LAUSR

where Ω is a bounded domain, p ∈ (1,+∞) and f is a C nonlinearity. This equation is the nonlinear version of the widely studied semilinear elliptic equation −∆u = f (u) in a bounded domain Ω ⊂ R. Stable solutions to semilinear equations have very recently been proved to be bounded, and therefore smooth, in dimension n ≤ 9 (see [2]). This result is optimal, since examples of unbounded stable solutions are well known in dimension n ≥ 10. Moreover, the results in [2] give a complete answer to a long-standing open problem about the regularity of extremal solutions to −∆u = λ f (u). We investigate the boundedness of stable solutions to (1) up to dimension n with n < p + 4p/(p − 1). If n ≥ p + 4p/(p − 1), examples of unbounded stable solutions are known even in the unit ball. In the radial case or under strong assumptions on the nonlinearity, stable solutions to (1) are proved to be bounded for n < p + 4p/(p − 1). In the thesis (see [7]), we prove a new L a priori estimate for stable solutions to (1), under a new condition on n and p, which is optimal in the radial case and more restrictive in the general one. However, it improves the known results in the