Abstract
In this paper, we prove the existence of a solution between a well-ordered subsolution and supersolution of a class of nonlocal elliptic problems and give some degree information. Using the method and bifurcation theory, we present the existence and multiplicity of positive solutions for the nonlocal problems with the changes of the parameter.
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1 Introduction
In this paper, we consider the following problem:
where \(\Omega\subseteq R^{N}\) is a smooth bounded domain, \(\gamma\in (0,+\infty)\), and \(a:[0,+\infty)\to(0,+\infty)\) is a continuous function with
Chipot and Lovat [1] considered the following model problem:
Here Ω is a bounded open subset in \(R^{N}\), \(N\geq1\), with smooth boundary Γ, and T is an arbitrary time. The diffusion coefficient a is a function from R into \((0, +\infty )\), which depends on the entire population in the domain rather than on the local density, and u describes the density of a population subject to spreading. If \(\gamma=2\), then we get the well-known Carrier equation. The fact that (1.1) appears in some applied mathematics attracts a lot of attention. With the aid of the Krasnoselskii fixed point theorem and the Schaefer fixed point theorem, by the monotonicity of \(f_{\lambda}\), Corrêa [2] considered the existence of positive solutions of (1.1) for \(\gamma \geq1\). By establishing a comparison principle, Corrêa et al. [3] proved the existence of positive solutions to (1.1) also for \(\gamma=1\). Under the assumption that \(A(x,u)\) (which is generalized from the nonlocal term \(a(s)\)) is bounded, there are some results on the existence of positive solutions and the existence of n distinct solutions; see [4, 5].
Another nonlocal elliptic equations are the Kirchhoff elliptic problems like
related to nonlinear vibrations of beams, where \(a: R\to R\) is a given function, and \(\|\cdot\|\) denotes the usual norm in \(H_{0}^{1}(\Omega)\). In this case, variational methods are used to consider the existence of the solutions to (1.2) because the nonlocal operator \(u\to a(\|u\|)\Delta u\) possesses a variational structure; see [6–13] and the references therein. Especially, for the Kirchhoff elliptic equations
Chen et al. [14] examined in detail the number of solutions admitted subject to the variations of parameters embedded in nonlinear terms. For the case \(a(t)\equiv1\), the existence and multiplicity of positive solutions for the elliptic equations has been extensively investigated; see [15–22]. Especially, Ambrosetti et al. [23] studied the equation
and established multiple results for different λ, where Ω is a bounded domain in \(R^{N}\) with \(0< q<1<p\leq2^{*}\) (\(2^{*}=\frac{2N}{N-2}\) if \(N\geq3\) and \(2^{*}=+\infty\) if \(N = 1, 2\)) and \(\lambda>0\).
Naturally, we hope that there are some interesting results for (1.1) that are similar to those in (1.3) and (1.4) in [14, 23] and references therein. Notice that the methods used in [8–11, 14–16, 19, 21–25] are the sub-suppersolution method, theory of topological degree, and the variational method. Unfortunately, the operator \(u\to a(\int_{\Omega }|u(x)|^{\gamma}\,dx)\Delta u\) has no variational structure. Up to now, the tools to study (1.1) are a fixed point result with Leray-Schauder condition and the Schaefer fixed point theorem. Very recently, Alves and Covei [26] established the sub-supersolution method, which can be used to study the existence of weak solutions for a large class of nonlocal problems.
Motivated by the works of [14, 23, 26, 27], in this paper, we study the existence and multiplicity of the classical positive solutions for (1.1). The paper is organized as follows. In Section 2, according to the idea in [26, 28], we prove the existence of solutions between well-ordered subsolution and supersolution to guarantee the existence of classical solution to (2.1) and give a formula to calculate the degree. Section 3 presents the existence and multiplicity of positive solutions to (3.1) λ when \(p>1>q>0\) or \(1>p>q>0\), which improves the results in [29], where the \(a(t)\) is bounded, or Ω is an annular region. In Section 4, when nonlinearity is linear at \(u=0\), by bifurcation theory we discuss the unbounded connected component for (4.1) λ and present sufficient and necessary conditions for the existence of positive solutions to (4.7) λ . In Section 5, in the case where the nonlinear term is singular at \(u=0\), we consider the existence of positive solutions to (5.1) λ . In Section 6, sufficient and necessary conditions of positive solutions to (5.1) are given to guarantee that positive solutions to (5.1) are in \(C[0,1]\) or \(C^{1}[0,1]\) when \(N=1\).
Notation
In this paper we use the following notation.
-
Let \(u:\overline{\Omega}\to R\) is continuous, and \(|u|_{\infty}=\max_{x\in\overline{\Omega}}|u(x)|\);
-
\(C(\overline{\Omega})=\{u:\overline{\Omega}\to R| u(x) \mbox{ is continuous on }\overline{\Omega}\}\) with norm \(\|u\|=|u|_{\infty}\);
-
\(C^{1}(\overline{\Omega})=\{u\in C(\overline{\Omega})|\nabla u(x) \mbox{ is continuous on }\overline{\Omega}\}\) with norm \(\|u\|=\max\{ |u|_{\infty},|\nabla u|_{\infty}\}\).
2 Sub-supersolution method
Now we consider the general problem
where \(\Omega\subseteq R^{N}\) is a smooth bounded domain, \(\gamma\in (0,+\infty)\), and \(a:[0,+\infty)\to(0,+\infty)\) is a continuous function with
Definition 2.1
The pair functions \(\alpha, \beta\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)\) are subsolution and supersolution of (2.1) if
and
where \(\chi(x, u)=\alpha(x)+(u-\alpha(x))^{+}-(u-\beta(x))^{+}\).
Definition 2.2
Let \(u, v\in C^{1}(\overline{\Omega})\). We say that \(u\prec v\) if \(u(x)< v(x)\) on Ω and \(u(x)\leq v(x)\) for all \(x\in\partial\Omega\), and if \(u(x)=v(x)\) for some \(x\in\Gamma\subseteq\partial\Omega\), then we write \(\frac{\partial u}{\partial n}|_{x\in\partial\Gamma}>\frac {\partial v}{\partial n}|_{x\in\Gamma}\).
Remark 2.1
\(S=\{u\in C^{1}(\overline{\Omega}):\alpha\prec u\prec\beta\}\) is an open set if \(\alpha\prec\beta\).
We say that an open set \(S\subseteq C^{1}(\overline{\Omega})\) is admissible for the degree if the compact operator A has no fixed point on its boundary ∂S and the set of fixed points of A in S is bounded. In this case, we define
where R is such that every fixed point u of A in S satisfies \(\| u\|< R\). By excision property this degree does not depend on R.
To be able to associate a degree with a pair of subsolution and supersolution, we have to reinforce the definition.
Definition 2.3
A subsolution α of (2.1) is said to be strict if every solution u of (2.1) such that \(\alpha\leq u\) satisfies \(\alpha\prec u\).
In the same way, a strict supersolution β of (2.1) is a supersolution such that every solution u of (2.1) such that \(u\leq\beta\) satisfies \(u\prec\beta\).
Definition 2.4
The function \(F : \Omega\times R\) is an \(L^{p}\)-Carathéodory function if
-
1.
\(F (\cdot, u)\) is measurable for all \(u\in\Omega\);
-
2.
\(F(x,\cdot)\) is continuous for a.e. \(x\in\Omega\);
-
3.
for all bounded set \(B\subseteq R^{N}\), there exists \(h_{B}\in L^{p}(\Omega)\) such that for a.e. \(x\in\Omega\) and all \(u\in B\),
$$\bigl\vert F(x, u)\bigr\vert \leq h_{B}(x). $$
Remark 2.2
The idea of the above definitions comes from [28].
If F is an \(L^{p}\)-Carathéodory function with \(p>N\), then the operator
is well defined, continuous, and maps bounded sets to bounded sets. Then the operator \(A: C^{1}(\overline{\Omega})\to C^{1}(\overline{\Omega })\) defined as
is completely continuous, and problem (2.1) is
Theorem 2.1
Let \(\Omega\subseteq R^{N}\) (\(N\geq1\)) be a smooth bounded domain, and \(\gamma\in(0,+\infty)\). Suppose that \(F:\Omega\times R\to R\) is a continuous function. Assume that α and β are the subsolution and supersolution of (2.1), respectively. If there exists \(h\in L^{p}(\Omega)\) (\(p>N\)) such that
Then problem (2.1) has at least one solution u such that, for all \(x\in\overline{\Omega}\),
If, moreover, \(\alpha(x)\) and \(\beta(x)\) are strict and satisfy \(\alpha\prec\beta\), then
is admissible for the degree, and
Proof
By \(L^{p}\)-theory there exists \(R>0\) greater than \(\max\{\| \alpha\|,\|\beta\|\}\) such that, for every F satisfying (2.2) and every solution of (2.1) with \(\alpha\leq u\leq\beta\), we have
Let
We will study the modified problem (\(\lambda>0\))
where \(\chi(x, u)=\alpha(x)+(u-\alpha(x))^{+}-(u-\beta(x))^{+}\).
Step 1. Every solution u of (2.3) is such that \(\alpha(x)\leq u(x)\leq\beta(x)\), \(x\in\overline{\Omega}\).
We prove that \(\alpha(x)\leq u(x)\) on Ω̅. By contradiction assume that \(\max_{x\in\overline{\Omega}}(\alpha (x)-u(x))=M>0\). Note that \(\alpha(x)-u(x)\not\equiv M\) on Ω̅ (\(\alpha (x)-u(x)\leq0\), \(x\in\partial\Omega\)). If \(x_{0}\in\Omega\) is such that \(\alpha(x_{0})-u(x_{0})=M\), then
This is a contradiction.
Now we prove that \(\beta(x)\geq u(x)\) on Ω̅. By contradiction assume that \(\max_{x\in\overline{\Omega}}(\beta -u(x))=-m<0\). Note that \(\beta(x)-u(x)\not\equiv-m\) on Ω̅ (\(\beta (x)-u(x)\geq0\), \(x\in\partial\Omega\)). If \(x_{0}\in\Omega\) is such that: \(\beta(x_{0})-u(x_{0})=-m\), then
This is a contradiction.
Consequently,
Step 3. Every solution of (2.3) is a solution of (2.1). Every solution of (2.3) is such that \(\alpha(x)\leq u(x)\leq\beta (x)\). Since F̅ satisfies (2.2), we also have that \(\|u\|< R\). Hence,
and u is a solution of (2.1).
Step 4. Problem (2.3) has at least one solution.
Define the operator
It is easy to see that N̅ is well defined, continuous, and maps bounded sets to bounded sets. Then the operator \(\overline{A}: C^{1}(\overline{\Omega})\to C^{1}(\overline{\Omega})\) defined as
is completely continuous.
By the hypothesis on F and the construction of F̅ there exists \(h\in L^{p}(\Omega)\) such that, for every \(u\in C^{1}(\overline {\Omega})\),
which guarantees that there exists \(K>0\) large enough such that, for all \(v\in\overline{A}(C^{1}(\overline{\Omega}))\),
Then there exists \(\overline{K}>\max\{\|\alpha\|,\|\beta\|\}\) large enough such that
and, by a classical result of degree theory [30],
Therefore, there exists \(u\in B_{C^{1}}(0,\overline{K})\) such that
Steps 2 and 3 yield that
Step 5. If \(\alpha(x)\) and \(\beta(x)\) are the strict subsolution and supersolution, then we have
Since \(\alpha(x)\) and \(\beta(x)\) are the strict subsolution and supersolution, A has no fixed point on ∂S, and so \(\operatorname{deg}(I-A, S,\theta)\) is well defined. Step 6 guarantees that A̅ has no fixed point in \(B(0, \overline{K})-\overline{S}\). Then
The proof is complete. □
Remark 2.3
If we do not define the topological degree, we may use \(C(\overline{\Omega})\) and obtain similar results.
Remark 2.4
The difference between our Theorem 2.1 and Theorem 1 in [26] is that \(F(x,u)\) can change sign and we get the existence of classical solutions to (2.1).
Remark 2.5
In the particular case \(N=1\), we can also allow \(p=1\), and it is classical that, in this case, A also is completely continuous.
Remark 2.6
The difference between Definition 2.1 and the definitions in [26] is that we define a special function χ and the classical sub-supersolutions and that in [26] the sub-supersolutions are in the sense of distribution.
In the following sections, we suppose that \(a(t):[0,+\infty)\) is continuous and increasing on \([0,+\infty)\) for convenience.
3 The existence of positive solutions with concave and convex nonlinearities
In this section, we consider the problem
where \(\gamma>0\), \(1>q>0\), \(p>0\), \(\Omega=\{x\in R^{N}||x|<1\}\).
In order to consider the existence of positive solutions for (3.1) λ , we list some previous results. Let \(\varphi_{1}\) be the eigenfunction corresponding to the principle eigenvalue of
It is found that \(\lambda_{1}>0\) and \(\varphi_{1}(x)>0\) for \(x\in\Omega \); see [16]. Moreover, there exist u, \(u^{*}\in C^{2}(\overline{\Omega })\) that satisfy
and
respectively. By [19] the following results are true:
Suppose that \(u_{\lambda}\) is a positive solution to (3.1) λ . Let
Then v satisfies
and the transform (3.3) will be used later.
Let
Obviously, K is cone in \(C^{1}(\overline{\Omega})\).
Using Theorem 2.1, we have following theorems.
Theorem 3.1
Assume that \(\frac{2N}{N-2}>p>1\) and \(\lim_{t\to+\infty}\frac{t^{(p-1)/\gamma}}{a(t)}=+\infty\). Then there exist \(\Gamma_{1}\geq\Gamma_{2}>0\) such that
-
(1)
(3.1) λ has at least two positive solutions if \(\lambda\in(0,\Gamma_{2})\);
-
(2)
(3.1) λ has at least one positive solution if \(\lambda =\Gamma_{1}\) and \(\lambda=\Gamma_{2}\);
-
(3)
(3.1) λ has no positive solutions if \(\lambda>\Gamma_{1}\).
Moreover,
Theorem 3.2
Assume that \(0< q< p<1\). Then (3.1) λ has at least one positive solution for \(\lambda\geq0\).
Now we consider
where \(a_{0}=\inf_{t\in[0,+\infty)}a(t)\).
Lemma 3.1
(see [16])
Assume that \(0< q<1\), \(p>1\). Then there exist \(\Gamma_{a_{0}}\) and \(C_{a_{0}}>0\) such that
-
(1)
(3.5) λ has at least two positive solutions if \(\lambda\in(0,\Gamma_{a_{0}})\);
-
(2)
(3.5) λ has at least one positive solution if \(\lambda =\Gamma_{a_{0}}\);
-
(3)
(3.5) λ has no positive solutions if \(\lambda>\Gamma_{a_{0}}\).
Moreover,
Lemma 3.2
(see [21])
Suppose that \(f:\Omega\times R^{+}\to R\) is a continuous function such that \(s^{-1}f(x,s)\) is strictly decreasing for \(s>0\) at each \(x\in\Omega\). Let \(w, v\in C(\overline{\Omega})\cap C^{2}(\Omega)\) satisfy:
-
(a)
\(\Delta w+f(x, w)\leq0\leq\Delta v + f(x,v)\) in Ω;
-
(b)
\(w, v>0\) in Ω, and \(w\geq v\) on ∂Ω;
-
(c)
\(\Delta v\in L^{1}(\Omega)\).
Then \(w\geq v\) in Ω̅.
Proof of Theorem 3.1
(1) We show that for \(\lambda\in(0,\Gamma_{a_{0}})\), (3.1) λ has at least one positive solution.
For \(u\in P\), we define the operator
where \(G(x,y)\) is the Green function for \(-\Delta u=h\).
For \(\lambda\in(0,\Gamma_{a_{0}})\), by [16] there exists a \(u_{\lambda }\in C^{1}(\overline{\Omega})\) such that
with \(\frac{\partial u_{\lambda}}{\partial n}<0\), \(x\in\partial \Omega\). Let \(\beta(x)=u_{\lambda}\) and \(b_{0}=\sup_{t\in[0,|\beta|^{\gamma }_{\infty}|\Omega|]}a(t)\). Since \(0<1<q\) and \(\lambda>0\), we can choose \(\varepsilon>0\) small enough such that
and
Let \(\alpha(x)=\varepsilon\varphi_{1}(x)\). Then
Therefore, by the strict monotonicity of a we have
and
with
which implies that α and β are the subsolution and supersolution of (1.2) with \(\alpha\prec\beta\). Now Theorem 2.1 implies that (3.1) λ has at least one positive solution \(u_{\lambda}\) with \(\alpha(x)\leq u_{\lambda}(x)\leq\beta(x)\), \(x\in\overline {\Omega}\). Moreover, if \(W=\{u\in K\subseteq C^{1}(\overline{\Omega})|\alpha\prec u\prec\beta\}\) and \(u\in[\alpha,\beta]\) is a solution to (1.2), then we have
which, together with the maximum principle, means that
that is,
A similar argument shows that
Now Theorem 2.1 guarantees that
Let
Obviously, \(\Gamma_{1}>0\).
(2) We show that \(\Gamma_{1}<+\infty\).
Assume that \(u_{\lambda}\) is a solution to (3.1) λ . Let \(c=a(\int_{\Omega}|u_{\lambda}|^{\gamma}\,dx)^{-\frac{1}{p-1}}\) and \(v=cu_{\lambda}\). Then we get (3.4). By Lemma 3.1 there exist \(C'>0\) and \(\Gamma'>0\) such that equation
has at least one positive solution for all \(0\leq\lambda\leq\Gamma '\) and
which, together with (3.4), implies that
and
Now we show that \(\{\int_{\Omega}|u_{\lambda}(x)|^{\gamma}\,dx: \lambda\in(0,\Gamma _{1})\}\) is bounded.
In fact, if \(\{\int_{\Omega}|u_{\lambda}(x)|^{\gamma}\,dx: \lambda\in (0,\Gamma_{1})\}\) is unbounded, then there exists a sequence \(\{ u_{\lambda_{n}}\}\) such that
Now (3.7) means that
Then
Integration on Ω yields that
Let \(s_{n}=\int_{\Omega}|u_{\lambda_{n}}(x)|^{\gamma}\,dx\). Then
which contradicts to
Since \(a(t)>0\) is continuous on \([0,+\infty)\) with \(\inf_{t\geq 0}a(t)=a_{0}>0\), the boundedness of \(\{\int_{\Omega}|u_{\lambda }(x)|^{\gamma}\,dx: \lambda\in(0,\Gamma_{1})\}\) means that
which, together with (3.8), means that
Hence,
From (3.9) we have
(3) We show that there exists \(u_{\Gamma_{1}}\) satisfying (3.1)\(_{\Gamma_{1}}\). By the definition of \(\Gamma_{1}>0\) there exists a sequence \(\lambda _{n}\to\Gamma_{1}\) and \(u_{\lambda_{n}}\) is a positive solution of (3.1)\(_{\lambda_{n}}\). From (3.10), there exists \(C_{1}>0\) such that
which guarantees that \(\{u_{\lambda_{n}}\}\) is relatively compact in \(C^{1}(\overline{\Omega})\). Then there exists \(u_{\Gamma_{1}}\in C^{1}(\overline{\Omega})\) such that
A standard bootstrap argument shows that \(u_{\Gamma_{1}}\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)\) is a nonnegative solution for (3.1)\(_{\Gamma_{1}}\).
(4) We show that for \(\lambda\in(0,\Gamma_{a_{0}})\), (3.1) λ has at least two positive solutions.
By (3.10) and the Green formula there exists \(C_{2}>0\) such that
Let \(R>\max\{C,C_{2}\}\) and \(\overline{\lambda}>\Gamma\). Let \(H(\tau ,u)=u-(-\Delta)^{-1}((\tau\lambda+(1-\tau)\overline{\lambda })u^{q}+u^{p})\) and \(B_{R}=\{u|\|u\|< R\}\). If there exist \(\tau_{0}\in[0,1]\) and \(u_{0}\in K\cap\partial B_{R}\) such that
then we have
which, together with (3.11) and (3.12), means that \(\|u\|=\max\{|u_{0}|_{\infty},|\nabla u_{0}|_{\infty}\}< R\). This contradicts to \(u_{0}\in(\partial B_{R})\cap K\). The homotopy of H,
Next, we claim that
In fact, suppose that there exist \(1\geq\mu_{0}\geq0\) and \(u_{0}\in\partial B_{R}\cap K\) such that \(H(0,u_{0})=\mu_{0}u_{0}\). Obviously, \(\mu_{0}>0\) and \(u_{0}\) satisfy
Let \(v=\mu_{0}^{-\frac{1}{p-1}}u_{0}\). Then v satisfies that
Since
this contradicts the definition of \(\Gamma_{1}\), which means that
Therefore, (3.13) is true, and so,
Now for \(\lambda\in(0,\Gamma_{a_{0}})\), we consider
From [16], (3.15) has one positive solution \(v_{0}\). Let \(r=\|v_{0}\|\). Let \(0< r<\min\{a(C^{\gamma}|\Omega|)^{\frac{1}{q-1}}\|v_{0}\|, \|e\|\} \). For \(\tau\in[0,1]\), define
We claim that
In fact, suppose \((\tau_{0},u_{0})\in[0,1]\times K\cap\partial B(0,r)\). Then
Let \(c=a(\int_{\Omega}u_{0}^{\gamma}\,dx)^{-\frac{1}{q-1}}\) and \(v=cu_{0}\). Then
By Lemma 3.2 we have
and so
which contradicts to \(\|u_{0}\|=r< a(C^{\gamma}|\Omega|)^{\frac {1}{q-1}}\|v_{0}\|\). From the homotopy of H it follows that
which, together with (3.14), implies that
Consequently, A has another fixed point \(u_{\lambda,1}\in(B_{R}-\overline{(W\cup B_{r})})\cap K\), that is, (3.1) λ has another positive solution \(u_{\lambda,1}\) for all \(\lambda\in(0,\Gamma_{a_{0}})\).
Consequently,
Similarly, (3.1)\(_{\Gamma_{2}}\) also has at least one positive solution. The proof is complete. □
Proof of Theorem 3.2
For given \(\lambda>0\), since \(0< q< p<1\), there exists \(K_{1}>0\) such that
that is,
Let \(\beta(x)=K_{1}e(x)\). Then
Let \(b_{0}=a(\int_{\Omega}\beta^{\gamma}(x)\,dx)\). Choose \(\varepsilon >0\) small enough such that
and
Let \(\alpha(x)=\varepsilon\varphi_{1}(x)\). Now (3.17), (3.18), and (3.19) guarantee that
and
which guarantees that α and β are the subsolution and supersolution to (3.1) λ . Now Theorem 2.1 implies that (3.1) λ has at least one positive solution for all \(\lambda \geq0\). The proof is complete. □
4 The existence of positive solutions when the nonlinearity is linear at \(u=0\)
In this section, we consider the problem
where \(\gamma>0\), and \(\Omega\subseteq R^{N}\) is a bounded smooth domain. Now we list following conditions for convenience:
- (H1):
-
\(f(x,u)\) is continuous on \(\overline{\Omega}\times(-\infty ,+\infty)\), and
$$ \lim_{|u|\to0+}\frac{f(x,u)}{u}=0 \quad \mbox{uniformly on } x \in \overline{\Omega}. $$(4.2)
For \(u\in C(\overline{\Omega})\), we define the operator
where \(G(x,y)\) is the Green function for \(-\Delta u=h\).
Of course, under these new notation, \((\lambda, u)\) solves (4.1) λ if and only if
Suppose that (H1) holds. It is easy to see that \(L:C(\overline {\Omega})\to C(\overline{\Omega})\) is a compact and continuous linear operator and \(H(\lambda, \cdot):C(\overline{\Omega})\to C(\overline{\Omega})\) is a compact and continuous nonlinear operator. Moreover, (4.2) guarantees that
Now, we state the following result.
Lemma 4.1
(see [31])
Let E be a Banach space. Suppose that L is a compact linear operator and that \(\lambda^{-1}\in\sigma(L)\) with odd multiplicity. If H satisfies condition (4.4), then the set
has a closed connected component \(C = C_{\lambda}\) such that \((\lambda ,0)\in C\) and
-
(i)
C is unbounded in \(\mathbb{R}\times E\), or
-
(ii)
there exists \(\hat{\lambda}\neq\lambda\) such that \((\hat {\lambda}; 0)\in C\) and \(\hat{\lambda}^{-1}\in\sigma(S)\).
Suppose that \(\lambda_{1}\) is the principle eigenvalue to the problem
It is well known that the first eigenfunction \(\phi_{1}\) associated to \(\lambda_{1}\) can be chosen positive. Moreover, \(\lambda_{1}\) is an eigenvalue with odd multiplicity.
By the global bifurcation theorem, (H1) guarantees that there exists a closed connected component \(C=C_{\lambda}\) of solutions for (4.1) λ that satisfies (i) or (ii).
Lemma 4.2
There exists \(\delta>0\) such that if \((\lambda, u)\in C\) with \(|\lambda-\lambda_{1}|+|u|<\delta\) and \(u\neq0\), then u has a defined sign, that is,
Proof
Take \(\{u_{n}\}\) in \(C(\overline{\Omega})\) and \(\lambda _{n}\to\lambda_{1}\) such that
Considering \(w_{n}=u_{n}/\|u_{n}\|\), we get
It is easy to check that
where K is a positive constant.
Since that \(\{u_{n}\}\) is bounded in \(C(\overline{\Omega})\), \(\{w_{n}\}\) is also bounded in \(C^{1}(\overline{\Omega})\). By the Arzelà-Ascoli theorem, \(\{w_{n}\}\) converges to some \(w\in C^{1}(\overline{\Omega})\), uniformly in Ω̅, under a convenient subsequence. Of course, \(\| w\|_{C(\overline{\Omega})} = 1\), and thus \(w\neq0\) in Ω.
Now, by (4.5) we know that \(\{u_{n}\}\) is a Cauchy sequence in \(C(\overline{\Omega})\) and
Letting \(n\to+\infty\), we have
that is,
Since \(w\neq0\), by spectral theory we must have
Without loss of generality, we can suppose that \(w(x)>0\) for all \(x\in \Omega\). Since w is the \(C^{1}(\overline{\Omega})\)-limit of \(\{w_{n}\}\), we must have \(w_{n}(x) > 0\) for all \(x\in\Omega\) and n large enough. Therefore, the sign of \(u_{n}\) coincides with that of \(w_{n}\) for n large enough. The proof is complete. □
Now we decompose C into \(C =C^{+}\cup C^{-}\), where
and
A simple computation gives that \(C^{+}=\{(\lambda,u)\in C | (\lambda ,-u)\in C^{-}\}\) and \(C^{+}\) is unbounded if and only if C is also unbounded.
Theorem 4.1
If (H1) holds, then there exists an unbounded closed connected component \(C=C_{\lambda}\) of solutions for (4.1) λ .
Proof
In fact, suppose that C is bounded, which implies that \(C^{+}\) is bounded and C contains \((\hat{\lambda},0)\), where \(\hat{\lambda}\neq\lambda _{1}\), \(\hat{\lambda}^{-1}\in\sigma(L)\).
In this way, we can take \(\{u_{n}\}\) in \(C(\overline{\Omega})\) and \(\lambda_{n}\to\hat{\lambda}\) such that
Considering \(w_{n}=u_{n}/\|u_{n}\|_{C(\overline{\Omega})}\), we know that it satisfies problem (4.6). Moreover, as in the proof of Lemma 4.2, under an adequate subsequence, \(\{w_{n}\}\) converges to w in \(C^{1}(\overline{\Omega})\), which is a nonzero solution of the eigenvalue problem
that is, w is an eigenfunction related to λ̂. Since \(\hat{\lambda}\neq\lambda_{1}\), w must change sign. Then, for n large, each \(w_{n}\) must change sign, and the same should hold for \(u_{n}=w_{n}\| u_{n}\|_{C(\overline{\Omega})}\), which contradicts to \((\lambda_{n}, u_{n})\in C^{+}\). The proof is complete. □
Now we consider the following special problem:
where \(\gamma>0\), Ω is a bounded smooth domain, and \(p>1\).
By Theorem 4.1 the connected component \(C^{+}\) of (4.7) λ is unbounded. Now we have following theorem for (4.7) λ .
Theorem 4.2
Suppose that \(p>\max\{\alpha\gamma+1,\gamma -1\}\). Then at least one positive solution of (4.7) λ exists if and only if \(\lambda>\lambda_{1}\).
Proof
First, we will show that for any \(\Lambda>0\), there exists \(r>0\) such that
From now on, we denote by \(\|\cdot\|\) the usual norm in \(H_{0}^{1}(\Omega )\), that is,
Indeed, suppose that (4.8) is false. Then, there are \(\{u_{n}\}\in H^{1}_{0}(\Omega)\) such that
Considering \(w_{n} =u_{n}/\|u_{n}\|\), it follows from (4.3) that
Since that \(\{w_{n}\}\) is bounded in \(H_{0}^{1}(\Omega)\), without loss of generality, we can suppose that there is \(w\in H^{1}_{0}(\Omega)\) satisfying
and
Taking \(v =u_{n}/\|u_{n}\|^{p-\alpha\gamma}\) as a test function, (4.9) is
Since \(p>\alpha\gamma+1\), letting \(n\to+\infty\), we derive
Since
we have
which implies that
By the Fatou lemma
Therefore, we should have \(w=0\). Thereby, \(\{w_{n}\}\) converges to 0 in \(L^{2}(\Omega)\). Taking \(v = w_{n}\) as a test function, we see that
Since \(\{\lambda_{n}\}\) is bounded from above by Λ and \(\frac{1}{c_{1}+c_{2}(\int_{\Omega}u_{n}(x)^{\gamma}\,dx)^{\alpha }}\int_{\Omega}u_{n}(x)^{p}w_{n}(x)\,dx\geq0\), we have
Taking the limit, we have that \(\|w_{n}\|\to0\), which contradicts to \(\| w_{n}\|= 1\) for all n. Then (4.8) is true, which, together with the boundedness of Ω, implies that
Next, we will show the nonexistence of solution for \(\lambda\leq \lambda_{1}\), proving that \(C^{+}\) does not intersect \([0,\lambda_{1}]\times H^{1}_{0}(\Omega)\). Indeed, suppose that
Using \(v=\phi_{1}\) as a test function in (4.6), we get
This is a contradiction.
Consequently, problem (4.7) λ has at least one positive solution if and only if \(\lambda>\lambda_{1}\). The proof is complete. □
5 The positive solutions for singular nonlocal elliptic problems
In this section, we consider the singular elliptic equation
where \(\gamma>0\), \(1>q>0\), Ω is a bounded domain in \(R^{N}\), \(N\geq2\), with \(C^{2,\beta}\) boundary ∂Ω, \(\beta\in (0,1)\), \(K\in C^{2,\beta}(\overline{\Omega})\), and \(0< q<1\), \(\mu\in(0,1)\).
Now we list some previous results for the following equation:
where \(K\in C(\overline{\Omega})\) and \(0< q<1\), \(\mu\in(0,1)\). Define
Theorem 5.1
(see [21])
Let \(K(x)<0\), \(x\in\overline{\Omega}\). Then
-
(i)
(5.2) λ has a unique solution \(u_{\lambda}\in E\) for any \(\lambda\in R\);
-
(ii)
\(u_{\lambda}\) is increasing with respect to λ;
-
(iii)
\(c_{1}d(x)\leq u_{\lambda}(x)\leq c_{2}d(x)\) for any \(x\in \Omega\) and some \(c_{1}\) and \(c_{2} > 0\) independent of x;
-
(iv)
\(u_{\lambda}\in C^{1,1-\mu}(\Omega)\).
Theorem 5.2
(see [21])
Let \(\min_{x\in{\overline{\Omega }}}K(x)>0\). Then
-
(i)
there exists \(\lambda^{*}>0\) such that (5.2) λ has at least one positive solution \(u_{\lambda}\in E\) for any \(\lambda >\lambda^{*}\);
-
(ii)
\(c_{1}d(x)\leq u_{\lambda}(x)\leq c_{2}d(x)\) for any \(x\in \Omega\) and some \(c_{1}\) and \(c_{2} > 0\) independent of x;
-
(iii)
\(u_{\lambda}\in C^{1,1-\mu}(\Omega)\).
Using Theorems 5.1 and 5.2, by Theorem 2.1 we have the following results for (5.1) λ .
Theorem 5.3
Let \(K(x)<0\) for all \(x\in\overline{\Omega }\). Then
-
(i)
(1.1) has at least one solution \(u_{\lambda}\in E\) for any \(\lambda\geq0\);
-
(ii)
\(c_{1}d(x)\leq u_{\lambda}(x)\leq c_{2}d(x)\) for any \(x\in \Omega\) and some \(c_{1}\) and \(c_{2} > 0\) independent of x, and \(u_{\lambda}\in C^{1,1-\mu}(\Omega)\).
Proof
(1) For \(\lambda\geq0\), we consider the problem
where \(n\in\{1,2,\ldots\}\).
Since \(\mu\in(0,1)\) and \(0< q<1\), there exists \(K_{1}>0\) such that
that is,
Let \(\beta(x)=K_{1}(e(x)+1)\). Then
Let \(b_{0}=a(\int_{\Omega}\beta(x)^{\gamma}\,dx)\). Since \(0< q<1\), there exists \(\frac{1}{n}>\varepsilon_{n}>0\) small enough such that
and
Let \(\alpha_{n}(x)=\varepsilon_{n}\varphi_{1}(x)\). Then, for \(u\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)\), (5.4), (5.5), and (5.6) imply that
and
which means that \(\alpha_{n}(x)\) and \(\beta(x)\) are the subsolution and supersolution to (5.3) n . Now Theorem 2.1 implies that (5.3) n has at least one positive solution \(u_{n}\).
Now we consider set \(\{u_{n}\}\). Since \(|u_{n}|_{\infty}\leq K_{1}|e+1|_{\infty}\), it follows that
and so
Lemma 3.2, together with (3.3), implies there exists \(c_{0}>0\) such that
where \(b_{0}=a(\int_{\Omega}\beta^{\gamma}(x)\,dx)\). Then
and
The same technique as in [32], Theorem 1.1, yields
Therefore, \(u_{n}\in C^{1,1-\mu}(\overline{\Omega})\). The sequence \(\{ u_{n}\}\) has a subsequence \(\{u_{n_{i}}\}\) such that
Now a straightforward calculation yields
(2) Suppose that \(u_{\lambda}\) satisfies (5.1) λ . Let \(c_{0}=a(\int_{\Omega}u_{\lambda}^{\gamma}(x)\,dx)>0\) and \(v(x)=a(\int _{\Omega}u_{\lambda}^{\gamma}(x)\,dx) u_{\lambda}(x)\), \(x\in\overline {\Omega}\). Then \(v(x)\) satisfies
Now Theorem 5.1 implies that there exist \(c_{1}\) and \(c_{2} > 0\) independent of x such that
and so
Moreover, \(v\in C^{1,1-\mu}(\Omega)\) implies that \(u\in C^{1,1-\mu }(\Omega)\). The proof is complete. □
Theorem 5.4
Let \(\min_{x\in{\overline{\Omega}}}K(x)>0\). Suppose that there exists \(\rho>\frac{1}{1-q}\) such that
Then
-
(i)
there exists \(\lambda^{*}>0\),such that (1.1) has at least one positive solution \(u_{\lambda}\in E\) for any \(\lambda>\lambda^{*}\);
-
(ii)
\(c_{1}d(x)\leq u_{\lambda}(x)\leq c_{2}d(x)\) for any \(x\in \Omega\) and some \(c_{1}\) and \(c_{2} > 0\) independent of x, and \(u_{\lambda}\in C^{1,1-\mu}(\Omega)\).
Proof
For \(\varphi_{1}\), by the Hopf maximum principle, there exist \(\delta_{0}>0\) and \(\Sigma\subset\Omega\) such that
Then there exists \(M_{1}>0\) such that
and there exists \(M_{2}>0\) such that
where \(K^{*}=\max_{x\in\overline{\Omega}}K(x)\). By (5.7), choose \(M>\max\{M_{1},M_{2}\}\) large enough such that
Combining (5.8) and (5.9), we have
and so
Therefore, for \(n>0\),
Let \(u^{*}_{1}(x)\) satisfy
Now by (5.7) it follows that
which implies that there exists \(T_{0}>0\) such that, for all \(t>T_{0}\),
Let
For \(\lambda>T_{1}\), let
It is easy to see that
For \(u\in C^{1}(\overline{\Omega})\), let
Then
Let
From (5.9) we know that \(b_{0}>1\). By (5.11), (5.13), and (5.14), for \(u\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)\), we have
and
Hence, \(\underline{u}_{\lambda}(x)\) and \(\overline{u}_{\lambda}(x)\) are the subsolution and supersolution of (5.3) n . Now Theorem 2.1 implies that for \(n\in\{1,2,\ldots\}\), (5.3) n has at least one solution \(u_{\lambda}\) with
Now we consider set \(\{u_{\lambda,n}\}\). From (5.14) we have
and
The same technique as in [32], Theorem 1.1, yields
Therefore, \(u_{n}\in C^{1,1-2\mu/(1+\mu)}(\overline{\Omega})\). The sequence \(\{u_{n}\}\) has a subsequence \(\{u_{n_{i}}\}\) such that
Now a straightforward calculation yields
(2) Suppose that \(u_{\lambda}\) satisfies (5.1) λ . Let \(c_{0}=a(\int_{\Omega}u_{\lambda}^{\gamma}(x)\,dx)>0\) and \(v(x)=a(\int _{\Omega}u_{\lambda}^{\gamma}(x)\,dx) u_{\lambda}(x)\), \(x\in\overline {\Omega}\). Then \(v(x)\) satisfies
Now Theorem 5.2 implies that there exist \(c_{1}\) and \(c_{2} > 0\) independent of x such that
and so
Moreover, \(v\in C^{1,1-\mu}(\Omega)\) implies that \(u\in C^{1,1-\mu }(\Omega)\). The proof is complete. □
6 Positive solutions for (1.1) when \(N=1\)
In this section, we consider the case \(N=1\):
and by using Theorem 2.1 we present sufficient and necessary conditions for the existence of positive solutions for (6.1).
Now we list some conditions for convenience:
- (H1):
-
\(f:(0,1)\times(0,+\infty)\rightarrow(0,+\infty)\) is continuous, and there exist λ, μ, δ (\(-\infty<\lambda<0<\mu<1\), \(0<\delta\leq1\)) such that for all \(x\in(0,1) \) and \(v\in(0,+\infty)\), we have
$$\begin{aligned}& c_{0}^{\mu}f(x,v)\leq f(x,c_{0}v)\leq c_{0}^{\lambda}f(x,v),\quad 0\leq c_{0}\leq \delta, \end{aligned}$$(6.2)$$\begin{aligned}& c_{0}^{\lambda}f(x,v)\leq f(x,c_{0}v)\leq c_{0}^{\mu}f(x,v),\quad c_{0}\geq1/ \delta. \end{aligned}$$(6.3)
Now we state a result on the existence of positive solutions for the following problem from [33]
Theorem 6.1
(see [33])
Suppose (H1) holds. Then necessary and sufficient conditions for the existence of positive solutions from \(C[0,1]\) for the boundary value problem (6.4) are
Theorem 6.2
(see [33])
Suppose (H1) holds. Then a necessary and sufficient condition for the existence of positive solutions from \(C^{1}[0,1]\) for the boundary value problem (6.4) is
Lemma 6.1
(see [34])
Suppose \(u\geq0\) is concave on \([0,1]\). Then
By using the idea of the proof in [22] sufficient and necessary conditions for the existence of positive solutions to (6.1) are obtained.
Theorem 6.3
Suppose (H1) holds. Then necessary and sufficient conditions for the existence of positive solutions from \(C[0,1]\) for the boundary value problem (6.1) are
Proof
Necessity. Suppose that \(u_{0}\) is a \(C[0,1]\) positive solution to (6.1). Let \(v(x)= a(\int_{0}^{1}u_{0}(t)^{\gamma}\,dt)u_{0}(x)\), and so v satisfies
It is easy to see that \(g(x,v)\) satisfies (H1), and Theorem 6.1 implies that
which is equivalent to (6.5).
Sufficiency. Now we consider
Choose \(m\geq2\) such that \(m(\mu-\lambda)>1\) and
Let
and
and, by the proof of [33], define
and
where
Let \(H^{*}=\max_{x\in[0,1]}H_{1}(x)\) and choose \(K_{1}>0\) large enough such that
Let \(\beta(x)=K_{1}H_{1}(x)\), \(x\in[0,1]\). Inequalities (6.2) and (6.3) guarantee that
and so
Let \(b_{0}=\sup_{t\in[0,\int_{0}^{1}\beta^{\gamma}(t)\,dt]}a(t)\). Let \(K_{2}\) be small enough such that
and
Let \(\alpha(x)=K_{2}h_{1}(x)\), \(x\in[0,1]\). Inequalities (6.2) and (6.3) guarantee that
and so
For \(u\in C[0,1]\), from (6.7) and (6.8) we have
and
Consequently, α and β are the subsolution and supersolution to (6.6) n . By Theorem 2.1, (6.1) n has at least one positive solution \(u_{n}\) with \(\alpha(x)\leq u_{n}(x)\leq\beta(x)\), \(x\in[0,1]\). Moreover, combining (6.2), (6.3), and (6.7), we have
which guarantees that \(\{u_{n}'(x)\}\) are uniformly bounded on \([\frac{1}{3k},1-\frac {1}{3k}]\subseteq(0,1)\), \(k\geq1\). Therefore, \(\{u_{n}(x)\}\) has a uniformly convergent subsequence on any \([\frac{1}{3k},1-\frac{1}{3k}]\subseteq(0,1)\), \(k\geq1\). By the diagonal method there exists a subsequence of \(\{u_{n}\}\) that converges uniformly \(u_{0}\) on any \([\frac{1}{3k},1-\frac{1}{3k}]\subseteq (0,1)\), \(k\geq1\). Without loss of generality, assume that
Obviously, \(u_{0}(x)\) is continuous on \((0,1)\), and
Since \(\alpha(0)=\beta(0)=\alpha(1)=\beta(1)=0\), we have
which means that \(u_{0}\) is continuous on \([0,1]\) with \(u_{0}(0)=u_{0}(1)=0\). The dominated convergence theorem guarantees that
which, together with the continuity of \(a(t)\), means that
Since
where
letting \(n\to+\infty\), we have
Differentiating the above equations yields that
which, together with \(u_{0}(0)=u_{0}(1)=0\), means that \(u_{0}\in C[0,1]\cap C^{2}(0,1)\) and \(u_{0}\) is a solution to (6.1). □
Theorem 6.4
Suppose (H1) holds. Then a necessary and sufficient condition for the existence of a positive solution from \(C^{1}[0,1]\) of (6.1) is
Proof
Necessity. Suppose that \(u_{0}\) is a positive solution. Let \(v=a(\int _{0}^{1}u_{0}^{\gamma}(t)\,dt)u_{0}(t)\), \(t\in[0,1]\), and so v satisfies
It is easy to see \(g(x,v)\) satisfies (H1), and Theorem 6.2 implies that
which is equivalent to (6.9).
Sufficiency. Let
By Lemma 6.1 we can see that \(h(x)\geq x(1-x)|h|_{\infty}\). From (6.9) we know that there exists \(N>0\) such that \(\int _{0}^{1}s(1-s)f(s,1-s)\,ds\leq N\). Then
where \(a_{1}=\min\{1, \| h\|\}\), \(a_{2}=\max\{1, N\}\).
Let \(k_{2}>0\) large enough such that
Inequalities (6.2) and (6.3) imply that
Let \(k_{1}< k_{2}\) be small enough such that
Inequalities (6.2) and (6.3) imply that
Consequently, for \(u\in C^{1}[0,1]\), (6.10) and (6.11) guarantee that
and
Moreover, for \(\alpha(x)\leq u\leq v(x)\), choose \(c>0\) large enough such that
So, from (6.2) and (6.3) we have
which, together with (6.9), guarantees that
By Theorem 2.1, (6.1) has at least one positive solution \(u\in C^{1}[0,1]\) with \(\alpha(x)\leq u(x)\leq\beta(x)\), \(x\in[0,1]\). □
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Acknowledgements
This research is supported by Young Award of Shandong Province (ZR2013AQ008) and the Fund of Science and Technology Plan of Shandong Province (2014GGH201010).
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Yan, B., Ma, T. The existence and multiplicity of positive solutions for a class of nonlocal elliptic problems. Bound Value Probl 2016, 165 (2016). https://doi.org/10.1186/s13661-016-0670-z
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DOI: https://doi.org/10.1186/s13661-016-0670-z