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

Given H: R^3 → R of class C^1 and bounded, we consider a sequence (u_n) of solutions of the H-system Δu = 2H(u)u_x ∧ u_y in the unit open disc D satisfying the boundary condition u_n = y_n on ∂D. In the first part of this paper, assuming that (u_n) is bounded in H^1 (D, R^3) we study the behavior of (u_n) when the boundary data γ_n shrink to zero. We show that either u_n → 0 strongly in H^1 (D, R^3) or u_n blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on R^2. 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.