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Exact multiplicity of positive solutions for a class of semilinear problems. (English) Zbl 0918.35049

By carefully analyzing the local behaviour of the positive solution curve at turning points, the authors precisely characterized the global bifurcation diagrams of an ODE arising from the radial positive solutions for a class of semilinear elliptic problems, and then established an exact multiplicity result of positive solutions in a ball. The prototype examples of the nonlinear terms are: \[ f(u)= u(u- b)(c-u),\tag{1} \] where \(0< 2b< c\), \[ f(u)= u^p- u^q,\tag{2} \] where \(0< p< q\), and \[ f(u)= u(u- b)/(1+ au^p),\tag{3} \] where \(a> 0\), \(b\geq 0\), and \(1< p\leq 2\). Many of these bifurcation diagrams were formally discussed by P. L. Lions [SIAM Rev. 24, 441-467 (1982; Zbl 0511.35033)], but proved rigorously for special nonlinearities in balls in this paper.

MSC:

35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0511.35033
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References:

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