Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions

Abstract

We study the fourth order nonlinear critical problem Δ 2 u = u 2 ∗ − 1 in the unit ball of R n ( n ≥ 5 ), subject to the Steklov boundary conditions u = Δ u − d u ν = 0 on ∂ B . We provide the exact range of the parameter d for which this problem admits a positive (radial) solution. We also show that the solution is unique in this range and in the class of radially symmetric functions. Finally, we study the behavior of the solution when d tends to the extremals of this range. These results complement previous results in [E. Berchio, F. Gazzola, T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations 12 (2007) 381–406].