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

Abstract

Given H:ℝ3→ℝ of class C1 and bounded, we consider a sequence (un) of solutions of the H-system Open image in new window in the unit open disc Open image in new window satisfying the boundary condition un=γn on ∂Open image in new window. In the first part of this paper, assuming that (un) is bounded in H1(Open image in new window,ℝ3) we study the behavior of (un) when the boundary data γn shrink to zero. We show that either un→0 strongly in H1(Open image in new window,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.