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Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity
Boundary Value Problems volume 2017, Article number: 38 (2017)
Abstract
In this paper, we consider a class of p-Kirchhoff type problems with a singularity in a bounded domain in \(R^{N}\). By using the variational method, the existence and multiplicity of positive solutions are obtained.
1 Introduction and main results
The purpose of this paper is to investigate the existence of multiple positive solutions to the following problem:
where \(\Delta_{p}u= \operatorname{div}(\vert \nabla u\vert ^{p-2}\nabla u)\), Ω is a smooth bounded domain in \(R^{N}\), \(0< r<1<p<q<p^{*}\) (\(p^{*}=\frac {Np}{N-p}\) if \(N>p\) and \(p^{*}=\infty\) if \(N\leq p \)), \(M(s)=as^{p-1}+b\) and \(a,b,\lambda>0\). \(f, g \in C(\overline{\Omega})\) are nontrivial nonnegative functions.
Problem (1.1) is related to the stationary problem introduced by Kirchhoff in [1]. More precisely, it is the model
where \(\rho,\rho_{0}, E, L\) are constants, which was proposed as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings to describe transversal oscillations of a stretched string. For more details and backgrounds, we refer to [2, 3].
The existence and multiplicity of solutions for the following problem:
on a smooth bounded domain \(\Omega\subset R^{N}\) has been studied in many papers. Liu and Zhao [4] proved (1.2) has at least two nontrivial weak solutions by Morse theory under some restriction on \(M(s)\) and \(h(x,u)\). In [5], the authors considered the following problem:
where \(M(s)=as+b\), \(1< q< p< r\leq p^{*}\), they proved the existence of multiplicity nontrivial solutions by using the Nehari manifold when the weight functions \(f(x)\) and \(g(x)\) change their signs. For more results, we refer to [6–10] and the references therein.
When \(p=2\) and \(N=3\), problem (1.1) reduces to the following singular Kirchhoff type problem:
where \(1< q \leq5 \), the existence of solutions for problem (1.4) has been widely studied (see [11–13]). When \(3< q < 5\) and \(\lambda=1\), Liu and Sun [11] proved that problem (1.4) has at least two positive solutions for \(\mu>0\) small enough. Liao et al. showed the multiplicity of positive solutions by the Nehari mainfold in the case of \(q=3\) in [12]. When q is a critical exponents, at least two positive solutions are obtained by variational and perturbation methods in [13].
However, the singular p-Kirchhoff type problems have few been considered, especially \(p\neq2, N\neq3\). Here we focus on extending the results in [11] and [12]. In fact, the extension is nontrivial and requires a more careful analysis. Our method is based on the Nehari manifold; see [6, 11, 14, 15].
Before starting our main theorems, we make use of the following notations:
• Let \(W^{1,p}_{0}(\Omega)\) be the Sobolev space with norm \(\Vert u\Vert =(\int_{\Omega} \vert \nabla u\vert ^{p} \,dx)^{\frac{1}{p}}\), the norm in \(L^{p}(\Omega)\) is denoted by \(\parallel\centerdot\parallel_{p}\);
• Let \(S_{z}\) be the best Sobolev constant for the embedding of \(W_{0}^{1,p}(\Omega)\) in \(L_{z}(\Omega)\) with \(0< z< p^{*}\). Then, for all \(u\in W_{0}^{1,p}(\Omega)\backslash\{0\}\),
In general, we say that a function \(u\in W_{0}^{1,p}(\Omega)\) is a weak solution of problem (1.1) if
for all \(\varphi\in W_{0}^{1,p}(\Omega)\). Thus, the functional corresponding to problem (1.1) is defined by
where \(\widehat{M}(s)=\int_{0}^{s}M(t) \,dt \).
To obtain the existence results, we introduce the Nehari manifold:
and we define
Now we split \(N_{\lambda}\) into three disjoint parts as follows:
Let \(\lambda^{*}=\) \(\max \{\frac{(1-r)\lambda_{1}(a)}{p^{\frac {p+1}{p}}},\frac{(1-r)\lambda_{2}}{p} \}\), where \(\lambda_{1}(a)\) and \(\lambda_{2}\) are given by
and
then we state the main theorems.
Theorem 1.1
Assume that \(p^{2}< q< p^{*}\) and \(N<2p\). Then, for each \(a>0\) and \(0<\lambda<\lambda^{*}\), the problem (1.1) has at least two positive solutions \(u_{\lambda}^{+}\in N_{\lambda }^{+} \) and \(u_{\lambda}^{-}\in N_{\lambda}^{-}\).
Define
then \(\Lambda>0\) is obtained by some \(\phi_{\Lambda}\in W_{0}^{1,p}(\Omega)\) with \(\int_{\Omega}g \vert \phi_{\Lambda} \vert ^{p^{2}} \,dx=1\). In particular,
and
where μ is an eigenvalue of (1.7), \(u\in W_{0}^{1,p}(\Omega)\) is nonzero and an eigenvector corresponding to μ such that
we write
and all distinct eigenvalues of (1.7)  denoted by \(0<\mu_{1}<\mu _{2}<\cdots\) , we have
where \(\mu_{1}\) is simple, isolated and can be obtained at some \(\psi \in E\) and \(\psi>0\) in Ω (see [16]).
Theorem 1.2
Assume that \(p^{2}=q< p^{*}\) and \(N<2p\). Then
-
(i)
for each \(a\geq\frac{1}{\Lambda}\) and \(\lambda>0\), the problem (1.1) has at least one positive solution;
-
(ii)
for each \(a<\frac{1}{\Lambda}\) and \(0<\lambda<\frac{1-r}{p}\hat {\lambda}\), where
$$\hat{\lambda}=\frac{bS_{1-r}^{\frac{1-r}{p}}(p^{2}-p)}{\Vert f\Vert _{\infty }(p^{2}+r-1)} \biggl(\frac{b\Lambda(p+r-1)}{(1-a\Lambda)(p^{2}+r-1)} \biggr)^{\frac {p+r-1}{p^{2}-p}}, $$the problem (1.1) has at least two positive solutions \(u_{\lambda }^{+}\in N_{\lambda}^{+} \), \(u_{\lambda}^{-}\in N_{\lambda}^{-}\) and
$$\lim_{a\to\frac{1}{\Lambda}^{-}}\inf_{u\in N_{\lambda}^{-}} J(u)=\infty. $$
This paper is organized as follows: In Section 2, we present some lemmas which will be used to prove our main results. In Section 3 and Section 4, we will prove Theorems 1.1 and 1.2, respectively.
2 Preliminaries
Lemma 2.1
(i) If \(q\geq p^{2}\), then the energy functional \(J(u)\) is coercive and bounded below in \(N_{\lambda}\);
(ii) if \(q< p^{2}\), then the energy functional \(J(u)\) is coercive and bounded below in \(W_{0}^{1,p}(\Omega)\).
Proof
(i) For \(u\in N_{\lambda}\), we have
By the Sobolev inequality,
Thus, \(J(u)\) is coercive and bounded below in \(N_{\lambda}\).
(ii) For \(u\in W_{0}^{1,p}(\Omega)\), we have
Thus, \(J(u)\) is coercive and bounded below in \(W_{0}^{1,p}(\Omega)\). □
Lemma 2.2
If \(q> p^{2}\) and \(0<\lambda<\max \{\lambda_{1}(a),\lambda_{2}\}\), then, for all \(a>0\),
-
(i)
the submanifold \(N_{\lambda}^{0}=\emptyset\);
-
(ii)
the submanifold \(N_{\lambda}^{\pm}\neq\emptyset\).
Proof
(i) Suppose \(N_{\lambda}^{0}\neq \emptyset\). Then, for \(u\in N_{\lambda}^{0}\), we have
and
By (2.1) and (2.2), for all \(u\in N_{\lambda}^{0}\), we have
and
Hence, if \(N_{\lambda}^{0}\) is nonempty, then the inequality \(\lambda\geq\max\{\lambda_{1}(a), \lambda_{2}\}\) must hold.
(ii) Fix \(u\in W_{0}^{1,p}(\Omega)\). Let
We see that \(h_{a}(0)=0\) and \(h_{a}(t)\rightarrow-\infty\) as \(t\rightarrow\infty\). Since \(q> p^{2}\) and
there is a unique \(t_{a,\max}>0\) such that \(h_{a}(t)\) reaches its maximum at \(t_{a,\max}\), increasing for \(t\in[0,t_{a,\max})\) and decreasing for \(t \in(t_{a,\max},\infty)\) with \(\lim_{t\rightarrow \infty}h_{a}(t)=-\infty\). Clearly, if \(tu \in N_{\lambda}\), then \(tu \in N_{\lambda}^{+}\) (or \(N_{\lambda}^{-}\)) if and only if \(h'_{a}(t)>0\) (or <0). Moreover,
and
On the other hand, since
there exist unique \(t^{+}\) and \(t^{-}\) such that \(0< t^{+}< t_{a,\max}< t^{-}\),
and
That is, \(t^{+}u\in N_{\lambda}^{+}\) and \(t^{-}u\in N_{\lambda}^{-}\). □
Lemma 2.3
(i) If \(q= p^{2} \) and \(a\geq\frac{1}{\Lambda}\), then, for all \(\lambda>0\), \(N_{\lambda}^{+}=N_{\lambda}\neq\emptyset\);
(ii) if \(q= p^{2}, a<\frac{1}{\Lambda}\) and \(0<\lambda<\hat{\lambda}\), then \(N_{\lambda}=N_{\lambda}^{+}\cup N_{\lambda}^{-} \) and \(N_{\lambda}^{\pm}\neq\emptyset\).
Proof
First, we show that \(N_{\lambda}^{+}=N_{\lambda}\).
Indeed, for all \(u \in N_{\lambda}\), we have
Therefore, \(u\in N_{\lambda}^{+}\).
Next, we declare \(N_{\lambda}^{+}\neq\emptyset\).
Fix \(u\in W_{0}^{1,p}(\Omega)\). Let
Obviously, \(\bar{h}(0)=0\) and \(\lim_{t \to\infty}\bar{h}(t)=\infty\). Since
we can deduce that \(\bar{h}(t)\) is increasing for \(t\in[0,\infty)\). Thus, there is a unique \(t^{+}>0\) such that \(\bar{h}(t^{+})=\lambda\int_{\Omega}f \vert u\vert ^{1-r} \,dx\) and \(\bar {h}'(t^{+})>0\). That is, \(t^{+}u \in N_{\lambda}^{+}\).
(ii) The proof is similar to Lemma 2.2, we omit it here. □
We write \(N_{\lambda}=N_{\lambda}^{+}\cup N_{\lambda}^{-}\) and define
then we have the following lemma.
Lemma 2.4
Suppose that \(q>p^{2}\) and \(0<\lambda<\lambda^{*}\), then we have
-
(i)
\(\alpha^{+}<0\);
-
(ii)
\(\alpha^{-}>C_{0}\), for some \(C_{0}>0\).
In particular \(\alpha^{+}=\inf_{u\in N_{\lambda}}J(u)\).
Proof
(i) Let \(u \in N_{\lambda}^{+}\), it follows that
and
Substituting this into \(J(u)\), we have
and then \(\alpha^{+}<0\).
(ii) Let \(u\in N_{\lambda}^{-}\). We divide the proof into two cases.
Case (A): \(\lambda^{*}= \frac{(1-r)\lambda_{2}}{p}\). Since \(u\in N_{\lambda}^{-}\), and by the Sobolev inequality,
which implies
Hence,
Thus, if \(0<\lambda<\frac{(1-r)\lambda_{2}}{p}\), then \(\alpha ^{-}>C_{0}>0\).
Case (B): \(\lambda^{*}=\frac{(1-r)\lambda_{1}(a)}{p^{\frac {p+1}{p}}}\). By (2.1), one has
which implies
Repeating the argument of case (A), we conclude if \(\lambda<\frac {(1-r)\lambda_{1}(a)}{p^{\frac{p+1}{p}}}\), then \(\alpha^{-}>C_{0}\) for some \(C_{0}>0\). □
Lemma 2.5
Suppose that \(q=p^{2},a<\frac {1}{\Lambda}\) and \(0<\lambda<\frac{1-r}{p}\hat{\lambda}\), then we have
-
(i)
\(\hat{\alpha}^{+}<0\);
-
(ii)
\(\hat{\alpha}^{-}>C_{0}\), for some \(C_{0}>0\).
In particular \(\hat{\alpha}^{+}=\inf_{u\in N_{\lambda}}J(u)\).
Proof
(i) Repeating the same argument of Lemma 2.4(i), we conclude that \(\hat{\alpha}^{+}<0\).
(ii) Let \(u\in N_{\lambda}^{-}\). By (1.6), one has
which implies that
Then we have
Thus, if \(\lambda<\frac{1-r}{p}\hat{\lambda}\), then \(\hat{\alpha }^{-}>C_{0}\) for some \(C_{0}>0\). □
Lemma 2.6
For each \(u\in N_{\lambda}^{+}\) (resp. \(u\in N_{\lambda}^{-}\)), there exist \(\varepsilon>0\) and a continuous function \(f:B(0;\varepsilon)\subset W_{0}^{1,p}(\Omega )\rightarrow R^{+}\) such that
where \(B(0;\varepsilon)=\{\omega\in W_{0}^{1,p}(\Omega):\Vert \omega \Vert <\varepsilon\}\).
Proof
For \(u\in N_{\lambda}^{+}\), define \(F: W_{0}^{1,p}(\Omega)\times R \to R\) as follows:
Since \(u\in N_{\lambda}^{+}\), it is easily seen that \(F(0,1)=0\) and \(F_{t}(0,1)>0\). Then by the implicit function theorem at the point \((0,1)\), we can see that there exist \(\varepsilon>0\) and a continuous function \(f:B(0;\varepsilon)\subset W_{0}^{1,p}(\Omega)\to R^{+}\) such that
In the same way, we can prove the case \(u\in N_{\lambda}^{-}\). □
Remark 2.1
3 Proof of Theorem 1.1
By Lemma 2.1 and the Ekeland variational principle [17], there exists a minimizing sequence \(\{u_{n}\}\subset N_{\lambda}^{+}\) such that
Note that \(J(\vert u_{n}\vert )=J(u_{n})\). We may assume that \(u_{n}\geq0\) in Ω. Using Lemma 2.1 again, we can see that there is a constant \(C_{1}>0\) such that, for all \(n\in N^{+}\), \(\Vert u_{n}\Vert \leq C_{1}\). Thus, there exist a subsequence (still denoted by \(\{u_{n}\}\)) and \(u_{\lambda}^{+}\) in \(W_{0}^{1,p}(\Omega)\) such that
Now we conclude that \(u_{\lambda}^{+} \in N_{\lambda}^{+}\) is a positive solution of (1.1). The proof is inspired by Liu and Sun [11]. In order to prove the claim, we divide the arguments into six steps.
Step 1: \(u_{\lambda}^{+}\) is not identically zero.
Indeed, it is an immediate conclusion of the following inequalities:
Step 2: There exists \(C_{2}\) such that up to a subsequence we have
In order to prove (3.1), it suffices to verify
Since \(u_{n}\in N_{\lambda}^{+}\),
It follows that
Suppose by contradiction that
Then, from (3.3) and (3.4), one has
Thus \(\Vert u_{n}\Vert ^{p}\) converges to a positive number A that satisfies
and
On the other hand, by (2.3), we have
which is impossible. Hence, (3.1) and (3.2) must hold.
Step 3: For nonnegative \(\varphi\in W_{0}^{1,p}(\Omega)\) and \(t>0\) small, we can find \(f_{n}(t):=f_{n}(t \varphi)\) such that \(f_{n}(0)=1\) and \(f_{n}(t)(u_{n}+t\varphi)\in N_{\lambda}^{+}\) for each \(u_{n}\in N_{\lambda}^{+}\) by Lemma 2.6. \(f'_{n+}(0)\in[-\infty, \infty ]\) is denoted by the right derivative of \(f_{n}(t)\) at zero. We claim that there exists \(C_{3}>0\) such that \(f'_{n+}(0)>-C_{3}\) for all \(n\in N^{+}\). Since \(u_{n}, f_{n}(t)(u_{n}+t\varphi)\in N_{\lambda}\), we deduce that
and
Thus
Then, dividing by \(t>0\) and letting \(t\to0\), we have
One deduces from (3.1)
Therefore, by the boundedness of \(\{u_{n}\}\), we conclude that \(\{ f'_{n+}(0)\}\) is bounded from below.
Step 4: Choose \(n^{*}\) large enough such that \(\frac{(1-r)C_{1}}{n}<\frac {C_{2}}{2}\) for all \(n>n^{*}\). Then we claim that there exists \(C_{4}\) such that \(f'_{n+}(0)< C_{4}\) for each \(n>n^{*}\). Without loss of generality, we may suppose \(f'_{n+}(0)\geq0\). Then from condition (ii), we have
Then, dividing by \(t>0\) and letting \(t\to0\), we deduce
From (3.5) and the choice of \(n^{*}\), we have
Namely,
Therefore, by the boundedness of \(\{u_{n}\}\), we conclude \(\{ f'_{n+}(0)\}_{n>n^{*}}\) is bounded from above.
Step 5: \(u_{\lambda}^{+}>0\) a.e. in Ω and for nonnegative \(\varphi\in W_{0}^{1,p}(\Omega)\), we have
Similar to the argument in Step 4, one can obtain
Since \(f(\vert u_{n}+t\varphi \vert ^{1-r}-\vert u_{n}\vert ^{1-r})\geq0, \forall t>0\), by Fatou’s lemma, we obtain
It follows from (3.7) and (3.8) that
for all \(n>n^{*}\).
Passing to the limit as \(n\to\infty\), one has
Then using Fatou’s lemma again, we infer that
Since \(u_{n} \to u_{\lambda}^{+}\) a.e. in Ω, we get \(u_{\lambda }^{+}\geq0\) a.e. in Ω. Thus, one infers from (3.9) that
On the other hand
Combining (3.10) and (3.11), we have
Thus, (3.6) can be obtained by inserting (3.12) into (3.9). Moreover, from (3.6), one has
Therefore, using the strong maximum principle for weak solutions (see [18]), we obtain \(u_{\lambda}^{+}> 0 \) a.e. in Ω.
Step 6: \(u_{\lambda}^{+}\) is a weak solution of (1.1), and \(u_{\lambda}^{+}\in N_{\lambda}^{+}\). By (3.12), we have \(u_{n}\to u_{\lambda}^{+}\) strongly in \(W_{0}^{1,p}(\Omega)\), and so \(u_{\lambda}^{+}\in N_{\lambda}^{+}\). Assume \(\phi\in W_{0}^{1,p}(\Omega)\) and \(\varepsilon>0\), define \(\Psi\in W_{0}^{1,p}(\Omega)\) by \(\Psi :=(u_{\lambda}^{+}+\varepsilon\phi)^{+}\). Then from Step 5 it follows
Since the measure of the domain of integration \([u_{\lambda }^{+}+\varepsilon\phi\leq0]\) tends to zero as \(\varepsilon\to0\), it follows \(\int_{[u_{\lambda}^{+}+\varepsilon\phi\leq0]} \vert \nabla u_{\lambda}^{+}\vert ^{p-2}\nabla u_{\lambda}^{+}\nabla\phi \,dx \to0\). Dividing by ε and letting \(\varepsilon\to0\), we have
Notice that ϕ is arbitrary, the inequality also holds for −ϕ, so it follows that \(u_{\lambda}^{+}\) is a weak solution of (1.1). Moreover, from (3.2) and (3.12), we deduce that \(u_{\lambda}^{+}\in N_{\lambda}^{+}\).
A similar argument shows that there exists another solution \(u_{\lambda }^{-}\in N_{\lambda}^{-}\).
4 Proof of Theorem 1.2
(i) By Lemma 2.3(i), we write \(N_{\lambda}=N^{+}_{\lambda}\) and define
Similar to Lemma 2.5(i), we have \(\theta^{+}<0\). Applying Lemma 2.2(i) and the Ekeland variational principle, we see that there exists a minimizing sequence \(\{u_{n}\}\) for \(J(u)\) in \(N^{+}_{\lambda}\) such that
Repeating the same argument as Theorem 1.1, we can see that \(u_{\lambda }\in N^{+}_{\lambda}\) is a positive solution of the problem (1.1).
(ii) Similar to the proof of Theorem 1.1, we know that the problem (1.1) has at least two positive solutions \(u_{\lambda}^{+}\in N_{\lambda }^{+}\) and \(u_{\lambda}^{-}\in N_{\lambda}^{-}\). Moreover, combining (2.4) with (2.5), we have
and
This completes the proof of Theorem 1.2.
Remark 4.1
The results of Theorems 1.1 and 1.2 extend the results of [11, 12]. The results from the cited work correspond to our results for the case \(p=2\) and \(N=3\). From these two references, we obtained the motivation for this paper.
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Acknowledgements
This work was supported by Young Award of Shandong Province (ZR2013AQ008), NNSF (61603226), the Fund of Science and Technology Plan of Shandong Province (2014GGH201010) and NSFC (11671237).
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Wang, D., Yan, B. Existence and multiplicity of positive solutions for p-Kirchhoff type problem with singularity. Bound Value Probl 2017, 38 (2017). https://doi.org/10.1186/s13661-017-0771-3
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DOI: https://doi.org/10.1186/s13661-017-0771-3