Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System

This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, Sμ1,μ2q1,q2−divq1x∇up−2∇u=μ1up−2u+α+1uα−1uvβ+1in Ω−divq2x∇vp−2∇v=μ2vp−2v+β+1uα+1vβ−1vin Ωu>0, v>0in Ωu=v=0on ∂Ω, where Ω⊂ℜNN≥3,2≤p<N, q1, q2∈W1,pΩ∩CΩ¯, α,β≥0, μ1, μ2, ≥0 and α,β>0 satisfy α+β=p∗−2 with p∗=pN/N−p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q1, q2 near their minima and the dimension N. Moreover, we use the well known Pohozaev identity for prove the nonexistence result.

Abstract licence: CC BY

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