摘要
We study the equation wtt + ?SN-1w-μwt-δw + h(t, ω)wp= 0,(t, ω) ∈ R × SN-1, and under some conditions we prove a monotonicity theorem for its positive solutions. Applying this monotonicity theorem,we obtain a Liouville-type theorem for some nonlinear elliptic weighted singular equations. Moreover, we obtain the necessary and sufficient condition for-div(|x|θ▽u) = |x|lup, x ∈ RN\{0} having positive solutions which are bounded near 0, which is also a positive answer to Souplet’s conjecture(see Phan and Souplet(2012)) on the weighted Lane-Emden equation-?u = |x|aup, x ∈ RN.
We study the equation wtt + ?SN-1w-μwt-δw + h（t, ω）wp= 0,（t, ω） ∈ R × SN-1, and under some conditions we prove a monotonicity theorem for its positive solutions. Applying this monotonicity theorem,we obtain a Liouville-type theorem for some nonlinear elliptic weighted singular equations. Moreover, we obtain the necessary and sufficient condition for-div（|x|θ▽u） = |x|lup, x ∈ RN\{0} having positive solutions which are bounded near 0, which is also a positive answer to Souplet’s conjecture（see Phan and Souplet（2012）） on the weighted Lane-Emden equation-?u = |x|aup, x ∈ RN.
基金
supported by National Natural Science Foundation of China(Grant Nos.11771428,11331010,11688101 and 11571339)
