Abstract
We study positive solutions to the two-point boundary value problem:
where \(\lambda >0\) is a parameter, \(h \in C^1((0,1],(0,\infty ))\) is a decreasing function, \(f \in C^1((0,\infty ),\mathbb {R}) \) is an increasing concave function such that \(\lim \limits _{s \rightarrow \infty }f(s)=\infty \), \(\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0\), \(\lim \limits _{s \rightarrow 0^+}f(s)=-\infty \) (infinite semipositone) and \(c \in C([0,\infty ),(0,\infty ))\) is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for \(\lambda \gg 1\).
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4 Appendix
4 Appendix
Throughout this appendix, assume c satisfies \((H_3)\), and \(c(z)=c(0)~;~z<0\). First we consider the boundary value problem:
We establish:
Lemma A.1
Let \(\bar{h}\in L^1(0,1).\) Then (A.1) has a unique solution. Further, \(T:L^1(0,1)\rightarrow C[0,1]\) such that \(u=T(\bar{h})\) is the solution of (A.1), is a completely continuous operator.
Proof
By integrating we obtain
where \(A\in \mathbb {R}\) is such that \(u'(1)+c(u(1))u(1)=0\), i.e., \(G(A)=0\) where
Since \(G:\mathbb {R}\rightarrow \mathbb {R}\) is increasing and \(\lim \limits _{z \rightarrow \infty }G(z)=\infty \), \(\lim \limits _{z \rightarrow -\infty }G(z)=-\infty \), \(\exists _1\) \(A\in \mathbb {R}\) such that \(G(A)=0\). Thus (A.1) has a unique solution \(u\equiv T(\bar{h})\). Note that \(|A|\le \int _0^1|\bar{h}|ds\). (If \(z>\int _0^1|\bar{h}|ds\) then \(G(z)>0\) while if \(z<-\int _0^1|\bar{h}|ds\) then \(G(z)<0\).) Hence (A.2) and (A.3) give:
So T maps bounded sets in \(L^1(0,1)\) into relatively compact subsets of C[0, 1]. It remains to show that T is continuous. Let \({\bar{h}_n}\subset L^1(0,1)\) be such that \(\bar{h}_n\rightarrow \bar{h}\) in \(L^1(0,1)\) and let \(u_n=T(\bar{h}_n)\), \(u=T( \bar{h})\). Then \(-(u''_n-u'')=\bar{h}_n-\bar{h}.\) So (A.4) gives \(|u_n-u|_{C^1}\le 2\Vert \bar{h}_n-\bar{h}\Vert _{L^1}\rightarrow 0\). Thus Lemma A.1 is proven. \(\square \)
Next, we consider the boundary value problem:
when c satisfies \((H_3)\) and the following hypotheses hold:
\((G_1)\):\(~~g:(0,1)\times (0,\infty )\rightarrow \mathbb {R}\) is continuous and \(\exists ~\Phi ,\Psi \in C^2(0,1)\cap C^1[0,1]\) with
\(~~~~~~~~~\Psi \le \Phi ;~(0,1)\), \(\inf \limits _{(0,1)}\frac{\Psi }{p}>0\) where \(p(t)=\min (t,1-t)\) such that
and
\((G_2)\):\(~~\exists ~\Gamma \in L^1(0,1)\) such that for all \(\zeta \in [\Psi ,\Phi ]\), \(|g(t,\zeta (t))|\le \Gamma (t);~(0,1).\)
We establish:
Lemma A.2
Let \((G_1)-(G_2)\) hold. Then (A.5) has a positive solution \(u\in [\Psi ,\Phi ]~;~[0,1]\).
Proof
For \(v\in C[0,1]\) define \(u=S(v)\) to be the unique solution of:
where \(\tilde{v}(s)=\min \big (\max (v(s),\Psi (s)),\Phi (s)\big ).\) Note that by \((G_2)\), \(\exists \) \(\Gamma \in L^1(0,1)\) such that \(|g(t,\tilde{v})|\le \Gamma (t))\) and so (A.6) has a unique solution by Lemma A.1. Since \(\tilde{T}:C[0,1]\rightarrow L^1(0,1)\) defined \(\tilde{T}(v)=\tilde{v}\) is continuous and \(S=T\circ \tilde{T}\) (here T is as defined in Lemma A.1), it follows that \(S:C[0,1]\rightarrow C[0,1]\) is completely continuous. Since S(C[0, 1]) is bounded, S has a fixed point u by the Schauder Fixed Point Theorem. We now claim that \(u\in [\Psi ,\Phi ]\). Suppose there exists \(t_0\in (0,1)\) such that \(u(t_0)<\Psi (t_0).\) Let \((\alpha ,\beta )\) be the largest open interval in (0, 1] containing \(t_0\) such that \(u<\Psi \) on \((\alpha ,\beta )\). Then \(u(\alpha )=\Psi (\alpha )\). If \(\beta <1\) then \(u(\beta )=\Psi (\beta )\), while if \(\beta =1\) then \(u(\beta )\le \Psi (\beta )\) and \(u'(\beta )\ge \Psi '(\beta )\) in view of the boundary conditions of u and \(\Psi \) at 1. Thus in either case we have
Further, since \(\tilde{u}=\Psi \) on \((\alpha ,\beta )\) we have
Multiplying (A.8) by \(u-\Psi \) and integrating we obtain
Hence using (A.7), we obtain \(u'=\Psi '~;~(\alpha ,\beta )\), i.e., \(u=\Psi ~;~(\alpha ,\beta )\) (since \(u(\alpha )=\Psi (\alpha )\)), a contradiction. Hence \(u(t)\ge \Psi (t)~;~(0,1)\). Similarly, we obtain \(u(t)\le \Phi (t)~;~(0,1)\). Thus \(u\in [\Psi ,\Phi ]~;~[0,1]\) and so \(\tilde{u}=u\), i.e., u is a solution of (A.5). Hence Lemma A.2 is proven. \(\square \)
Finally, we consider (1.1) when for some \(l>0, \alpha>0, \gamma >0\) such that \(\alpha +\gamma <1\), h and f satisfy:
- \((\tilde{H}_1)\):
-
: \(h:(0,1]\rightarrow (0,\infty )\) is \(L^1(0,1)\) and \(\limsup \limits _{s \rightarrow 0^+}s^\gamma h(s)<\infty .\)
- \((\tilde{H}_2)\):
-
: \(f:(0,\infty )\rightarrow \mathbb {R}\) is \(C^1\), \(\lim \limits _{s \rightarrow \infty }f(s)=\infty \), \(\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0\), \(\lim \limits _{s \rightarrow 0^+}f(s)=-\infty \) and \(~~~~~~~~~\lim \limits _{s \rightarrow 0^+}s^{1+\alpha }f'(s)=l\).
We establish:
Lemma A.3
Let \((\tilde{H}_1)-(\tilde{H}_2)\) hold. Then (1.1) has a maximal positive solution for \(\lambda \gg 1\).
Proof
By Theorem 1.3 in [19], the Dirichlet boundary value problem:
has a positive solution w for \(\lambda \gg 1\). Clearly this w satisfies the role of \(\Psi \) in \((G_1)\) since \(w'(1)\le 0\). Let e be the unique positive solution of
and choose \(z(t)=M(\lambda )e(t)~;~[0,1]\) where \(M(\lambda )\gg 1\) so that
where \(\tilde{f}(s)=\max \limits _{(0,s]}f(t)\) (which is possible since \(\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0\)). Then
Also \(z(0)=0\) and
since c is increasing. Now choose \(M(\lambda )\gg 1\) so that \(z\ge \Psi (=w)\). Then z satisfies the role of \(\Phi \) in \((G_1)\) and hence \((G_1)\) hold. Also, for \(\zeta \in [\Psi ,\Phi ]\), \(\exists ~C>0\) such that
Note that \(\Gamma \in L^1(0,1)\) since \(\inf \limits _{(0,1)}\frac{\Psi }{p}>0\) and \(\limsup \limits _{s\rightarrow 0^+}s^\gamma h(s)<\infty \). So \((G_2)\) holds and by Lemma A.2, (1.1) has a positive solution for \(\lambda \gg 1\). To show the existence of a maximal solution, let u be a positive solution of (1.1). Then
This implies
Since \(\lim \limits _{s\rightarrow \infty }\frac{f(s)}{s}=0=\lim \limits _{s\rightarrow \infty }\frac{\tilde{f}(s)}{s}\) , \(\exists ~c_\lambda >0\) such that
and hence
This implies u is bounded by a supersolution of the form \(M(\lambda ) e(t)\) for \(M(\lambda )\gg 1\), and the existence of the maximal positive solution follows. \(\square \)
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Hai, D.D., Muthunayake, A. & Shivaji, R. A uniqueness result for a class of infinite semipositone problems with nonlinear boundary conditions. Positivity 25, 1357–1371 (2021). https://doi.org/10.1007/s11117-021-00820-x
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DOI: https://doi.org/10.1007/s11117-021-00820-x