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Asymptotic behavior of positive solutions of a Dirichlet problem involving combined nonlinearities. (English) Zbl 1206.35135

Summary: We study the behavior of positive solutions of the following Dirichlet problem
\[ \begin{cases} -\Delta_pu= \lambda u^{s-1}+ u^{q-1} &\text{in }\Omega, \\ u_{|\partial \Omega}=0, \end{cases} \]
when \(s\rightarrow p^-\). Here \(p >1\), \(s\in\,]1,p]\) and \(q>p\) with \(q\leq\frac{Np}{N-p}\) if \(N>p\).

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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