Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin

Dedicated to Louis Nirenberg for his 85th birthday Abstract. We examine the regularity of the extremal solution of the non-linear eigenvalue problem ∆2u = λf(u) on a general bounded domain Ω in R N, with the Navier boundary condition u = ∆u = 0 on ∂Ω. We establish energy estimates which show that for any non-decreasing convex and superlin-ear nonlinearity f with f(0) = 1, the extremal solution u ∗ is smooth provided N ≤ 5. If in addition lim inf t→+∞ f(t)f ′′(t) (f ′)2(t)> 0, then u ∗ is regular for N ≤ 7, while if γ: = lim sup t→+∞ f(t)f ′′(t) (f ′)2(t) < +∞, then the same holds for N < 8 γ. It follows that u ∗ is smooth if f(t) = et and N ≤ 8, or if f(t) = (1 + t)p and N < 8p p−1 We also show that if f(t) = (1 − t)−p, p> 1 and p 6 = 3, then u ∗ is smooth for N ≤ 8p p+1. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., u ∗ is smooth for N ≤ 12 when f(t) = et [11], and for N ≤ 8 when f(t) = (1 − t)−2 [9] (see also [22])