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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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##### References:
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