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Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities. (English) Zbl 1490.35186

Summary: This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of \(p-q\) type and singular nonlinearities \[ \begin{cases}\begin{aligned} - \mathcal{L}_{p,q} u & = \lambda \frac{f(u)}{u^\gamma}, \,u>0 && \quad\text{ in } \Omega, \\ u & = 0 && \quad\text{ on } \partial\Omega, \end{aligned}\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with \(C^2\) boundary, \(N \geq 1\), \(\lambda >0\) is a real parameter, \[ \mathcal{L}_{p,q} u : = \mathrm{div}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u, \] \(1<p<q< \infty\), \(\gamma \in (0,1)\), and \(f\) is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by H. Amann [SIAM Rev. 18, 620–709 (1976; Zbl 0345.47044)], we prove existence of three positive solutions in the positive cone of \(C_\delta(\overline{\Omega})\) and in a certain range of \(\lambda\).

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
47H10 Fixed-point theorems

Citations:

Zbl 0345.47044
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References:

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