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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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