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

Given a C1, bounded, real valued mapping H on R3, we consider a sequence (u^{n}) of solutions of the H-system $\Delta u=2H(u)u_{x}\wedge u_{y}$ in the unit open disc D satisfying the boundary condition u^{n}=\gamma^{n} on ∂D. In the first part of this paper, assuming that (u^{n}) is bounded in the Sobolev space H1(D,R3), we study the behavior of (u^{n}) when the boundary data \gamma^{n} shrink to zero. We show that either u^{n} tends to zero strongly in H1 or u^{n} blows up at least one H-bubble, 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 the article we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum and 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.