Abstract
Let a: [0, +∞) → [0, +∞) be an increasing continuous function with a(t) = 0 if and only if t = 0 and limt→+∞a(t) = +∞, Ω ⊂ ℝN be a bounded domain having the segment property and T[u,u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u: Ω → ℝ. Let p ≥ 1, μi ≠ 0 be real numbers, [Formula: see text], 1 ≤ i ≤ p and [Formula: see text] Put [Formula: see text] and [Formula: see text] Under certain hypotheses on Gi, we show that the minimization problem [Formula: see text] has a solution. Moreover, due to the well-known theorem on generalized multipliers involving the Robinson constraint qualification condition, the solution of the preceeding minimization problem is a weak solution of the corresponding Euler–Lagrange equation (1.1)–(1.2) below. We emphasize that no Δ2-condition on A or [Formula: see text] is imposed. One application to mechanics is given.
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