Some sharp inequalities related to Trudinger-Moser inequality

This paper deals with improvements of the Trudinger-Moser inequality related to the operator Q(V)(u) := -Delta(n)u + V(x)vertical bar u vertical bar(n-2)u, where n >= 2 and the potential V : R-n -> R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E := {u is an element of W-1,W-n (R-n) : integral(Rn) V(x)vertical bar u vertical bar(n)dx < infinity} endowed with the norm parallel to u parallel to := [integral(Rn) (vertical bar del u vertical bar(n) + V(x)vertical bar u vertical bar(n))dx](1/n) and we prove that if (u(k)) is a sequence in E such, that parallel to u(k)parallel to = 1, u(k) -> u not equivalent to 0 in E and 0 < p < p(n)(u) := beta(n) (1-parallel to u parallel to(n))(-1/(n-1)), then sup (k)integral(Rn) psi(p vertical bar u(k)vertical bar(n/(n-1))) dx < infinity, (*) where psi(t) := e(t) - Sigma(n-2)(i-0) t(/)(i)i!, beta(n) := nw(n-1)(1/(n-1))) and omega(n-1) is the measure of the unit sphere in R-n. Furthermore, p(n)(u) is sharp in the sense that there exists a sequence (u(k)) subset of E satisfying parallel to u(k)parallel to = 1 and u(k) -> u not equivalent to 0 in E such that the supremum (*) is infinite for p >= p(n) (u). As an application of the previous result we prove the following sharp form of the Trudinger-Moser inequality for the subspace E. Considering l(alpha) := sup({u is an element of E : parallel to u parallel to = 1}) integral(Rn) psi o v(alpha) (u) dx, where v(alpha) (u) := beta(n) (1 + alpha parallel to u parallel to(n)(n))(1/(n-1)vertical bar)u vertical bar(n/(n-1)), assuming some conditions of symmetry on V it is established (1) for 0 <= alpha < lambda(1)(V) we have l(alpha) < infinity, (2) for alpha >= lambda(1)(V), l(alpha) = infinity and (3) moreover, we prove that for 0 <= alpha < lambda(1)(V), an extremal function for l(alpha) exists. Here lambda(1)(V) denotes the first eigenvalue of Q(V)(u). (C) 2018 Elsevier Inc. All rights reserved.