Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on H-0(1) (Omega) subset of L-q with q = 2n/n-2, alpha > 0 and n >= 4: [GRAPHICS] integral(Omega) a(x, u)vertical bar del u vertical bar(2) - lambda integral(Omega) vertical bar u vertical bar(2) where a(x, s) presents a global minimum alpha at (x(0), 0) with x(0) is an element of Omega. In order to describe the concentration of u(x) around x(0), one needs to calibrate the behavior of a(x, s) with respect to s. The model case is [GRAPHICS] integral(Omega) (alpha + vertical bar x vertical bar(beta) vertical bar u vertical bar(k))vertical bar del u vertical bar(2) - lambda integral(Omega) vertical bar u vertical bar(2). In a previous paper dedicated to the same problem with lambda = 0, we showed that minimizers exist only in the range beta < kn/q, which corresponds to a dominant non-linear term. On the contrary, the linear influence for beta >= kn/q prevented their existence. The goal of this present paper is to show that for 0 < lambda <= alpha lambda(1) (Omega), 0 <= k <= q - 2 and beta > kn/q+2, minimizers do exist. (C) 2018 Elsevier Inc. All rights reserved.