Abstract
In this paper, we study the following nonlocal problem:
where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\) with \(N\ge 3\), \(a,b>0\), \(1< q<2\) and \(\lambda >0\) is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as \(b\searrow 0\).
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1 Introduction and main results
In this paper, we are concerned with the multiplicity of positive solutions for the nonlocal problem
where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\) \((N\ge 3)\), \(a,b>0\), \(1< q<2\) and \(\lambda >0\) is a parameter.
In the past two decades, the following Kirchhoff type problem on a bounded domain
has attracted great attention of many researchers. The Kirchhoff type problem is often viewed as nonlocal due to the presence of the term \(b\int _{\Omega } |\nabla u|^{2}\,dx\) which implies that such a problem is no longer a pointwise identity. By using the variational method, there are many interesting results of positive solutions to (1.2), see e.g. [1, 4, 5, 9, 11] and the references therein.
If we replace \(b\int _{\Omega } |\nabla u|^{2}\,dx\) with \(-b\int _{\Omega } |\nabla u|^{2}\,dx\), then (1.2) turns out to be the following new nonlocal one:
This kind of problem involving negative nonlocal term not only presents some interesting difficulties different from Kirchhoff type problem but also has its own physical and mechanical motivation, see [8, 14]. Yin and Liu [16] considered problem (1.3) when \(f(x,u)=|u|^{p-2}u\) with \(2< p<2^{*}\) and showed the existence of two nontrivial solutions. Based on [16], Wang and Yang [15] further obtained the existence of infinitely many sign-changing solutions. In [17], the authors extended the results of [16] to a general case of nonlinear terms. For \(f(x,u)=\lambda |u|^{-\gamma }\) with \(0<\gamma <1\), [6] got the multiplicity of positive solutions to (1.3). In [10], we proved that problem (1.3) possesses at least one positive solution when \(N=3\), \(f(x,u)=\lambda f(x)|u|^{p-2}u\) with \(3< p<4\) and \(f(x)\in L^{\frac{{6}}{{6-p}}}(\Omega )\) may change sign. In particular, Duan et al. [3] and Lei et al. [7] proved that there exists \(\lambda _{*}>0\) such that, for each \(\lambda \in (0,\lambda _{*})\), problem (1.1) has two positive solutions by using the minimization argument and the mountain pass theorem.
From the works described before, it is important and interesting to ask whether the multiplicity of positive solutions to problem (1.1) can be established by other methods? In the present paper, we shall give an affirmative answer. The main technique applied here is a separation argument for the Nehari-type set of problem (1.1), which has been firstly introduced by Tarantello [13] and later refined by Sun and Li [12].
Let \(H:= H_{0}^{1}(\Omega )\) and \(L^{s}(\Omega )\) be the standard Sobolev spaces endowed with the standard norms \(\|\cdot \|\) and \(|\cdot |_{p}\), respectively. Denote by → and ⇀ the strong and weak convergence, respectively. We use \(o_{n}(1)\) to denote a quantity such that \(o_{n}(1)\to 0\) as \(n\to \infty \). C and \(C_{i}\) denote various positive constants which may vary from line to line. We say that \(I\in C^{1}(H,\mathbb{R})\) satisfies the Palais–Smale condition at level \(c\in \mathbb{R}\) (\((\text{PS})_{c}\) in short) if any sequence \(\{u_{n}\}\subset H\) such that \(I(u_{n})\to c\) and \(I'(u_{n})\to 0\) in \(H^{-1}\) as \(n\to \infty \) has a convergent subsequence. S denotes the best constant in the Sobolev embedding \(H\hookrightarrow L^{2^{*}}(\Omega )\), that is,
Associated with problem (1.1), we define the energy functional
Then \(I_{b} \in C^{1}(H,\mathbb{R})\). Recall that a function \(u\in H\) is called a weak solution to (1.1) if, for any \(\phi \in H\), there holds
Define the Nehari type set of (1.1)
and then decompose \(\Lambda _{b}\) into three subsets:
It is important to notice that there exists a norm gap in \(\Lambda _{b}\):
Set
Our main results are as follows.
Theorem 1.1
Assume that \(\lambda \in (0,T_{b})\), then problem (1.1) has at least two positive solutions \(u_{*} \in \Lambda _{b}^{+}\), \(\tilde{u}_{*} \in \Lambda _{b}^{-}\) with \(\|u_{*}\|<\|\tilde{u}_{*}\|\).
Moreover, as a by-product of our arguments, we regard b as a parameter and obtain the blow-up behavior of the solution \(\tilde{u}_{b} \in \Lambda _{b}^{-}\) and the asymptotic behavior of the other one \(u_{b} \in \Lambda _{b}^{+}\) of problem (1.1) as \(b\searrow 0\). Namely, we have the following theorem.
Theorem 1.2
Assume that \(\{b_{n}\}\) is a sequence satisfying \(b_{n}\searrow 0\) as \(n\to \infty \). Then there exists a subsequence, still denoted by \(\{b_{n}\}\), such that
(i) \(\|\tilde{u}_{b_{n}}\|\to \infty \) as \(n\to \infty \).
(ii) \({u}_{b_{n}}\to {u}_{0}\) in H as \(n\to \infty \), where \({u}_{0}\) is a positive solution of the problem
Remark 1.3
Compared with [3, 7], we adapt a new method to show the existence and multiplicity of positive solutions to problem (1.1). In particular, we obtain the blow-up and the asymptotic behavior of these solutions. As far as we know, such phenomena about the solutions to (1.1) are first observed, which reveals some relationship between the nonlocal problem (1.1) and the classical semilinear problem (1.5).
The paper is organized as follows. In Sect. 2, we present some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1.1 and 1.2, respectively.
2 Preliminaries
Lemma 2.1
Let \(\lambda \in (0,T_{b})\). Then and \(\Lambda _{b}^{0}=\{0\}\).
Proof
For any \(u\in H\), \(u\neq 0\), we define
It is easy to see that \(g(t)\) attains its maximum value at \(t_{\max }=[\frac{{a(2-q)}}{{b(4-q)\|u\|^{2}}}]^{1/2}\) with
We note that, by Hölder’s inequality, for \(\lambda \in (0,T_{b})\), there holds
It follows that there are two and only two positive constants \(t^{+}=t^{+}(u)\) and \(t^{-}=t^{-}(u)\) such that
Equivalently, we obtain \(t^{+}u\in {{\Lambda ^{-}_{b}}}\) and \(t^{-}u\in {{\Lambda ^{+}_{b}}}\).
Next, we prove that \(\Lambda _{b}^{0}=\{0\}\). Arguing by contradiction, we assume that there exists \(w\in \Lambda _{b}^{0}\) satisfying \(w\neq 0\). Then we have \(a(2-q)\|w\|^{2}-b(4-q)\|w\|^{4}=0\). This yields \(b\|w\|^{2}=\frac{{a(2-q)}}{{4-q}}\). For \(\lambda \in (0,T_{b})\), it follows from \(w\in \Lambda _{b}\) and Hölder’s inequality that
which makes no sense. This ends the proof. □
Lemma 2.2
Given \(u\in \Lambda _{b}^{\pm }\), there exist \(\rho _{u}>0\) and a differential function \(g_{\rho _{u}}:B_{\rho _{u}}(0)\to \mathbb{R}^{+}\) defined for \(w\in H\), \(w\in B_{\rho _{u}}(0)\) satisfying
and
Proof
We only give the proof for the case \(u\in \Lambda _{b}^{-}\). In a similar way, one can prove the other case \(u\in \Lambda _{b}^{+}\). Fix \(u\in \Lambda _{b}^{-}\) and define \(F:\mathbb{R}^{+}\times H \to \mathbb{R}\) by
By \(u\in \Lambda _{b}^{-}\subset \Lambda _{b}\), we easily see that \(F(1,0)=0\) and
Then, we are able to use the implicit function theorem for F at the point \((1,0)\) and get \(\overline{\rho }>0\) and a differential functional \(g=g(w)>0\) defined for \(w\in H\), \(\|w\|<\overline{\rho }\) such that
Thanks to the continuity of g, we can take \(\rho >0\) possibly smaller (\(\rho <\overline{\rho }\)) such that, for any \(w\in H\), \(\|w\|<{\rho }\), there holds
Moreover, for any \(\phi \in H\), \(r>0\), it follows from
that
Consequently, we derive
This completes the proof. □
Lemma 2.3
If \(\lambda \in (0,T_{b})\), then we have
(i) the functional \(I_{b}\) is coercive and bounded from below on \(\Lambda _{b}\);
(ii) \(\inf_{\Lambda _{b}^{+}\cup \Lambda _{b}^{0}}I_{b}=\inf_{ \Lambda _{b}^{+}}I_{b} \in (-\infty ,0)\).
Proof
(i) For \(u\in \Lambda _{b}\), by Hölder’s inequality, we have
and the conclusion (i) follows.
(ii) For \(u\in \Lambda _{b}^{+} \), there holds
This together with Lemma 2.1 gives that \(\inf_{\Lambda _{b}^{+}\cup \Lambda _{b}^{0}}I_{b}= \inf_{\Lambda _{b}^{+}}I_{b}<0\). Moreover, from (i) we infer that \(\inf_{\Lambda _{b}^{+}\cup \Lambda _{b}^{0}}I_{b} \neq -\infty \). Therefore, \(\inf_{\Lambda _{b}^{+}\cup \Lambda _{b}^{0}}I \in (-\infty ,0)\). □
Lemma 2.4
For all \(\lambda >0\), \(I_{b}\) satisfies the \((PS)_{c}\) condition at any level \(c<\frac{{a^{2}}}{{4b}}\).
Proof
The proof is similar to that of [16, Lemma 2]. We omit the details. □
3 Proof of Theorem 1.1
Lemma 3.1
Assume that \(\lambda \in (0,T_{b})\), then problem (1.1) has a positive solution \(u_{b}\) with \(u_{b} \in \Lambda _{b}^{+}\).
Proof
It is easily verified that the sets \(\Lambda _{b}^{+}\cup \Lambda _{b}^{0}\) and \(\Lambda _{b}^{-}\) are closed. Applying the Ekeland variational principle, we can derive a minimizing sequence \(\{u_{n}\}\subset \Lambda _{b}^{+}\cup \Lambda _{b}^{0}\) satisfying that
and
Noting that \(I_{b}(|u|)=I_{b}(u)\), we may suppose that \(u_{n} \ge 0\) in Ω. By Lemma 2.3, \(\{u_{n}\}\) is bounded in H, and so we can assume
In what follows we prove that \(u_{b}\) is a positive solution to (1.1). The proof will be divided into four steps.
textbfStep 1: \(u_{b}\neq 0\).
By contradiction, we suppose that \(u_{b}=0\). Since \(u_{n}\in \Lambda _{b}^{+}\cup \Lambda _{b}^{0}\), we see that, for n large,
As a consequence, we derive
which is a contradiction to (3.1). Thus, \(u_{b}\neq 0\).
textbfStep 2: There exists a constant \(C_{1}>0\) such that
To prove that, it suffices to verify
By \(u_{n} \in \Lambda _{b}^{+}\cup \Lambda _{b}^{0}\),
Suppose to the contrary that
Then we can assume \(\|u_{n}\|^{2}\to A>0\) as \(n\to \infty \), where A satisfies
Combining this with \(\{u_{n}\} \subset \Lambda _{b}\), we have
It follows that
which leads to a contradiction
when \(\lambda \in (0,T_{b})\). Thus, (3.3) holds.
textbfStep 3: \(I'_{b}(u_{n}) \to 0\) in \(H^{-1}\).
Let \(0 <\rho <\rho _{n} \equiv \rho _{u_{n}}\), \(g_{n}\equiv g_{u_{n}}\), where \(\rho _{u_{n}}\) and \(g_{u_{n}}\) are given as in Lemma 2.2 with \(u=u_{n}\). Let \(w_{\rho }=\rho u\) with \(\|u\|=1\). Fix n and set \(z_{\rho }=g_{n}(w_{\rho })(u_{n}-w_{\rho })\). By \(z_{\rho }\in \Lambda _{b}^{+}\), we have from (3.2) that
Then, by the mean value theorem,
Hence, we derive
and thus,
from which it follows that
By Step 2, Lemma 2.2, and the boundedness of \(\{u_{n}\}\), one sees that
and
Therefore, for fixed n, we deduce by taking \(\rho \to 0\) in (3.4) that
which provides that \(I'_{b}(u_{n})\rightarrow 0\) as \(n\to \infty \).
textbfStep 4: \(u_{b}\) is a positive solution of problem (1.1) and \(u_{b}\in \Lambda _{b}^{+}\).
It follows from Step 3, Lemmas 2.3 and 2.4 that, along a subsequence, \(u_{n}\to u_{b}\) in H with \(I_{b}(u_{b})<0\) and \(I'_{b}(u_{b})=0\). Hence, \(u_{b}\ge 0\) is a weak solution to problem (1.1) satisfying \(u_{b}\in \Lambda _{b}^{+}\). The standard elliptic regularity argument and the strong maximum principle imply that \(u_{b}\) is positive. Thus we complete the proof of Lemma 3.1. □
Lemma 3.2
Assume that \(\lambda \in (0,T_{b})\), then problem (1.1) has a positive solution \(\tilde{u}_{b}\) with \(\tilde{u}_{b} \in {{\Lambda ^{-}_{b}}}\).
Proof
Similar to the proof of Lemma 3.1, one can construct a bounded and nonnegative sequence \(\{\tilde{u}_{n}\}\subset \Lambda _{b}^{-}\) satisfying that
Without loss of generality, we may suppose that \(0\in \Omega \). Take a cut-off function \(\varphi (x)\in C_{0}^{\infty }(\Omega )\) satisfying \(0 \leq \varphi \leq 1\) in Ω and \(\varphi (x) \equiv 1\) near zero. Define
It is known that (see [2])
Firstly, we prove the following upper bound for \(\inf \nolimits _{\Lambda _{b}^{-}}I_{b}\):
where \(u_{b}\) is the first positive solution obtained in the previous subsection. By \(u_{b}\in \Lambda _{b}^{+}\) and (1.4), we easily see that \(a-b\|u_{b}\|^{2}>0\). Since \(u_{b}\) is a positive solution of (1.1), we also have
from which it follows that
To proceed, set \(w_{\varepsilon }=u_{b}+Rv_{\varepsilon }\) with \(R>1\). By (3.7), we have
Let \(h(t)\) be defined as in Lemma 2.1. As can be seen from the proof of Lemma 2.1, we have that \(h(t_{\varepsilon })=\lambda \int _{\Omega }|{\frac{w_{\varepsilon }{} }{{\|w_{\varepsilon }\|}}}|^{q}\,dx\) and \(h'(t_{\varepsilon })<0\), where \(t_{\varepsilon }=t^{+} ({\frac{w_{\varepsilon }}{{\|w_{\varepsilon } \|}}} )\). From the structure of h and \(\int _{\Omega }|{\frac{w_{\varepsilon }}{{\|w_{\varepsilon }\|}}}|^{q}\,dx>0\), it follows that \(t_{\varepsilon }\) is uniformly bounded by a suitable positive constant \(C_{1}\), \(\forall R\ge 1\) and \(\forall \varepsilon >0\).
On the other hand, we can infer from (3.8) that there exists \(\varepsilon _{1}>0\) such that
Thus, we can find \(R_{1}\ge 1\) such that \(\|w_{\varepsilon }\|>C_{1}\), \(\forall R\ge R_{1}\), and \(\forall \varepsilon \in (0,\varepsilon _{1})\).
Let
Note that \(H-\Lambda _{b}^{-}=E_{1}\cup E_{2}\) and \(\Lambda _{b}^{+} \subset E_{1}\). Since \(u_{b} \in \Lambda _{b}^{+}\), by the continuity of \(t^{+}(u)\), one sees that \(u_{b}+tR_{1}v_{\varepsilon }\) for \(t\in (0,1)\) must intersect \(\Lambda _{b}^{-}\), and consequently
Hence (3.5) will follow if we show that
By the mean value theorem, we can get \(\delta (x)\in [0,1]\) satisfying
for any \(x\in \Omega \). By (3.6), (3.7), and (3.9),
which implies that there exists \(t_{1}>0\) small enough such that
Thus, we only need to consider the case of \(t\ge t_{1}\). Since
we deduce that (3.5) holds.
Secondly, we claim that \(\tilde{u}_{b}\neq 0\). If, to the contrary, \(\tilde{u}_{b}=0\), from \(\{\tilde{u}_{n}\}\subset \Lambda _{b}^{-}\) it then follows that
As a consequence, we obtain \(\|\tilde{u}_{n}\|^{2}\to \frac{a}{b}\) as \(n\to \infty \). Furthermore,
which contradicts (3.5). Hence, the claim holds. This time we can proceed as in the proof of Lemma 3.1 and deduce that \(\tilde{u}_{b}\) is a positive solution of problem (1.1) with \(\tilde{u}_{b}\in \Lambda _{b}^{-}\). The proof is complete. □
Proof of Theorem 1.1
This is an immediate consequence of (1.4), Lemmas 3.1 and 3.2. □
4 Proof of Theorem 1.2
Proof of Theorem 1.2
For any sequence \(\{b_{n}\}\) with \(b_{n}\searrow 0\), we can use Theorem 1.1 to obtain sequences \(\{u_{b_{n}}\}\subset {{\Lambda _{b_{n}}^{+}}}\) and \(\{\tilde{u}_{b_{n}}\} \subset {{\Lambda _{b_{n}}^{-}}}\) corresponding to positive solutions to problem (1.1) with \(b=b_{n}\) when \(\lambda \in (0,T_{b_{n}})\).
By \(\tilde{u}_{b_{n}} \in {{\Lambda _{b_{n}}^{-}}}\) and (1.4), we see that
and conclusion (i) follows.
Next, we prove conclusion (ii) of Theorem 1.2. Note that
for all \(n\in \mathbb{N}\). Then, by Hölder’s inequality, we have
Since \(1< q<2\), it follows that \(\{u_{b_{n}}\}\) is bounded in H. As a consequence, there exists a subsequence of \(\{b_{n}\}\) (still denoted by \(\{b_{n}\}\)) such that \(u_{b_{n}}\rightharpoonup u_{0}\) in H as \(n\to \infty \). Furthermore, we have that, for all \(v\in H\),
which implies that \(u_{0}\) is a positive solution to problem (1.5). To complete the proof, we only need to show that \(u_{b_{n}}\to u_{0}\) in H. This follows easily from
as \(n\to \infty \). Theorem 1.2 is thus proved. □
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Shi, Z., Qian, X. New multiplicity of positive solutions for some class of nonlocal problems. Bound Value Probl 2021, 55 (2021). https://doi.org/10.1186/s13661-021-01531-8
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DOI: https://doi.org/10.1186/s13661-021-01531-8