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2010, 9(5): 1421-1438. doi: 10.3934/cpaa.2010.9.1421

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

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I–06123 Perugia

2. 

Dipartimento di Matematica, Università degli Studi della Calabria, Ponte Pietro Bucci 31B, I–87036 Arcavacata di Rende, Italy

Received  September 2009 Revised  December 2009 Published  May 2010

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.

Citation: Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421
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