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

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Abstract

Given H:ℝ3→ℝ of class C 1 and bounded, we consider a sequence (u n) of solutions of the H-system in the unit open disc satisfying the boundary condition u n=γ n on ∂. In the first part of this paper, assuming that (u n) is bounded in H 1(,ℝ3) 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(,ℝ3) or u n 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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10 10123, Torino, Italy

    Paolo Caldiroli

  2. Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze, 206 33100, Udine, Italy

    Roberta Musina

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  1. Paolo Caldiroli
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  2. Roberta Musina
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Correspondence to Paolo Caldiroli.

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Communicated by C.A. Stuart

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Caldiroli, P., Musina, R. The Dirichlet Problem for H-Systems with Small Boundary Data: BlowUp Phenomena and Nonexistence Results. Arch. Rational Mech. Anal. 181, 1–42 (2006). https://doi.org/10.1007/s00205-005-0398-x

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