Noncoercive Problems in Nonlinear Elasticity with Stored Energies Having Fast or Slow Growth: Orlicz–Sobolev Space Setting V. K. LE 1 Communicated by E. Zuazua Abstract. This paper is concerned with a noncoercive unilateral problem in nonlinear elasticity with stored energy function having fast or slow growth. We prove an existence result for the associated minimization problem by using a recession approach in Orlicz–Sobolev space. Key Words. Nonlinear elasticity, Orlicz–Sobolev space, recession function, property (P). 1. Introduction and Problem Settings This paper describes some existence results for noncoercive unilateral problems in nonlinear elasticity where the stored energy can have very fast or slow (nonpolynomial) growth and the external force may also depend on the displacement. We consider a hyperelastic body B which occupies an open bounded domain WÌ R 3 with smooth boundary. Let f = f (x, u) be the external body force acting on B; u is the deformation vector. For simplicity, we assume that the surface force acting on ¶W is zero; the case where this force is nonzero can be treated similarly. On the other hand, we do not assume that any portion of ¶W is fixed or has a prescribed deformation. This causes the noncoercivity of the problem. We assume the usual orientation-preserving and locally invertibility condition, det(ry (x)) > 0, for a.e. x ˛W, all admissible deformations y. 1 Associate Professor, Department of Mathematics and Statistics, University of Missouri, Rolla, Missouri. JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 119, No. 1, pp. 83–103, October 2003 (g2003) 83 0022-3239=03=1000-0083=0 g 2003 Plenum Publishing Corporation