2025/05/15 by Sho Katayama, Yasuhito Miyamoto, Katayama, Sho +1 · 1 citation
Mathematics · #35B05 #35B33 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary: 35J60 #advanced mathematical theories #secondary 34D05
paper · pdf · doi:10.48550/arxiv.2505.10503
openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
We are concerned with positive radial solutions of the inhomogeneous elliptic equation Δu+K(|x|)up+μf(|x|)=0 on ℝN, where N≥ 3, μ>0 and K and f are nonnegative nontrivial functions. If K(r)∼ rα, α>-2, near r=0, K(r)∼ rβ, β>-2, near r=∞ and certain assumptions on f are imposed, then the problem has a unique positive radial singular solution for a certain range of μ. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if p is between the critical Sobolev exponent pS(α) and Joseph-Lundgren exponent pJL(α). Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for pS(α)-2.