Communications on Pure and Applied Analysis (CPAA)

Nonexistence for $p$--Laplace equations with singular weights

Pages: 1421 - 1438, Volume 9, Issue 5, September 2010      doi:10.3934/cpaa.2010.9.1421

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Patrizia Pucci - Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I–06123 Perugia, Italy (email)
Raffaella Servadei - Dipartimento di Matematica, Università degli Studi della Calabria, Ponte Pietro Bucci 31B, I–87036 Arcavacata di Rende, Italy (email)

Abstract: Aim of this paper is to give some nonexistence results of nontrivial solutions for the following quasilinear elliptic equations with singular weights in $R^n\setminus \{0\}$

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{a-2}u+\lambda | u|^{q-2}u+h(|x|)f(u) = 0 $ and

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{p^*_\alpha-2}u+\lambda | u|^{q-2}u+h(|x|)f(u)= 0, $

where $1 < p < n$, $\alpha \in [0,p]$, $a \in [p,p^*_\alpha]$, $p_\alpha^*= p(n-\alpha)/(n-p)$, $\lambda, \mu \in R$ and $q \ge 1$, while $h: R^+ \to R^+_0$ and $f: R\to R$ are given continuous functions. The main tool for deriving nonexistence theorems for the equations is a Pohozaev--type identity. We first show that such identity holds true for weak solutions $u$ in $H^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the first equation and for weak solutions $u$ in $D^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the second equation. Then, under a suitable growth condition on $f$, we prove that every weak solution $u$ has the required regularity, so that the Pohozaev--type identity can be applied. From this identity we derive some nonexistence results, improving several theorems already appeared in the literature. In particular, we discuss the case when $h$ and $f$ are pure powers.

Keywords:  Quasilinear singular elliptic equations, singular weights, Hardy–Sobolev embeddings.
Mathematics Subject Classification:  Primary: 35J15; Secondary: 35J70.

Received: September 2009;      Revised: December 2009;      Available Online: May 2010.