The Dirichlet Problem for H-Systems with Small Boundary Data: BlowUp Phenomena and Nonexistence Results

Given H:ℝ³→ℝ of class C¹ and bounded, we consider a sequence (uⁿ) of solutions of the H-system in the unit open disc satisfying the boundary condition uⁿ=γⁿ on ∂. In the first part of this paper, assuming that (uⁿ) is bounded in H¹(,ℝ³) we study the behavior of (uⁿ) when the boundary data γⁿ shrink to zero. We show that either uⁿ→0 strongly in H¹(,ℝ³) or uⁿ blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ². Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large' solution at a mountain pass level when the boundary datum is small.