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An exact multiplicity theorem involving concave-convex nonlinearities and its application to stationary solutions of a singular diffusion problem. (English) Zbl 0992.34014
The authors study the exact multiplicity of positive solutions to the boundary value problem $$u''(x)=f(x,u(x))$$, $$u(-L)=u(L)=0$$, where $$L>0$$ is a parameter, $$f\in C^2(0,\infty)\cap C[0,\infty)$$ satisfies (H1) $$f(u)>0$$ for $$u\geq 0$$, (H2) $$\lim_{u\to\infty} f(u)/u=m_\infty$$ with $$0<m_\infty \leq\infty$$, and (H3) there exists a constant $$C\geq 0$$ such that $$f''(u)<0$$ for $$0<u<C$$ and $$f''(u)>0$$ for $$u>C$$. The authors apply the time map $$T(\alpha)= \int^\alpha_0 (F(\alpha)- F(u))^{-1/2}du$$ to study the problem. Under one additional hypothesis on $$uf'(u)/f(u)$$, they prove that $$\lim_{\alpha\to 0}T (\alpha)=0$$, $$\lim_{\alpha \to\infty} T(\alpha):= L_\infty\geq 0$$. Moreover, if set $$\widetilde L=\max_{\alpha \in(0,\infty)} T(\alpha)$$ and let $$L_\infty>0$$, then the problem has exactly two positive solutions for $$L_\infty <L<\widetilde L$$, exactly one positive solution for $$0<L\leq L_\infty$$ and $$L=\widetilde L$$, and no positive solutions for $$L>\widetilde L$$.
Reviewer: Ruyun Ma (Lanzhou)

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 82D10 Statistical mechanics of plasmas
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