田中, 敏

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TitleExistence of solutions of two point boundary value problemswith concave and convex nonlinearities (Dynamics ofFunctional Equations and Related Topics)Author(s)田中, 敏Citation数理解析研究所講究録 (2002), 1254: 190-192Issue Date2002-04URL http://hdl.handle.net/2433/41890RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityExistence of solutions of two point boundary value problems withconcave and convex nonlinearities(Satoshi Tanaka)Department of Electrical EngineeringHachinohe National College of Technology( ) .2(E) $u”+\lambda a(x)f(u)=0$ , $0\leq x\leq 1$ ,. , $\lambda>0$ , $a\in C^{1}[0,1],$ $a(x)>0$ for $\mathrm{O}\leq x\leq 1,$ $f\in$$C(-\infty, \infty),$ $f(s)>0$ for $s>0,$ $f$ $(0, \infty)$ Lipschitz , $f(-s)=-f(s)$for $s>0$ , $s_{0}>0$ , (0, so]-L $f(s)$ , , $f(s)/s$. ,(C) $\lim_{sarrow\infty}\frac{f(s)}{s}=\infty$, $\lim_{sarrow+\mathrm{O}}\frac{f(s)}{s}=\infty$.$f$$f(s)=|s|^{q-1}v+|s|^{p-1}v$, $0<q<1<p$.(B) $u(0)=u(1)=0$. $(\mathrm{E})-(\mathrm{B})$ , 1, 2 .1. $(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$ $\lambda_{0}>0$ :(i) $0<\lambda<\lambda_{0}$ $(\mathrm{E})-(\mathrm{B})$ 2 ,(ii) $\lambda=\lambda_{\mathrm{O}}$ $(\mathrm{E})-(\mathrm{B})$ 1 ,(iii) $\lambda>\lambda_{0}$ $(\mathrm{E})-(\mathrm{B})$ .2. $\lambda_{0}$ 1 . $\{\lambda_{k}\}_{k=1}^{\infty}$ :$0<\lambda_{\mathrm{O}}<\lambda_{1}<\cdots<\lambda_{k}<\lambda_{k+1}<\cdots$ , $\lim_{karrow\infty}\lambda_{k}=\infty$,, $k\in\{1,2, \ldots\}$ , (i), (ii) :(i) $\lambda\in(0, \lambda_{k})$ , $(\mathrm{E})-(\mathrm{B})$ $u$ $u’(0)>0$ ) $(0,1)$ $k$, 2 ,(ii) $\lambda=\lambda_{k}$ , $(\mathrm{E})-(\mathrm{B})$ $u$ $u’(0)>0$ $(0, 1)$ $k$, 1 .1254 2002 190-192190$\lim_{karrow\infty}\lambda_{k}=\infty$ , 2 $\lambda>0$ $(\mathrm{E})-(\mathrm{B})$.Ambrosetti-Brezis-Cerami [1]$\{$$\triangle u+\lambda(|u|^{q-1}u+|u|^{p-1}u)=0$ , $x\in\Omega$ ,$u=0$ , $x\in\partial\Omega$ ,$0<q<1<p$, 1 2. $\Omega\subset \mathrm{R}^{N}$ ., Ouyang-Shi[3](P1) $\{$$\Delta u+\lambda f(u)=0$ , $x\in B^{N}$ ,$u=0$ , $x\in\partial B^{N}$ ,, $f(s)=s^{q}+s^{p}(0<p<1<q)$ ,$(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$ $\Lambda>0$(i) $0<\lambda<\Lambda$ (P1) 2 ,(ii) $\lambda=\Lambda$ (P1) ? 1 ,(iii) $\lambda>\Lambda$ (P1) ., $N\geq 3,$ $B^{N}=\{x\in \mathrm{R}^{N} : |x|\leq 1\}$ .$(\mathrm{E})-(\mathrm{B})$ [1] [3] $N=1$ . ,$(\mathrm{E})-(\mathrm{B})$ , [1] [3] ., $(a(x)\equiv 1)$ , $N=1,$ $f(s)=s^{q}+s^{p}(0<p<1<q)$S\’anchez-Ubilla[4] , $(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$ $\Lambda>0$:(i) $0<\lambda<\Lambda$ $(\mathrm{E})-(\mathrm{B})$ 2 ,(ii) $\lambda=\Lambda$ $(\mathrm{E})-(\mathrm{B})$ 1 ,(iii) $\lambda>\Lambda$ $(\mathrm{E})-(\mathrm{B})$ ., S\’anchez-Ubilla[4].(I) $u(0)=0$ , $u’(0)=\mu$. - [2] , $(\mathrm{E})-(\mathrm{I})$$u(x;\lambda, \mu)$ $[0, 1]$ . , $f$$u(x;\lambda, \mu)=-u(x;\lambda, -\mu)$ , $\mu>0$ .$\lambda$ 0 $\infty$ , , $\mu$ 0 $\infty$ $u(t;\lambda, \mu)$, 1, 2. , .$b(x)$$b(x) \equiv\lambda a(x)\frac{f(u(x,\lambda,\mu))}{u(x,\lambda,\mu)}.\cdot$191, (E)$u”(x;\lambda,\mu)+b(x)u(x;\lambda,\mu)=0$. Sturm , $b(x)>0$$u(x;\lambda,\mu)$ , $b(x)>0$ $u(x;\lambda,\mu)$., $b(x)$ , $\lambda$ 0 $\infty$ , $u(x;\lambda,\mu)$., $\muarrow+0$ $\muarrow\infty$ $u(x;\lambda,\mu)$ .$E[u](x)= \frac{[u’(x)]^{2}}{2}+\lambda a(x)F(u(x))$. ,$F(v)= \int_{0}^{v}f(s)ds\geq 0$ , $v\in \mathrm{R}$. , $F(v)=F(|v|)>0$ for $v\in(-\infty, \infty)\backslash \{0\},$ $F(0)=0$, $F(v)$$(0, \infty)$ .$E[u]$$\frac{\mu^{2}}{2}A_{*}\leq E[u(\cdot ; \lambda,\mu)](x)\leq\frac{\mu^{2}}{2}A^{*}$ , $0\leq x\leq 1$.$A_{*}= \exp(-\int_{0}^{1}\frac{[a’(s)]_{-}}{a(s)}ds)$ , $A^{*}= \exp(\int_{0}^{1}\frac{[a’(s)]_{+}}{a(s)}ds)$. , $\muarrow 0$ $|u(x;\lambda, \mu)|arrow 0$ , $\muarrow\infty$$\max\{|u(x;\lambda,\mu)|, |u’(x;\lambda,\mu)|\}arrow 0$ . , (C) $\muarrow 0$$b(x)arrow\infty$ , $\muarrow\infty$ $[0, 1]$ $b(x)arrow \mathrm{o}\mathrm{o}$, $\mu>0$ $u(x;\lambda,\mu)$., 1, 2 .REFERENCES[1] A. Ambrosetti, $\mathrm{H}$ Brezis and G. Cerami, Combined effects of concave and convex nonlineari-ties in some elliptic problems, J. hnct. Anal. 122 (1994), 519-543.[2] M. Naito and Y. Naito, Solutions with prescribed numbers of zeros for nonlinear second orderdifferential equations, Funkcial. Ekvac. 37 (1994), 505-520.[3] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for aclass of semilinear problem,$\mathrm{I}\mathrm{I}$ , J. Differential Equations 158 (1999), 94-151.[4] J. Sichez and P. Ubilla, One dimensional elliptic equation with concave and convex nonlin-earities, Electrvyn. J. Diff. Eqns. 2000 (2000), 1-9.192

Existence of solutions of two point boundary value problems with concave and convex nonlinearities (Dynamics of Functional Equations and Related Topics)

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Existence of solutions of two point boundary value problems with concave and convex nonlinearities (Dynamics of Functional Equations and Related Topics)

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