Abstract
In this paper, we consider the generalized logistic equation with nonlocal reaction term
Using the bifurcation and sub-supersolution method, we obtain the non-existence, existence, and uniqueness of positive solutions for different parameters on the nonlocal terms. Our works about the nonlocal elliptic problem improve the results in the previous literature.
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1 Introduction
In this paper, we consider the nonlocal elliptic boundary value problem
Here Ω is a bounded domain in \(\mathbb{R}^{N}\), \(N\geq 2\), with \(C^{2,\beta }\) boundary ∂Ω, λ, \(b\in \mathbb{R}\), \(r>0\), \(\beta \in (0,1)\), and \(f(u)\) is a polynomial denoted by
where
This type of problem was studied initially by Delgado et al. in [8], where they proposed the equation
and the corresponding steady-state problem
Here \(u(x,t)\) represents the density of a species in time \(t>0\) and at the point \(x\in \Omega \), the habitat of the species that is surrounded by inhospitable areas, λ is the growth rate of species, term \(-f(u)\) describes the limiting effect of crowding in the population. In this paper the authors proved the existence of an unbounded continuum of positive solutions of (1.3), presented some non-existence results, and discussed the local and global behavior of the continuum.
The introduction of nonlocal terms in the equation and in the boundary conditions models a number of processes in different fields such as mathematical physics, mechanics of deformable solids, mathematical biology, and many others (see [1, 2, 4, 5, 10–12, 16, 20]).
Obviously, problem (1.1) is a generalization of problem (1.3). In this paper, we present some results on the existence of an unbounded continuum of positive solutions of (1.1), the local and global behavior of the continuum, and prove the non-existence of positive solutions also.
The paper is organized as follows. In Sect. 2 we give some lemmas which show the relationship among the solution, sub-solution, and super-solution and the relationship between the solution and the nonlinear term f and prove the existence of an unbounded continuum of positive solutions of (1.1). Section 3 is devoted to proving the non-existence results and a priori bounds of positive solutions of (1.1). In Sect. 4 we presents some conditions for the existence of positive solutions for (1.1) and prove local and global behavior of the continuum of positive solutions of (1.1). Some ideas come from [13, 14].
Throughout our paper, we always suppose that (1.2) is true.
2 Bifurcation results
In order to discuss (1.1), we consider the following equation:
where \(\lambda \in \mathbb{R}\).
Denote by \(\varphi_{1}\) an eigenfunction corresponding to the principle eigenvalue \(\lambda_{1}\) of
From [9] and [15], \(\varphi_{1}\) belongs to \(C^{2,\beta }(\overline{\Omega })\), \(\varphi_{1}>0\) in Ω, and \(\lambda_{1}>0\). Moreover, assume that \(\|\varphi_{1}\|_{\infty }=1\).
Lemma 2.1
(See [3])
There exists a positive solution of (2.1) if and only if \(\lambda >\lambda_{1}\). Moreover, if it exists, the solution is unique, and we denote it by \(\theta_{\lambda }\). Furthermore, the following inequalities hold:
Lemma 2.2
Assume that u is the unique positive solution to (2.1) for \(\lambda >\lambda_{1}\). Then
where
and we have denoted by \(\beta_{1}\) the unique solution of the following linear problem in Ω under the homogeneous Dirichlet boundary condition:
and
Lemma 4.3 in [9] proved the relationship between the solution u of problem (2.1) when \(f(u)=u\) and the first eigenfunction \(\varphi_{1}\) of problem (2.2). Lemma 2.2 obtains a similar result for the case \(f(u)=\sum_{i=1}^{n}a_{i}u^{k_{i}}\). Since the proof is the same as that in [9], we omit it.
From Lemma 2.2, we obtain the following corollary directly.
Corollary 2.1
Assume that u is the unique positive solution to (2.1) for \(\lambda >\lambda_{1}\). There exist two positive constants \(\delta >0\) and \(K>0\) such that
Lemma 2.3
Assume that \(\theta_{\lambda }\) is the unique positive solution to (2.1) for \(\lambda >\lambda_{1}\).
-
(1)
If \(\underline{u}>0\) is a strict sub-solution of (2.1), then \(\underline{u}\leq \theta_{\lambda }\).
-
(2)
If \(\overline{u}>0\) is a strict super-solution of (2.1), then \(\theta_{\lambda }\leq \overline{u}\).
Since \(u^{-1}u(\lambda -f(u))=\lambda -f(u)\) is decreasing, it is easy to get the proof from Lemma 2.3 in [19], and we omit it.
We consider the Banach space \(X:=C_{0}(\overline{\Omega })\), denote \(B_{\rho }:=\{u\in X:\|u\|_{\infty }<\rho \}\). Define
and the map
where \(u_{+}=\max \{u(x),0\}\) and \((-\Delta)^{-1}\) is the inverse of the operator −Δ under homogeneous Dirichlet boundary conditions. Agmon–Douglas–Nirenberg theorem, embedding theorem, and strong maximum theorem(see [17]) guarantee that \((-\Delta)^{-1}\) is positive and compact. It is clear that u is a nonnegative solution of (1.1) if and only if \(K_{\lambda }u=0\).
Now we give the main result of this section.
Theorem 2.1
The value \(\lambda =\lambda_{1}\) is the only bifurcation point from the trivial solution for (1.1). Moreover, there exists a continuum \(C_{0}\) of nonnegative solutions of (1.1) unbounded in \(\mathbb{R}\times X\) emanating from \((\lambda_{1}, 0)\). Furthermore,
-
(i)
if \(b\leq 0\), the direction of bifurcation is supercritical.
-
(ii)
Assume \(b>0\).
-
(a)
If \(r < 1\), the direction of bifurcation is subcritical.
-
(b)
If \(r>1\), then the direction of bifurcation is supercritical.
-
(c)
Assume \(r=1\), and denote
$$ b_{0}= \frac{a_{1}\int_{\Omega }\varphi_{1}^{3}\,dx}{\int_{\Omega }\varphi_{1}\,dx \int_{\Omega }\varphi_{1}^{2}\,dx}. $$
-
(a)
If \(b>b_{0}\) (resp. \(b< b_{0}\)), the direction of bifurcation is subcritical (resp. supercritical).
Recall that we say that the direction of bifurcation is subcritical (resp. supercritical) if there exists a neighborhood V of \((\lambda_{1}, 0)\) such that for every solution \((\lambda, u)\in V\) satisfies \(\lambda <\lambda_{1}\) (resp. \(\lambda >\lambda_{1}\)), see [8].
In order to prove this result, we use the Leray–Schauder degree of \(K_{\lambda }\) on \(B_{\rho }\) with respect to zero, denoted by \(\operatorname{deg}(K_{\lambda }, B_{\rho } )\), and the index of the isolated zero of \(K_{\lambda }\), denoted by \(i(K_{\lambda }, u)\).
Lemma 2.4
If \(\lambda <\lambda_{1}\), then \(i(K_{\lambda },0)=1\).
Lemma 2.5
If \(\lambda >\lambda_{1}\), then \(i(K_{\lambda },0)=0\).
Since the proof is the same as that of Lemmas 2.3 and 2.4 in [8], we omit it.
Proof of Theorem 2.1
From Lemmas 2.4 and 2.5 and bifurcation theorem (see [18]), the same proof as that of Theorem 2.2 in [8] guarantees the existence of an unbounded continuum \(C_{0}\) of positive solutions of (1.1). Moreover, conclusion (i) is true.
We only give the proof of (ii).
(a) Assume now that \(b>0\) and the existence of a sequence \((\lambda _{n}, u_{n})\in C_{0}\) of positive solutions of (1.1) such that \(\lambda_{n}\geq \lambda_{1}\) and \(\|u_{n}\|_{\infty }\rightarrow 0\) as \(n\rightarrow \infty \). By the property of f, there is \(\delta_{1}>0\) such that
Take \(M>0\) such that
Since \(r<1\), choose n large enough such that \(u^{r}_{n}>Mu_{n}\), and then
which implies that \(u_{n}\) is a strict super-solution of the following system:
Using Lemma 2.1, we get (2.7) has a unique positive solution \(\theta_{n}\) and
Lemma 2.3 implies that
Integrating the above inequality yields that
And then
Using (2.5), one has
for n large enough. And then
which together with (2.6) implies
an absurdum.
(b) Assume now that \(b>0\), \(r>1\) and the existence of a sequence \((\lambda_{n}, u_{n})\in C_{0}\) of positive solutions of (1.1) such that \(\lambda_{n}\leq \lambda_{1}\) and \(\|u_{n}\|_{\infty }\rightarrow 0\) as \(n\rightarrow \infty \). Without loss of generality, assume that \(\|u_{n}\|_{\infty }\leq \delta \) defined in Corollary 2.1. Take \(\varepsilon >0\) such that
where K is defined in Corollary 2.1.
For n large we have \(u_{n}^{r}<\varepsilon u_{n}\), and then
which implies that \(u_{n}\) is a strict sub-solution of the following problem:
By Lemma 2.1, we get (2.9) has a unique positive solution \(\theta_{n}\). Moreover, from Corollary 2.1, we have
for n large enough. By Lemma 2.3, we have
Integrating the above inequality yields that
which together with (2.8) implies that
an absurdum.
(c) Assume that \(b>0\) and \(r=1\). In this case, we apply the Crandall–Rabinowitz theorem (see [7]). Then there exist \(\varepsilon >0\) and two regular functions \(\lambda (s)\), \(u(s)\), \(s\in (-\varepsilon, \varepsilon)\), such that in a neighborhood of \((\lambda_{1}, 0)\), the positive solutions are \(u(s)\), \(s\in (0, \varepsilon)\). We can write
and
where \(\lambda_{2}\in \mathbb{R}\), \(\varphi_{2}\in C^{2}(\overline{ \Omega })\). It is evident that the sign of \(\lambda_{2}\) determines the bifurcation direction. Substituting these expansions into (1.1) and identifying the terms of order one in s yield
Multiplying by \(\varphi_{1}\) and integrating in Ω, we conclude that
This finishes the proof. □
3 A priori bounds and non-existence results of (1.1)
In this section, we obtain a priori bounds of the solutions for \(b>0\) as well as non-existence results of (1.1).
Theorem 3.1
Assume that \(b>0\) and \(r<1\). Let \(u_{\lambda }\) be a positive solution of (1.1) such that \(\lambda \in K\subset \mathbb{R}\) a compact set. Then there exists a constant \(L_{0}>0\) such that
for a constant independent of \(\lambda \in K\). Moreover, there exists a constant \(L_{1}\) such that, if
(1.1) does not possess any positive solution.
Proof
Since K is compact, there is a positive constant \(k_{0}>0\) such that
Moreover, since \(u_{\lambda }\) is a positive solution of (1.1), we have, using Lemma 2.1 and Hölder’s inequality, that
Step 1. We show that there exists \(c_{1}>0\) such that
In fact, suppose to the contrary that there exists \(\{u_{\lambda_{n}} \}\) such that
Replacing λ in (3.1) by \(\lambda_{n}\), one has
Integrating inequality (3.3) in Ω yields that
i.e.,
By the property of \(f(u)\), one has
which together with (3.4) implies that
for n large enough. This is a contradiction because \(r<1\).
Step 2. We show that there exists a constant \(L_{0}>0\) such that
Since Step 1 holds, (3.1) guarantees that
Let \(L_{0}:=f^{-1}(k_{0}+bc_{1}^{r}|\Omega |^{1-r})\). We conclude
Step 3. We show that there exists a constant \(L_{1}\) such that, if
(1.1) does not possess any positive solution.
Now define a function
From the property of f and \(0< r<1\), one has
which together with \(g(0)=0\) implies that there is \(s_{0}\geq 0\) such that
Let
Assume that \(u_{\lambda }\) is a positive solution to (1.1) for \(\lambda \in \mathbb{R}\). From (3.4), we have
which means that
Consequently, (1.1) has no positive solution if \(\lambda < L_{1}\).
The proof is complete. □
Theorem 3.2
Assume that \(b>0\), \(k_{n}< r\), where \(k_{n}\) is the index of the last term of polynomial f. Let \(u_{\lambda }\) be a positive solution of (1.1) such that \(\lambda \in K\subset \mathbb{R}\) a compact set. Then there exists a constant \(L_{0}\) such that
for a constant independent of \(\lambda \in K\). Moreover, there exists a constant \(L_{1}\) such that
and if
(1.1) does not possess any positive solution.
Proof
Since K is compact, there is \(k_{1}\) such that
Moreover, using now the lower bound in Lemma 2.1, we get that
Step 1. We show that there exists a constant \(c_{1}>0\) such that
In fact, suppose to the contrary that there exists \(\{\lambda_{n}\} \subseteq K\) such that
By the property of \(f(u)\), there is \(M>0\) such that
for u large enough. Integrating (3.5) in Ω yields that
that is, by Hölder’s inequality and (3.6)
for n large enough.
This is a contradiction to \(k_{n}< r\).
Step 2. We show that there exists a constant \(L_{0}>0\) such that
Since K is compact, there exists \(k_{0}\) such that
From (3.5) and Step 1, one has
Let
Then \(L_{0}\) satisfies Step 2.
Step 3. We show that there exists a constant \(L_{1}\) such that, if
(1.1) does not possess any positive solution.
Now define
Since \(k_{n}< r\), one has
which together with \(g_{1}(0)=0\) implies that there exists \(s_{1} \geq 0\) such that
Let
Assume that \(u_{\lambda }\) is a positive solution to (1.1) for \(\lambda \in \mathbb{R}\). Integrating (3.5) in Ω yields that
i.e.,
And then
that is,
Hence
Consequently, Step 3 is true.
Moreover, we consider
It is easy to prove that \(h(s)\) gets its maximum at \(s=(\frac{b}{nAq})^{ \frac{1}{1-q}}\) and
Let
Obviously, we have
Since \(0<\frac{k_{i}}{r}<1\), \(i=1,2,\ldots ,n\), one has that
By the definition \(g_{1}\) in (3.7), we have
which implies that
On the other hand, since \(L_{1}>\lambda_{1}\), we have
The proof is complete. □
Theorem 3.3
Assume that \(b>0\), \(r=1\). Let \(u_{\lambda }\) be a positive solution of (1.1) such that \(\lambda \in K\subset \mathbb{R}\) a compact set.
-
(1)
If \(k_{n}=1=r\) and \(b<1/|\Omega |\) or \(b\int_{\Omega } \varphi_{1}\,dx>1\), then there exist a priori bounds of the solution of (1.1). Moreover, if \(b<1/|\Omega |\) and \(\lambda \leq 0\) or \(b\int_{\Omega }\varphi_{1}\,dx>1\) and \(\lambda \geq \lambda_{1}\), then (1.1) does not possess a positive solution.
-
(2)
If \(k_{n}>1=r\), then there exists a constant \(L_{0}>0\) such that
$$ \Vert u_{\lambda } \Vert _{\infty }\leq L_{0} $$
for a constant independent of λ. Moreover, there exists a constant \(L_{1}\) such that, if
(1.1) does not possess any positive solution.
Conclusion (i) is Proposition 3.1 in [8] and the proof of (ii) is similar to that in Theorem 3.1, and we omit it.
4 Existence and uniqueness results
In this section, first we introduce the method of sub-supersolution to some nonlocal elliptic problems.
Consider a continuous operator \(B:L^{\infty }(\Omega)\rightarrow \mathbb{R}\) and \(f:\Omega \times \mathbb{R}^{2}\rightarrow \mathbb{R}\) a continuous function and the general problem
where Ω is a bounded domain in \(\mathbb{R}^{N}\), \(N\geq 2\), with \(C^{2,\beta }\) boundary ∂Ω.
Definition 4.1
(See [6])
We say that the pair \((\underline{u}, \overline{u})\) with \(\underline{u}, \overline{u}\in H^{1}(\Omega)\cap L^{\infty }(\Omega)\) is a pair of sub-supersolutions of (4.1) if
-
(a)
\(\underline{u}\leq \overline{u}\) in Ω and \(\underline{u} \leq 0\leq \overline{u}\) on ∂Ω;
-
(b)
\(-\Delta \underline{u}-f(x,\underline{u},B(u))\leq 0\leq -\Delta \overline{u}-f(x,\overline{u},B(u))\) in the weak sense for all \(u\in [\underline{u}, \overline{u}]\).
Lemma 4.1
(See [6])
Assume that there exists a pair of sub-supersolutions of (4.1). Then there exists a solution \(u\in H^{1}(\Omega)\cap L^{\infty }(\Omega)\) of (4.1) such that \(u\in [\underline{u}, \overline{u}]\).
Now we give the main theorems.
Theorem 4.1
Assume that \(b<0\). Then (1.1) has a positive solution if and only if \(\lambda >\lambda_{1}\). Moreover, if there exists the unique positive solution, denoted by \(u_{\lambda,b}\), then
Proof
By Theorem 2.1 we know the existence of an unbounded continuum \(C_{0}\) of positive solutions bifurcating from the trivial solution at \(\lambda =\lambda_{1}\). Assume that \((\lambda, u_{\lambda })\in C_{0}\). Now we show that \(\lambda >\lambda_{1}\).
In fact, if \(\lambda \leq \lambda_{1}\), then
which implies that \(u_{\lambda }\) is a sub-solution (2.1).
Choose a constant K large enough such that
Obviously, \((u_{\lambda },K)\) is a pair of sub-supersolutions to (2.1). Then (2.1) has a positive solution for \(\lambda \leq \lambda_{1}\). This is a contradiction to Lemma 2.1.
We know that positive solutions do not exist for \(\lambda \leq \lambda _{1}\), hence we conclude that if \((\lambda,u)\in C_{0}\), we have \(\lambda >\lambda_{1}\).
Moreover, if \((\lambda,u)\in C_{0}\), since \(b<0\), we have
and Lemma 2.3 implies that \(u_{\lambda }\leq \theta_{\lambda }\), where \(\theta_{\lambda }\) is a solution to (2.1). Lemma 2.1 guarantees that (2.1) has a unique positive solution \(\theta_{\lambda }\) for all \(\lambda >\lambda_{1}\), which together with the unboundedness of \(C_{0}\) implies that (1.1) has at least one positive solution \(u_{\lambda,b}\) for all \(\lambda >\lambda_{1}\).
We show now the uniqueness.
Assume that there exist two positive solutions \(u\neq v\) for \(b<0\). If \(\int_{\Omega }u^{r}\,dx=\int_{\Omega }v^{r}\,dx\), u and v satisfy
where \(k=b\int_{\Omega }u^{r}\,dx=b\int_{\Omega }v^{r}\,dx<0\). This is a contradiction to Lemma 2.1.
So, assume that for instance
then
and then by Lemma 2.3 we get \(u>v\), an absurdum.
On the other hand, we have that
and then \(f(u_{\lambda,b})<\lambda \). So, as \(b\rightarrow -\infty \), we get
Moreover, Lemma 2.1 implies
and
we conclude that
This implies that \(\|u_{\lambda,b}\|_{\infty }\rightarrow 0\). □
Theorem 4.2
Assume that \(b>0\), \(0< r<1\). Then there exists \(\lambda_{\ast }<\lambda_{1}\) such that (1.1) possesses at least a positive solution if and only if \(\lambda \geq \lambda_{\ast }\). Moreover,
Proof
Define
We know by Theorems 2.1 and 3.1 that \(-\infty <\lambda_{\ast }<\lambda _{1}\).
Step 1. We show that (1.1) has at least one positive solution for all \(\lambda >\lambda_{*}\).
Take \(\lambda >\lambda_{\ast }\), then there exists \(\mu \in [ \lambda_{\ast }, \lambda)\) such that (1.1) possesses at least a positive solution, denoted by \(u_{\mu }\). Choose K large enough such that
Let \((\underline{u},\overline{u})=(u_{\mu },K)\). Since \(u_{\mu }\) is a positive solution of (1.1) and (4.2) is true, we have
-
(a)
\(\underline{u}=u_{\mu }< K=\overline{u}\) in Ω and \(\underline{u}=u_{\mu }=0< K=\overline{u}\) on ∂Ω;
-
(b)
$$\begin{aligned}& -\Delta \underline{u}-\underline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \underline{u})\biggr) \\& \quad =u_{\mu }\biggl(\mu +b \int_{\Omega }u^{r}_{\mu }\,dx-f(u_{ \mu }) \biggr)-u_{\mu }\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(u_{\mu })\biggr) \\& \quad =bu_{\mu } \int_{\Omega }\bigl(u^{r}_{\mu }-u^{r} \bigr)\,dx+(\mu -\lambda)u_{ \lambda } \\& \quad \leq 0, \quad x\in \Omega, \forall u\in [\underline{u},\overline{u}] \end{aligned}$$
and
$$\begin{aligned} -\Delta \overline{u}-\overline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \overline{u})\biggr) =&-0-K \biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(K)\biggr) \\ =&K\biggl(-\lambda -b \int_{\Omega }u^{r}\,dx+f(K)\biggr) \\ \geq& K\bigl(-\lambda -bK \vert \Omega \vert ^{r}+f(K)\bigr) \\ >&0, \quad x\in \Omega, \forall u\in [\overline{u},\overline{u}], \end{aligned}$$
which implies that \((\underline{u},\overline{u})\) is a pair of sub-supersolutions to (1.1). Theorem 4.2 guarantees that (1.1) has at least one positive solution for all \(\lambda >\lambda_{*}\).
Step 2. We show that, for \(\lambda =\lambda_{*}\), (1.1) has a positive solution.
By the definition of \(\lambda_{*}\), there exists \(\{\lambda_{n}\}\) such that \(\lambda_{n}\geq \lambda_{\ast }\) and \(\lambda_{n}\rightarrow \lambda_{\ast }\). Thanks to the bounds of Theorem 3.1, we have that \(u_{n}\rightarrow u_{\ast }\geq 0\), \(u_{\ast }\) is a solution for \(\lambda =\lambda_{\ast }\). Since \(\lambda_{\ast }<\lambda_{1}\) and \(\lambda_{1}\) is the unique bifurcation point from the trivial solution, we conclude that \(u_{\ast }>0\).
Step 3. We show that
Since u is bounded and
and then taking \(b\rightarrow 0\), we have that \(\lambda \geq \lambda _{1}\), that is,
Now we prove
It suffices to show that, for any \(\lambda <\lambda_{1}\), there exists \(b>0\) big enough such that (1.1) possesses at least one positive solution.
In fact, for any \(\lambda <\lambda_{1}\), there exists \(b>0\) large enough such that for the function
For above b, take \(K>1+|\lambda |+\frac{1}{2}\|\varphi_{1}\|_{ \infty }\) large enough such that
Let \(\underline{u}=\frac{1}{2}\varphi_{1}(x)\) and \(\overline{u}=K\). From (4.3) and (4.4), we have
-
(a)
\(\underline{u}=\frac{1}{2}\varphi_{1}< K=\overline{u}\) in Ω and \(\underline{u}=\frac{1}{2}\varphi_{1}(x)=0< K= \overline{u}\) on ∂Ω;
-
(b)
$$\begin{aligned} -\Delta \underline{u}-\underline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \underline{u})\biggr) =& \frac{1}{2}\lambda_{1}\varphi_{1}-\frac{1}{2} \varphi _{1}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f\biggl(\frac{1}{2} \varphi_{1}\biggr)\biggr) \\ =&\frac{1}{2}\varphi_{1}\biggl((\lambda_{1}- \lambda)-b \int_{\Omega }u^{r}\,dx+f\biggl( \frac{1}{2} \varphi_{1}\biggr)\biggr) \\ < &0, \quad x\in \Omega, \forall u\in [\underline{u},\overline{u}] \end{aligned}$$
and
$$\begin{aligned} -\Delta \overline{u}-\overline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \overline{u})\biggr) =&-0-K \biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(K)\biggr) \\ =&K\biggl(-\lambda -b \int_{\Omega }u^{r}\,dx+f(K)\biggr) \\ \geq& K\bigl(-\lambda -bK^{r} \vert \Omega \vert +f(K)\bigr) \\ >&0,\quad x\in \Omega, \forall u\in [\overline{u},\overline{u}], \end{aligned}$$
which implies that \((\underline{u},\overline{u})\) is a pair of sub-supersolutions to (1.1). Theorem 4.2 guarantees that (1.1) has at least one positive solution for all \(\lambda <\lambda_{1}\).
The proof is complete. □
Theorem 4.3
Assume that \(b>0\) and \(k_{n}< r\). There exists \(\lambda^{\ast }>\lambda_{1}\) such that (1.1) possesses at least a positive solution if and only if \(\lambda \leq \lambda^{\ast }\). Moreover,
Proof
Assume that \(b>0\) and \(k_{n}< r\). Define now
We know by Theorems 2.1 and 3.2 that \(\lambda_{1}<\lambda^{\ast }<+ \infty \).
Step 1. We prove now that there exists a positive solution for all \(\lambda \in (-\infty,\lambda^{\ast })\).
Indeed, take \(\lambda \in [\lambda_{1},\lambda^{\ast })\), then there exists \(\mu \in (\lambda_{1}, \lambda^{\ast }]\) such that (1.1) possesses at least a positive solution, denoted by \(u_{\mu }\). Choose ε small enough such that \(\varepsilon \varphi_{1}< u_{ \mu }\) for all \(x\in \Omega \) such that
Let \(\underline{u}=\varepsilon \varphi_{1}\) and \(\overline{u}=u_{ \mu }\). Since \(u_{\mu }\) is a positive solution of (1.1), from (4.5), one has
-
(a)
\(\underline{u}=\varepsilon \varphi_{1}< u_{\mu }\) in Ω and \(\underline{u}=0=\overline{u}\) on ∂Ω;
-
(b)
$$\begin{aligned} -\Delta \underline{u}-\underline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \underline{u})\biggr) =& \varepsilon \lambda_{1}\varphi_{1}-\varepsilon \varphi_{1}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(\varepsilon \varphi_{1})\biggr) \\ =&\varphi_{1}\varepsilon \biggl((\lambda_{1}-\lambda)-b \int_{\Omega }u^{r}\,dx+f( \varepsilon \varphi_{1})\biggr) \\ < &0, \quad x\in \Omega, \forall u\in [\underline{u},\overline{u}] \end{aligned}$$
and
$$\begin{aligned}& -\Delta \overline{u}-\overline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \overline{u})\biggr) \\& \quad =u_{\mu }\biggl( \mu +b \int_{\Omega }u^{r}_{\mu }\,dx-f(u_{ \mu }) \biggr)-u_{\mu }\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(u_{\mu })\biggr) \\& \quad =u_{\mu }\biggl(\mu -\lambda +b \int_{\Omega }\bigl(u^{r}_{\mu }-u^{r} \bigr)\,dx\biggr) \\& \quad >0, \quad x\in \Omega, \forall u\in [\overline{u},\overline{u}], \end{aligned}$$
which implies that \((\underline{u},\overline{u})\) is a pair of sub-supersolutions to (1.1). Theorem 4.2 guarantees that (1.1) has at least one positive solution for all \(\lambda \in [\lambda_{1},\lambda _{*})\).
Now, Theorem 2.1 implies \(C_{0}\) is supercritical, which implies that there exists \((\lambda,u)\in C_{0}\) with \(\lambda_{0}>\lambda_{1}\). For any \(\lambda <\lambda_{1}\), let \(K=[\lambda,\lambda_{0}]\). Theorem 3.2 guarantees that \(\|u\|_{\infty }\leq L_{0}\) for all \(\lambda \in K\), which together with the unboundedness of \(C_{0}\) implies that there is u such that \((\lambda,u)\in C_{0}\).
Step 2. We show that, for \(\lambda =\lambda^{*}\), (1.1) has a positive solution.
Taking a sequence of positive solutions \((\lambda_{n}, u_{n})\) of (1.1) such that \(\lambda_{n}\leq \lambda^{\ast }\) and \(\lambda_{n}\rightarrow \lambda^{\ast }\). Thanks to the bounds of Theorem 3.2, we have that \(u_{n}\rightarrow u^{\ast }\geq 0\), \(u^{\ast }\) is a solution for \(\lambda =\lambda^{\star }\). Since \(\lambda^{\ast }>\lambda_{1}\) and \(\lambda_{1}\) is the unique bifurcation point from the trivial solution, we conclude that \(u^{\ast }>0\).
Step 3. We show that
Observe that since \(\lambda_{1}<\lambda^{\ast }\leq L_{1}\) defined in Theorem 3.2 and \(\lim_{b\rightarrow \infty }L_{1}=\lambda_{1}\), we concluded that
Now we prove
It suffices to show that, for any \(\lambda >\lambda_{1}\), there exists \(b>0\) small enough such that (1.1) possesses at least a positive solution. For \(\lambda >\lambda_{1}\), take \(\tilde{\Omega }\supset \Omega \) and consider \(\tilde{\varphi _{1}}\) and \(\tilde{\lambda _{1}}\) the positive eigenfunction and eigenvalue associated with Ω̃. Choose K large enough such that
Choose \(b>0\) small enough such that
Choose \(\varepsilon >0\) small enough such that \(\varepsilon \varphi _{1}< K\tilde{\varphi }_{1}\) and
Let \(\underline{u}(x)=\varepsilon \varphi_{1}(x)\) and \(\overline{u}(x)=K \tilde{\varphi }_{1}(x)\) for \(x\in \overline{\Omega }\). From (4.6) and (4.7), we have
-
(a)
\(\underline{u}=\varepsilon \varphi_{1}< K\tilde{\varphi }_{1}= \overline{u}(x)\) in Ω and \(\underline{u}=0< K\tilde{\varphi } _{1}=\overline{u}\) on ∂Ω;
-
(b)
$$\begin{aligned} -\Delta \underline{u}-\underline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \underline{u})\biggr) =& \varepsilon \lambda_{1}\varphi_{1}-\varepsilon \varphi_{1}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(\varepsilon \varphi_{1})\biggr) \\ =&\varepsilon \varphi_{1}\biggl((\lambda_{1}-\lambda)-b \int_{\Omega }u^{r}\,dx+f( \varepsilon \varphi_{1})\biggr) \\ < &0,\quad x\in \Omega, \forall u\in [\underline{u},\overline{u}] \end{aligned}$$
and
$$\begin{aligned} -\Delta \overline{u}-\overline{u}\biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f( \overline{u})\biggr) =&K\tilde{ \lambda }\tilde{\varphi }_{1}-K \tilde{\varphi }_{1}\biggl( \lambda +b \int_{\Omega }u^{r}\,dx-f(K \tilde{\varphi }_{1})\biggr) \\ =&K\tilde{\varphi }_{1}\biggl(\tilde{\lambda }-\lambda -b \int_{\Omega }u ^{r}\,dx+f(K\tilde{\varphi }_{1})\biggr) \\ >&0,\quad x\in \Omega, \forall u\in [\overline{u},\overline{u}], \end{aligned}$$
which implies that \((\underline{u},\overline{u})\) is a pair of sub-supersolutions to (1.1). Theorem 4.2 guarantees that (1.1) has at least one positive solution for \(\lambda >\lambda_{1}\).
The proof is complete. □
Theorem 4.4
Assume that \(b>0\) and \(r=1< k_{n}\). There exists \(\lambda_{\ast }<\lambda_{1}\) such that (1.1) possesses at least a positive solution if and only if \(\lambda \geq \lambda_{\ast }\). Moreover,
The proof of Theorem 4.4 is similar to that of Theorem 4.3, we omit it.
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Zhou, J., Gao, G. & Yan, B. Study of a generalized logistic equation with nonlocal reaction term. Bound Value Probl 2018, 150 (2018). https://doi.org/10.1186/s13661-018-1066-z
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DOI: https://doi.org/10.1186/s13661-018-1066-z