Abstract
The proof consists of several steps. Let (uk) be a minimizing sequence for (6.7). Thus there is an M > 0 such that
for all k \( \epsilon \mathbb{N}\). In fact, if V1 \(\epsilon \ \mathrm{M_1}({v_0}, {w_0}) \rm \ and \ W_1 \epsilon \ \mathrm{M_1}({w_0}, {v_0})\) such that V1 satisfies (6.5) (i) and W1 satisfies (6.5) (iv), setting \(\hat{U} =\left\{\begin{array}{lll} {v_1,\quad x_1 \leq m_1,} \\ {w_0, \quad m_1 \leq + 1\leq {x_1} \leq m_4 - 1}\\{W_1, \quad m_4\leq {x_1}} \end{array}\right.\) with the usual interpolation in between, J1 \( (\hat{U})\) furnishes an upper bound for J1 (uk) independently of m and l.
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Rabinowitz, P.H., Stredulinsky, E.W. (2011). The Proof of Theorem 6.8. In: Extensions of Moser–Bangert Theory. Progress in Nonlinear Differential Equations and Their Applications, vol 81. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8117-3_7
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DOI: https://doi.org/10.1007/978-0-8176-8117-3_7
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