Abstract
We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a (p-1)-superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions.
Highlights
Let ⊂ RN be a bounded domain with a C2-boundary ∂ and let 1 < p < +∞
Μ ∈ (0, 1) and f : × R −→ R is a Carathéodory perturbation of the singular term (that is, for all x ∈ R, z −→ f (z, x) is measurable and for almost all z ∈, x −→ f (z, x) is continuous)
We are looking for positive solutions and we prove the existence of at least two positive smooth solutions
Summary
In particular in [29] the authors deal with superlinear singular problems. Our work here complements that of [27], where the authors deal with the resonant case, that is, in [27] the perturbation f (z, ·) is ( p −1)-linear. The present work and [27] cover a broad class of parametric nonlinear singular Dirichlet problems. We mention the parametric work of Aizicovici et al [2] on singular Neumann problems. Nonparametric singular Dirichlet problems were examined by Canino–Degiovanni [4], Gasinski–Papageorgiou [6] and Mohammed [25]. In [4,25] we have existence but not multiplicity while in [6] we have multiplicity results (the methods of proofs in all these papers are different)
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