2015/10/26 by Justino Sánchez, Sánchez, Justino, Vicente Vergara +1 · 1 citation
Mathematics · #34C20 #35J62 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35B33 #Secondary: 34C37 #math.AP #msc:34C20 #msc:34C37 #msc:35B33 #msc:35J62
paper · pdf · doi:10.48550/arxiv.1510.07669
20 pages, 1 figure
arxiv created 2015/10/26 · arxiv updated 2015/10/28
We consider the problem (1) \begincases Sk(D2u)= λ(1-u)q in B,
u <0 in B,
u=0 on ∂ B, \endcases where B denotes the unit ball in ℝn, n>2k (k∈ ℕ), λ>0 and q > k. We study the existence of negative bounded radially symmetric solutions of (1). In the critical case, that is when q equals Tso's critical exponent q=((n+2)k)/(n-2k)=:q^*(k), we obtain exactly either one or two solutions depending on the parameters. Further, we express such solutions explicitly in terms of Bliss functions. The supercritical case is analysed following the ideas develop by Joseph and Lundgren in their classical work [27]. In particular, we establish an Emden-Fowler transformation which seems to be new in the context of the k-Hessian operator. We also find a critical exponent, defined by qJL(k)= \begincases k((k+1)n-2(k-1)-2√(2[(k+1)n-2k]))/((k+1)n-2k(k+3)-2√(2[(k+1)n-2k])), n>2k+8,
∞, 2k < n ≤ 2k+8, \endcases which allows us to determinate the multiplicity of the solutions to (1) int the two cases q^*(k)≤ q < qJL(k) and q≥ qJL(k). Moreover, we point out that, for k=1, the exponent qJL(k) coincides with the classical Joseph-Lundgren exponent.