A class of supercritical Sobolev type inequalities with logarithm and related elliptic equations

Abstract

In this paper, we first deduce the following Sobolev inequality with logarithmic term:(0.1)sup⁡{∫B|u|2⁎|ln⁡(τ+|u|)||x|βdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<∞, where β>0, τ≥0 are constants, B is the unit ball in RN, N≥3, and 2⁎=2N/(N−2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0<β<min⁡{N/2,N−2} and 1≤τ<∞. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem:(0.2){−Δu=u2⁎−1(ln⁡(τ+u))|x|β+g(|x|,u),u>0inB,u=0on∂B, where g(r,u)∈C([0,1)×R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity:(0.3){−Δu=u2⁎−1(ln⁡(τ+u))|x|β,u>0inB,u=0on∂B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent |x|β is replaced by any non-negative constant.