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Abstract We study the non-linear minimization problem on H 0 1 (Ω) ⊂ L q with q = 2 n n − 2 , α > 0 and n ≥ 4 : inf u ∈ H 0 1 (Ω) ‖ u ‖ L q = 1 ⁡ ∫ Ω a (x , u) | ∇ u | 2 − λ ∫ Ω | u | 2 where a (x , s) presents a global minimum α at (x 0 , 0) with x 0 ∈ Ω. 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 inf u ∈ H 0 1 (Ω) ‖ u ‖ L q = 1 ⁡ ∫ Ω (α + | x | β | u | k) | ∇ u | 2 − λ ∫ Ω | u | 2. In a previous paper dedicated to the same problem with λ = 0 , we showed that minimizers exist only in the range β < k n / q , which corresponds to a dominant non-linear term. On the contrary, the linear influence for β ≥ k n / q prevented their existence. The goal of this present paper is to show that for 0 < λ ≤ α λ 1 (Ω) , 0 ≤ k ≤ q − 2 and β > k n / q + 2 , minimizers do exist. [ABSTRACT FROM AUTHOR]

We study the minimizing problem where is a smooth bounded domain of , and p a positive discontinuous function. We prove the existence of a minimizer under some assumptions. [ABSTRACT FROM AUTHOR]

Abstract: We study a non-linear minimization problem on with : We show that minimizers exist only for the range which corresponds to a dominant non-linear term. However, for the influence of linearity prevents their existence. [Copyright &y& Elsevier]