In this talk we will discuss semistable solutions of the boundary value problem Lu+f(u)=0 in Ω, u=0 on ∂Ω, where Lu:=∂_i(a^{ij}∂u_j) is uniformly elliptic. By semistability we mean that the lowest Dirichlet eigenvalue of the linearized operator at u is nonnegative. The basic problem (which has a long history) is to obtain a priori L^∞ bounds on a solution under minimal assumptions on f(t). A basic and standard assumption is that u>0 in Ω and f ∈ C^2 is positive, nondecreasing, and superlinear at infinity, i.e. f(0) > 0, f´≥ 0 and f(t)/t tends to ∞ as t tends to ∞. For radially symmetric solutions, an L^∞ bound for u is known for n ≥ 9. On the other hand there exists unbounded semistable solutions when n≥10 for f(u)=e^u. This problem, like many other semilinear elliptic problems studied in recent years, seems to be related to minimal surface stability but this still remains mysterious.