2015/08/19 by Asadollah Aghajani, Aghajani, Asadollah
Mathematics · #35B65 #35J61 #35K57 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B65 #msc:35J61 #msc:35K57
paper · pdf · doi:10.48550/arxiv.1508.04723
15 pages
arxiv created 2015/08/19 · arxiv updated 2015/08/20
We consider the semilinear elliptic equation -L u = f(u) in a general smooth bounded domain Ω⊂ Rn with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is a C2 positive, nondecreasing and convex function in [0,∞) such that (f(t))/(t)→∞ as t→∞. We prove that if u is a positive semistable solution then for every 0≤β<1 we have f(u)∫0uf(t)f"(t)~e^2β∫0t√((f"(s))/(f(s)))ds~dt∈ L1(Ω), by a constant independent of u. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori L∞ bound in dimensions n≤ 9, under the extra assumption that \limsupt→∞ \fracf(t)f"(t)f'(t)2 < (2)/(9-2√(14))≅ 1.318. Also, we establish a priori L∞ bound when n≤ 5 under the very weak assumption that, for some ε>0, \liminft→∞ \frac(tf(t))2-εf'(t) > 0 or \liminft→∞ \fract2f(t)f"(t)f'(t)(3)/(2)+ε > 0.