Abstract
In this paper, we consider the following p-Kirchhoff equation:
where \(f(x,u)=\lambda g(x)|u|^{q-2}u+h(x)|u|^{r-2}u,1< q< p< r< p^{*}\) (\(p^{*}=\frac{Np}{N-p}\) if \(N\ge p,p^{*}=\infty\) if \(N\le p\)). Using variational methods, we prove that, under proper assumptions, there exist \(\lambda_{0},\lambda_{1}>0\) such that problem (P) has a solution for all \(\lambda\in[0,\lambda_{0})\) and has a sequence of solutions for all \(\lambda\in[0,\lambda_{1})\).
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1 Introduction and main results
In this paper, we consider the following p-Kirchhoff equation:
where \(M,V\) are continuous functions, \(f(x,u)=\lambda g(x)|u|^{q-2}u+h(x)|u|^{r-2}u\ (1< q< p< r< p^{*})\) is concave and convex, and
Since the pioneering work of Lions [1], much attention has been paid to the existence of nontrivial solutions, multiplicity of solutions, ground state solutions, sign-changing solutions, and concentration of solutions for problem (1.1). For example, for the following Kirchhoff equation:
Li and Ye [2] and Guo [3] showed the existence of a ground state solution for problem (1.2) with \(N=3\), where the potential \(V(x)\in C({\mathbb {R}}^{3})\) and it satisfies \(V(x)\le\liminf_{|y|\to+\infty}V(y)\triangleq V_{\infty}<+\infty\). Sun and Wu [4] investigated the existence and non-existence of nontrivial solutions with the following assumption: \(V(x)\ge0\) and there exists \(c>0\) such that \(\operatorname{meas}\{x\in{\mathbb {R}}^{N}:V(x)< c\}\) is nonempty and has finite measure. Wu [5] proved that problem (1.2) has a nontrivial solution and a sequence of high energy solutions where \(V(x)\) is continuous and satisfies \(\inf V(x) \ge a_{1}>0\) and for each \(M>0\), \(\operatorname{meas}\{x\in{\mathbb {R}}^{N}:V(x)\le M\}<+\infty\). Nie and Wu [6] treated (1.2) where the potential is a radial symmetric function. Chen et al. [7] considered equation (1.2) when \(f(x,u)=\lambda a(x)|u|^{q-2}u+b(x)|u|^{r-2}u\ (1< q< p=2< r<2^{*})\).
Moreover, for p-Kirchhoff-type problem of the following form:
Cheng and Dai [8] proved the existence and non-existence of positive solutions, where \(M(t)\) satisfies
(M) There exists \(\sigma\in(0,1)\) such that \(\hat{M}(t)\ge \sigma[M(t)]t\), here \(\hat{M}(t)=\int_{0}^{t}M(s) \,ds\).
Furthermore, the authors in [9] dealt with problem (1.3) for the special case \(M(t)=t\) and \(p=2\). Recently, Chen and Zhu [10] considered problem (1.3) for \(M(t)=t^{\tau}\) and \(f(u)=|u|^{m-2}u+\mu|u|^{q-2}u\). Similar consideration can be found in [11–13].
However, p-Kirchhoff problem in the following form:
or p-Kirchhoff problem like (1.1) seems to be considered by few researchers as far as we know. Alves et al. [14] and Corrêa and Figueiredo [15] established the existence of a positive solution for problem (1.4) by the mountain pass lemma, where M is assumed to satisfy the following conditions:
- (\(\mathrm{H}_{1}\)):
-
\(M(t)\ge m_{0}\) for all \(t\ge0\).
- (\(\mathrm{H}_{2}\)):
-
\(\hat{M}(t)\ge[M(t)]^{p-1}t\) for all \(t\ge0\), where \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).
In [16], Liu established the existence of infinitely many solutions to a Kirchhoff-type equation like (1.1). They treated the problem with M satisfying (\(\mathrm{H}_{1}\)) and
- (\(\mathrm{H}_{3}\)):
-
\(M(t)\le m_{1}\) for all \(t>0\).
Very recently, Figueiredo and Nascimento [17] and Santos Junior [18] considered solutions of problem (1.1) by minimization argument and minimax method, respectively, where \(p=2\) and M satisfies (\(\mathrm{H}_{1}\)) and
- (\(\mathrm{H}_{4}\)):
-
The function \(t\mapsto M(t)\) is increasing and the function \(t\mapsto\frac{M(t)}{t}\) is decreasing.
Subsequently, Li et al. [19] investigated the existence, multiplicity, and asymptotic behavior of solutions for problem (1.4), where M could be zero at zero, i.e., the problem is degenerate.
Note that \(M(t)=a+bt\) does not satisfy (\(\mathrm{H}_{2}\)) for \(p=2\) and (\(\mathrm{H}_{3}\)) for all \(1< p< N\). Moreover, \(M(t)=a+bt^{k}\) fails to satisfy (\(\mathrm{H}_{2}\)), (\(\mathrm{H}_{3}\)) for all \(k>0\), and (\(\mathrm{H}_{4}\)) for all \(k>1\). In this paper, we will assume proper conditions on M, which cover the typical case \(M(t)=a+bt^{k}\) and the degenerate case. Furthermore, our assumption on the potential V is totally different from all the previous works which were concerned with Kirchhoff-type problems to the best of our knowledge. The assumption on V is related to the functions \(g,h\) in the nonlinearity f. The potential V is not necessarily radial and can be unbounded or decaying to zero as \(|x|\to+\infty\) according to different functions g and h. See assumptions \((\mathrm{V})\) and (\(\mathrm{M}_{1}\))–(\(\mathrm{M}_{5}\)) below.
Before stating our main results, we introduce some function spaces and then present two embedding theorems, which is important to investigating our problem. For any \(s\in(1,+\infty)\) and any continuous function \(K(x):{\mathbb {R}}^{N}\to{\mathbb {R}},K(x)\ge0,\not \equiv0\), we define the weighted Lebesgue space \(L^{s}({\mathbb {R}}^{N},K)\) equipped with the norm
Throughout the article we assume \(V(x)\) satisfies
- \((\mathrm{V})\) :
-
\(V(x)\in C({\mathbb {R}}^{N})\), \(V(x)\ge0\), and \(\{x\in{\mathbb {R}}^{N}:V(x)=0\}\subset B_{R_{0}}\) for some \(R_{0}>0\), where \(B_{R_{0}}=\{x| |x|\le R_{0},x\in{\mathbb {R}}^{N}\}\).
The natural functional space to study problem (1.1) is X with respect to the norm
The following theorem is due to Lyberopoulos [20]. Denote \(B_{R}=\{ x|x\in{\mathbb {R}}^{N},|x|\le R\}\) and \(B_{R}^{C}={\mathbb {R}}^{N}\backslash B_{R}\).
Theorem 1.1
Let \(p< r< p^{*}\), \(V(x)\) satisfies \((\mathrm{V})\), \(h(x)\in C({\mathbb {R}}^{N})\), and \(h(x)\ge0,\not\equiv0\) such that
where
Then the embedding \(X\hookrightarrow L^{r}({\mathbb {R}}^{N},h)\) is continuous. Furthermore, if \(\mathcal{M}=0\), then the embedding is compact.
Theorem 1.2
Let \(1< q< p\), \(V(x)\) satisfies \((\mathrm{V})\), \(g(x)\in C({\mathbb {R}}^{N})\), \(g(x)\ge0,\not\equiv0\) such that
where
Then the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is continuous. Furthermore, if \(\mathcal{L}=0\), then the embedding is compact.
Proof
This theorem can be seen as a corollary of Theorem 2.3 in [21]. Here we give a detailed proof for the readers convenience. Let \(\varphi_{R}\in C_{0}^{\infty}({\mathbb {R}}^{N})\) be a cut-off function such that \(0\le\varphi_{R}\le1\), \(\varphi_{R}(x)=0\) for \(|x|< R\), \(\varphi_{R}(x)=1\) for \(|x|>R+1\), and \(|D\varphi_{R}(x)|\le C\). For any fixed \(R>R_{0}\), we write \(u=\varphi_{R} u+(1-\varphi_{R})u\). Then it follows from Hölder’s inequality that
Furthermore, by the Sobolev embedding theorem, we have
Combining (1.9) with (1.10), we obtain the continuity of the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\).
In the following, we prove the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is compact. Let \(\mathcal{L}=0\) and suppose that \(u_{n}\rightharpoonup0\) weakly in X. Then \(\|u_{n}\|_{X}\) is bounded. Hence it follows from (1.9) that for any \(\varepsilon>0\), there exists \(R>0\) sufficiently large such that
Moreover, by the Rellich–Kondrachov theorem, \(\|(1-\varphi_{R}) u_{n}\|_{L^{q}({\mathbb {R}}^{N},g)}\to0\), and so there exists \(n(\epsilon)\in\mathbb{N}\) such that, for all \(n\ge n(\epsilon)\),
Hence, for any \(\epsilon>0\), there exist R and n sufficiently large such that
which implies the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is compact. □
In the rest of the paper, we assume
- \((\mathrm{A})\) :
-
The function V satisfies \((\mathrm{V})\) and the functions \(M,g,h\) are continuous and nonnegative such that \(\mathcal {M}=\mathcal{L}=0\), where \(\mathcal{M}\) and \(\mathcal{L}\) are defined by (1.7) and (1.8), respectively.
By Theorems 1.1 and 1.2, if \(\mathcal{M}=\mathcal{L}=0\), then the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) and \(X\hookrightarrow L^{r}({\mathbb {R}}^{N},h)\) is compact for \(1< q< p< r< p^{*}\). Let \(S_{q}\) and \(S_{r}\) be the best embedding constants, then
Since X is a reflexive and separable Banach space, it is well known that there exist \(e_{j}\in X\) and \(e_{j}^{*}\in X^{*}\) \((j=1,2,\ldots)\) such that
-
(1)
\(\langle e_{i},e_{j}^{*}\rangle=\delta_{ij}\), where \(\delta_{ij}=1\) for \(i=j\) and \(\delta_{ij}=0\) for \(i\neq j\).
-
(2)
, .
Set
Motivated by [8, 19], we make the following assumptions on M:
- (\(\mathrm{M}_{1}\)):
-
There exists \(\sigma>0\) such that
$$\hat{M}(t)\ge\sigma\bigl[M(t)\bigr]^{p-1}t $$holds for all \(t\ge0\), where \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).
- (\(\mathrm{M}_{2}\)):
-
\(M(t)\ge m_{0}>0\) for all \(t\ge0\).
- (\(\mathrm{M}_{3}\)):
-
\(M(t)\) is nonnegative and increasing for all \(t\ge0\).
- (\(\mathrm{M}_{4}\)):
-
There exists \(\rho>0\) such that
$$\frac{\sigma}{p}\bigl[M\bigl(\rho^{p}\bigr)\bigr]^{p-1}> \frac{1}{r}S_{r}^{-r/p}\rho^{r-p}, $$where \(S_{r}\) is the best embedding constant of \(X\hookrightarrow L^{r}({\mathbb {R}}^{n},h)\).
- (\(\mathrm{M}_{5}\)):
-
There exists \(\gamma_{1}>0\) such that
$$\frac{\sigma}{p}\bigl[M\bigl(\gamma_{1}^{p}\bigr) \bigr]^{p-1}\gamma_{1}^{p}\ge\frac{\beta _{1}^{r}\gamma_{1}^{r}}{4r}, $$where
$$\beta_{1}=\sup_{u\in Z_{1}, \Vert u \Vert =1} \biggl( \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx \biggr)^{1/r}. $$
The main results of our paper read as follows.
Theorem 1.3
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)). Suppose also \(p<\sigma r\) and \(1< q< p< r< p^{*}\). Then there exists \(\lambda_{0}>0\) such that problem (1.1) has a solution for all \(\lambda\in[0,\lambda_{0})\).
Theorem 1.4
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)). Suppose also \(p<\sigma r\) and \(1< q< p< r< p^{*}\). Then there exists \(\lambda_{1}>0\) such that problem (1.1) has a sequence \(\{u_{n}\}\) of solutions in X with \(J(u_{n})\to\infty\) as \(n\to\infty\) for all \(\lambda\in[0,\lambda_{1})\).
Remark 1.5
Set \(M(t)=a+bt^{k}\ (a,b,k>0)\). Then we can easily deduce that M satisfies (\(\mathrm{M}_{1}\)) for all \(p>1\) and \(0<\sigma\le\frac{1}{(p-1)k+1}\).
Remark 1.6
Let \(M(t)=a+b\ln(1+t)\ (a,b>0,t\ge0)\). Assume \(p>1,b(p-1)< a\), then by direct calculation, one has
Consequently, M satisfies (\(\mathrm{M}_{1}\)) for \(0<\sigma\le1-\frac{b(p-1)}{a}\).
Remark 1.7
Clearly, assumptions (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)) or (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{5}\)) cover the degenerate case.
2 Proofs of the main results
The associated energy functional to equation (1.1) is
For any \(v\in C_{0}^{\infty}({\mathbb {R}}^{N})\), we have
We say that \(\{u_{n}\}\) is a \((PS)_{c}\) sequence for the functional J if
where \(X^{*}\) denotes the dual space of X. If every \((PS)_{c}\) sequence of J has a strong convergent subsequence, then we say that J satisfies the \((PS)\) condition.
The proof of Theorem 1.3 mainly relies on the following mountain pass lemma in [22] (see also [23]).
Lemma 2.1
Let E be a real Banach space and \(J\in C^{1}(E,\mathbb{R})\) with \(J(0)=0\). Suppose
\((\mathrm{H}_{1})\) there are \(\rho,\alpha>0\) such that \(J(u)\ge\alpha\) for \(\|u\|_{E}=\rho\);
\((\mathrm{H}_{2})\) there is \(e\in E\), \(\|e\|_{E}> \rho\) such that \(J(e)< 0\). Define
Then
is finite and \(J(\cdot)\) possesses a \((PS)_{c}\) sequence at level c. Furthermore, if J satisfies the \((PS)\) condition, then c is a critical value of J.
In the following, we shall verify J satisfies all conditions of the mountain pass lemma.
Lemma 2.2
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)). Suppose also \(p<\sigma r\). Then any \((PS)_{c}\) sequence of J is bounded.
Proof
Let \(\{u_{n}\}\) be any \((PS)_{c}\) sequence of J and satisfy (2.3).
By (\(\mathrm{M}_{1}\)) and \((\mathrm{A})\), we have
Case 1. If (\(\mathrm{M}_{2}\)) holds. Then we deduce from (2.4) that
Hence \(\{u_{n}\}\) is bounded.
Case 2. If (\(\mathrm{M}_{3}\)) holds. Let \(\tau_{0}>0\) be fixed. If \(\| u_{n}\|^{p}\ge\tau_{0}\), then
which implies \(\{u_{n}\}\) is bounded. □
Lemma 2.3
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{4}\)). Then there are \(\rho,\alpha>0\) such that \(J(u)\ge\alpha\) for \(\|u\|=\rho \).
Proof
Case 1. (\(\mathrm{M}_{2}\)) is satisfied. It follows from (1.11), (2.1), and (\(\mathrm{M}_{1}\))–(\(\mathrm{M}_{2}\)) that
Denote \(\phi(t)=At^{p-q}-B\lambda-Ct^{r-q}\) with
Obviously, \(\phi(t)\) attains its maximum
at
Let \(\lambda_{0}=\frac{A(r-p)}{B(r-q)}t_{0}^{p-q}\), \(\rho=t_{0}\), and \(\alpha=t_{0}^{q}\phi(t_{0})\). Then \(J(u)\ge\alpha>0\) for \(\|u\|=\rho\) and \(\lambda\in[0,\lambda_{0})\).
Case 2. (\(\mathrm{M}_{4}\)) is fulfilled. Let \(\|u\|=\rho\). Then, by (1.11), (2.1), and (\(\mathrm{M}_{1}\)), there hold
where \(A(\rho)=\frac{\sigma}{p}[M(\rho^{p})]^{p-1}\) and \(B,C\) is defined by (2.8). In view of (\(\mathrm{M}_{4}\)), \(J(u)\ge\alpha>0\) for all \(0<\lambda<\lambda _{0}=\frac{1}{B}[A(\rho)\rho^{p-q}-C\rho^{r-q}]\). □
Lemma 2.4
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and \(p<\sigma r\). Then there is \(e\in X\) with \(\|e\|>\rho\) such that \(J(e)<0\).
Proof
By integrating (\(\mathrm{M}_{1}\)), we obtain
Hence, for \(\|tu\|^{p}\ge t_{1}\),
Consequently, \(J(tu)<0\) if \(t\ge R\) for some \(R>0\) sufficiently large. □
Lemma 2.5
Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)). Then any \((PS)_{c}\) sequence of J has a strong convergent subsequence.
Proof
Let \(\{u_{n}\}\) be any \((PS)_{c}\) sequence of J and satisfy (2.3). By Lemma 2.2, \(\{u_{n}\}\) is bounded. Passing to a subsequence if necessary, we have
Denote \(P_{n}=\langle J'(u_{n}),u_{n}-u\rangle\) and
We can easily obtain that
Since
we can deduce that
Case 1. (\(\mathrm{M}_{2}\)) holds. Using the standard inequality in \({\mathbb {R}}^{N}\) given by
or
we obtain from (2.12) that \(\|u_{n}-u\|\to0\) as \(n\to\infty\).
Case 2. If (\(\mathrm{M}_{3}\)) holds, then due to the degenerate nature of (1.1), two situations must be considered: either \(\inf_{n}\| u_{n}\|>0\) or \(\inf_{n}\|u_{n}\|=0\).
Case 2-1: \(\inf_{n}\|u_{n}\|>0\). Then we can deduce from (2.12)–(2.14) that \(\|u_{n}-u\|\to0\) as Case 1.
Case 2-2: \(\inf_{n}\|u_{n}\|=0\). If 0 is an accumulation point for the sequence \(\{\|u_{n}\|\}\), then there is a subsequence of \(\{ u_{n}\}\) (not relabelled) such that \(u_{n}\to0\). Hence \(0=J(0)=\lim_{n\to \infty}J(u_{n})= c\). By Lemma 2.3, \(c>0\). This is impossible. Consequently, 0 is an isolated point of \(\{\|u_{n}\|\}\). Therefore, there is a subsequence of \(\{u_{n}\}\) (not relabelled) such that \(\inf_{n}\| u_{n}\|>0\), and we can proceed as before.
This completes the proof. □
Proof of Theorem 1.3
The conclusion follows by Lemmas 2.2–2.5 immediately. □
To get multiplicity result of problem (1.1), we need the following fountain theorem.
Lemma 2.6
(Fountain theorem [24])
Let X be a Banach space with the norm \(\|\cdot\|\), and let \(X_{i}\) be a sequence of subspace of X with \(\dim X_{i}<\infty\) for each \(i\in\mathbb{N}\). Further, set
Consider an even functional \(\Phi\in C^{1}(X,\mathbb{R})\). Assume, for each \(k\in\mathbb{N}\), there exist \(\rho_{k}>\gamma_{k}>0\) such that
- \((\Phi_{1})\) :
-
\(a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}\Phi(u)\le0\);
- \((\Phi_{2})\) :
-
\(b_{k}:=\inf_{u\in Z_{k},\|u\|=\gamma_{k}}\Phi(u)\to +\infty, k\to+\infty\);
- \((\Phi_{3})\) :
-
Φ satisfies the \((PS)_{c}\) condition for every \(c>0\).
Then Φ has an unbounded sequence of critical values.
Proof of Theorem 1.4
Obviously the functional J is even. It remains to verify that J satisfies \((\Phi_{1})\)–\((\Phi_{3})\) in Lemma 2.6.
It follows from (2.10) that
for positive constants \(C_{1},C_{2}\) and for all \(t\ge0\). Hence
Since all norms are equivalent on the finite dimensional space \(Y_{k}\), we have, for all \(u\in Y_{k}\),
where \(C_{3},C_{4}\) are positive constants. Therefore \(a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}J(u)<0\) for \(\|u\|=\rho_{k}\) sufficiently large. This gives \((\Phi_{1})\).
Denote \(\beta_{k}=\sup_{u\in Z_{k},\|u\|=1} (\int_{{\mathbb {R}}^{N}}h|u|^{r} \,dx )^{1/r}\). Since \(Z_{k+1}\subset Z_{k}\), we deduce that \(0\le\beta_{k+1}\le\beta_{k}\). Hence \(\beta_{k}\to\beta_{0}\ge0\) as \(k\to+\infty\). By the definition of \(\beta_{k}\), there exists \(u_{k}\in Z_{k}\) with \(\|u_{k}\|=1\) such that
for all \(k\ge1\). Therefore there exists a subsequence of \(\{u_{k}\}\) (not relabelled) such that \(u_{k}\rightharpoonup u\) in X and \(\langle u,e_{j}^{*}\rangle=\lim_{k\to\infty}\langle u_{k},e_{j}^{*}\rangle=0\) for all \(j\ge1\). Consequently, \(u= 0\). This implies \(u_{k}\rightharpoonup0\) in X and so \(u_{k}\to0\) in \(L^{r}({\mathbb {R}}^{N},h)\). Thus \(\beta_{0}=0\). The proof of \((\Phi_{2})\) is divided into the following two cases.
Case 1: (\(\mathrm{M}_{2}\)) holds. For any \(u\in Z_{k}\), there holds
Set
Then
for all \(\lambda\in(0,\lambda_{1})\) and \(\|u\|=\gamma_{k}\). Hence \((\Phi_{2})\) is fulfilled.
Case 2: (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{5}\)) hold. For \(\|u\|=\rho\), we have
Set
Then by (\(\mathrm{M}_{5}\))
for all \(\lambda\in(0,\widetilde{\lambda}_{1})\) and \(\|u\|=\widetilde {\gamma}_{k}\). Hence \((\Phi_{2})\) is fulfilled.
By Lemma 2.5, we obtain \((\Phi_{3})\). Consequently, the conclusion follows by the fountain theorem. □
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Huang, J. Existence and multiplicity of solutions for a p-Kirchhoff equation on \({\mathbb {R}}^{N}\). Bound Value Probl 2018, 124 (2018). https://doi.org/10.1186/s13661-018-1045-4
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DOI: https://doi.org/10.1186/s13661-018-1045-4