: https://doi.org/10.1186/s13661-019-1237-6
MSC
- 35J60
- 35J92
- 58E05
Keywords
- p-biharmonic
- Kirchhoff type
- Variational method