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On some results of Moser and of Bangert. (English) Zbl 1149.35341
Summary: A new proof is given of results in [V. Bangert, Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 95–138 (1989; Zbl 0678.58014)] on the existence of minimal (in the sense of Giaquinta and Giusti) heteroclinic solutions of a nonlinear elliptic PDE \[ \sum_{i=1}^n{\partial\over\partial x_ i} F_{p_i}(x,u,Du)-F_u(x,u,Du)=0.\tag{1} \] Bangert’s work is based on an earlier paper of J. Moser [Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 229–272 (1986; Zbl 0609.49029)]. Unlike Bangert (op. cit.), the proof here is variational in nature, and involves the minimization of a ‘renormalized’ functional. It is meant to be the first step towards finding locally vs. globally minimal solutions of the PDE.
For Part II, see Adv. Nonlinear Stud. 4, No. 4, 377–396 (2004; Zbl 1229.35047).

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47J30 Variational methods involving nonlinear operators
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References:
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