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

In this paper, we first deduce the following Sobolev inequality with logarithmic term: sup{integral(B) vertical bar u vertical bar(2)*vertical bar ln (tau + vertical bar u vertical bar)vertical bar(vertical bar x vertical bar beta) dx : u is an element of H-0,rad(1)(B), parallel to del u parallel to(L2(B)) = 1} < infinity, (0.1) B (0.1) where beta > 0, tau >= 0 are constants, B is the unit ball in R-N, 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 < beta < min{N/2, N - 2} and 1 <= tau < infinity. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem: {-Delta u = u(2)*(-1)(ln(tau + u))(vertical bar x vertical bar beta) + g(vertical bar x vertical bar, u), u > 0 in B, (0.2) u = 0 on partial derivative B, where g(r, u) is an element of C([0, 1) x R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity: {- Lambda u= u(2)*(-1)(ln(tau + u))(vertical bar x vertical bar beta), u > 0 in B, (0.3) u=0 on partial derivative B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent vertical bar x vertical bar(beta) is replaced by any non-negative constant. (C) 2022 Elsevier Inc. All rights reserved.