Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data

  • Received: 01 October 2019 Revised: 01 February 2020
  • 35R06, 5J70, 35A01

  • In this paper, we main consider the non-existence of solutions $ u $ by approximation to the following quasilinear elliptic problem with principal part having degenerate coercivity:

    $ \begin{align*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{(p-1)\theta}}\right)+|u|^{q-1}u = \lambda, \; &x\in\Omega, \\ u = 0, \; &x\in\partial\Omega, \end{array} \right. \end{align*} $

    provided

    $ \begin{align*} q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}, \end{align*} $

    where $ \Omega $ is a bounded smooth subset of $ \mathbb{R}^N(N>2) $, $ 1<p<N $, $ q>1 $, $ 0\leq\theta<1 $, $ \lambda $ is a measure which is concentrated on a set with zero $ r $ capacity $ (p<r\leq N) $.

    Citation: Maoji Ri, Shuibo Huang, Canyun Huang. Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data[J]. Electronic Research Archive, 2020, 28(1): 165-182. doi: 10.3934/era.2020011

    Related Papers:

  • In this paper, we main consider the non-existence of solutions $ u $ by approximation to the following quasilinear elliptic problem with principal part having degenerate coercivity:

    $ \begin{align*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{(p-1)\theta}}\right)+|u|^{q-1}u = \lambda, \; &x\in\Omega, \\ u = 0, \; &x\in\partial\Omega, \end{array} \right. \end{align*} $

    provided

    $ \begin{align*} q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}, \end{align*} $

    where $ \Omega $ is a bounded smooth subset of $ \mathbb{R}^N(N>2) $, $ 1<p<N $, $ q>1 $, $ 0\leq\theta<1 $, $ \lambda $ is a measure which is concentrated on a set with zero $ r $ capacity $ (p<r\leq N) $.



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