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Published in:
2018 | OriginalPaper | Chapter
An Overview on Singular Nonlinear Elliptic Boundary Value Problems
Authors : Francesca Faraci, George Smyrlis
Published in: Applications of Nonlinear Analysis
Publisher: Springer International Publishing
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Abstract
We give a survey of old and recent results concerning existence and multiplicity of positive solutions (classical or weak) to nonlinear elliptic equations with singular nonlinear terms of the form where Ω is a bounded domain in \(\mathbb {R}^N\) (N ≥ 2) with sufficiently smooth boundary ∂Ω, Δ
p
u = div(|∇u|p−2∇u) (1 < p < ∞), f : Ω × [0, +∞) → [0, +∞) is a Carathéodory function and γ > 0. In some cases and in order to control more carefully the nonlinearity, we need to multiply the singular term u
−γ or f(⋅, u) by positive parameters. The main difficulty which arises in the study of such problems is the lack of differentiability of the corresponding energy functional which represents an obstacle to the application of classical critical point theory.
$$\displaystyle \left \{ \begin {array}{ll} -\varDelta _p u= f(x,u)+ u^{-\gamma }, & \mbox{ in }\ \varOmega \\ u>0, & \mbox{ in }\ \varOmega \\ u=0, & \mbox{ on }\ \partial \ \varOmega , \end {array} \right . $$