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

Abstract

<p style='text-indent:20px;'>In this paper, we main consider the non-existence of solutions <inline-formula><tex-math id="M1">$ u $</tex-math></inline-formula> by approximation to the following quasilinear elliptic problem with principal part having degenerate coercivity:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> $ \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*} $ </tex-math></disp-formula></p><p style='text-indent:20px;'>provided</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> $ \begin{align*} q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}, \end{align*} $ </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">$ \Omega $</tex-math></inline-formula> is a bounded smooth subset of <inline-formula><tex-math id="M3">$ \mathbb{R}^N(N>2) $</tex-math></inline-formula>, <inline-formula><tex-math id="M4">$ 1<p<N $</tex-math></inline-formula>, <inline-formula><tex-math id="M5">$ q>1 $</tex-math></inline-formula>, <inline-formula><tex-math id="M6">$ 0\leq\theta<1 $</tex-math></inline-formula>, <inline-formula><tex-math id="M7">$ \lambda $</tex-math></inline-formula> is a measure which is concentrated on a set with zero <inline-formula><tex-math id="M8">$ r $</tex-math></inline-formula> capacity <inline-formula><tex-math id="M9">$ (p<r\leq N) $</tex-math></inline-formula>.</p>