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Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. (English) Zbl 1109.35344

Summary: This paper deals with a class of singular semilinear elliptic Dirichlet boundary value problems \[ \begin{cases} \Delta u+\lambda u^{\beta}+p(x)u^{-\gamma}=0 \text{ in } \Omega,\\ u >0\text{ in }\Omega,\\ u =0 \text{ on } \partial\Omega, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb R^{N}\) with smooth boundary \(\partial\Omega\), \(N\geq 3\), \(p(x)\) is a nontrival nonnegative \(L^2(\Omega)\) function, and \(\beta,\gamma\) are constants satisfying \(0<\gamma<1<\beta<2^*-1\) with \(2^*:=2N/(N-2)\) where the combined effects of a superlinear and a singular term allow us to establish some existence and multiplicity results.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

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