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Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type. (English) Zbl 1211.35111
Summary: We are concerned with the generalized Lane-Emden-Fowler equation \(-\Delta u=\lambda f(u)+a(x)g(u)\) in \(\Omega\), subject to the Dirichlet boundary condition \(u_{\partial\Omega}=0\), where \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\), \(\lambda\in\mathbb R\), \(a\) is a nonnegative Hölder function, and \(f\) is positive and nondecreasing such that the mapping \(f(s)/s\) is nonincreasing in \((0,\infty)\). Here, the singular character of the problem is given by the nonlinearity \(g\) which is assumed to be unbounded around the origin. We distinguish two different cases which are related to the sublinear (respectively linear) growth of \(f\) at infinity.

MSC:
35J60 Nonlinear elliptic equations
35B50 Maximum principles in context of PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
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