Multiplicity of Solutions of the Weighted p-Laplacian with Isolated Singularity and Diffusion Suppressed by Convection

Abstract

In this paper, the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection $ - {\rm{div}}\left( {{{\left| x \right|}^\alpha }{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + \lambda {1 \over {{{\left| x \right|}^\beta }}}{\left| {\nabla u} \right|^{p - 2}}\nabla u \cdot x = {\left| x \right|^\gamma }g\left( {\left| x \right|} \right)\,\,\,\,{\rm{in}}\,\,B\backslash \left\{ 0 \right\}$ subject to nonlinear Robin boundary value condition ${\left| x \right|^\alpha }{\left| {\nabla u} \right|^{p - 2}}\nabla u \cdot \vec n = A - \rho u\,\,\,{\rm{on}}\,\,\partial B,$ where λ > 0, B ⊂ ℝ ^N (N ≥ 2) is the unit ball centered at the origin, α > 0, p > 1, β ∈ ℝ, γ > −N, g ∈ C([0, 1]) with g(0) > 0, A ∈ ℝ, ρ > 0 and ${\vec n}$ is the unit outward normal. The same problem with diffusion promoted by convection, namely λ ≤ 0, has already been discussed by the last two authors (Song-Yin (2012)), where the existence, nonexistence and classification of singularities for solutions are presented. Completely different from [Song, H. J. and Yin, J. X., Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection, Math. Meth. Appl. Sci., 35, 2012, 1089–1100], in the present case λ > 0, namely the diffusion is suppressed by the convection, non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point. At the same time, the singular solutions may exist only if the diffusion dominates the convection.

Keywords

Weighted p-Laplacian / Multiplicity of solutions / Isolated singularities / Convection

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