Differential and Integral Equations

Multiplicity results for a degenerate quasilinear elliptic equation in half-space

R. B. Assunção, P. C. Carrião, and O. H. Miyagaki

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In this work, we prove a multiplicity result for a class of quasilinear elliptic equation involving the subcritical Hardy-Sobolev exponent, and singularities both in the operator and in the non-linearity. Precisely, we study the problem \[ \left\{\begin{array}{rcll} -\mbox{div} & = & [|x_N|^{-ap}|\nabla_u|^{p-2}\nabla u]+\lambda|x_N|^{-(a+1-c)p}|u|^{p-2}u \\ & = & |x_N|^{-ap}|u|^{q-2}u+f & \mbox{in }\mathbb{R}^N_+ \\ u & = & 0 & \mbox{on } \partial\mathbb{R}^N_+ \end{array}\right. \]where we denote $ x=(x_1,x_2,\dots,x_N)=(x',x_N) \in \mathbb R^{N-1}\times \mathbb R $, $ \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N > 0 \right\} $, $ \partial \mathbb R_+^N= \left\{ x \in \mathbb R^N : x_N = 0 \right\} $, and we consider $ 1 < p < N $, $ 0 \leqslant a < (N-p)/p $, $ a < b < a+1 $, $c=0 $, $ d \equiv a+1-b $, $ q = q(a,b) \equiv Np/(N - pd) $ (the Hardy-Sobolev critical exponent), $ \lambda \in \mathbb R $ is a parameter, and $ f \in \big( L_b^q(\mathbb R_+^N) \big)^{*} $, the dual space of the weighted Lebesgue space. We prove an existence result for the case $ f \equiv 0 $ and a multiplicity result in the case $ \lambda = 0 $ for non-autonomous perturbations~$ f \not\equiv 0.$

Article information

Differential Integral Equations, Volume 22, Number 7/8 (2009), 753-770.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J62: Quasilinear elliptic equations
Secondary: 35D30: Weak solutions 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09] 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Assunção, R. B.; Carrião, P. C.; Miyagaki, O. H. Multiplicity results for a degenerate quasilinear elliptic equation in half-space. Differential Integral Equations 22 (2009), no. 7/8, 753--770. https://projecteuclid.org/euclid.die/1356019546

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