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Existence and bifurcation results for a class of nonlinear boundary value problems in $$(0,\infty{})$$. (English) Zbl 0749.34016
When $$\sigma>0$$ and $$r$$: $$(0,\infty)\to R$$ is measurable, positive a.e. on $$(\delta_ 1,\delta_ 2)\subset(0,\infty)$$ and satisfies some other assumptions, then the problem (1) $$-u''-r(x)| u|^ \sigma u=\lambda u$$ in $$(0,\infty)$$, $$u(0)=0$$, $$\lim_{x\to\infty} u(x)=0$$ has for each $$\lambda<0$$ a nonnegative bounded distributive solution $$u_ \lambda\in W_ 0^{1,2}(0,\infty)\cap C^{0,1/2}([0,\infty))$$, $$u_ \lambda\not\equiv 0$$. If $$r$$ is continuous in $$(0,\infty)$$, then $$u_ \lambda(x)>0$$ in $$(0,\infty)$$ and $$u_ \lambda\in C^ 2(0,\infty)$$ is a classical solution of (1). Further there is a sequence $$\lambda_ n\to 0-$$ such that the corresponding solutions $$u_{\lambda_ n}\to 0$$ as $$n\to\infty$$ in one of the space $$L_ 2$$, $$L_ p$$, $$L_ \infty$$, $$W^{1,2}$$.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
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