2015/08/26 by A. Aghajani, Aghajani, Asadollah
Computer Science · Mathematics · #35B65 #35J60 #35K57 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1508.06450
openalex publication_date 2015/08/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We consider the semilinear elliptic equation -\Δ u =\λ f(u) in a\nsmooth bounded domain \Ω of Rn with Dirichielt boundary condition,\nwhere f is a C1 positive and nondeccreasing function in [0,\∞)\nsuch that \(f(t))/(t)\→\∞ as t\→\∞. When\n\Ω is an arbitrary domain and f is not necessarily convex, the\nboundedness of the extremal solution u* is known only for n= 2,\nestablished by X. Cabr 'e citeC1. In this paper, we prove this for higher\ndimensions depending on the nonlinearity f. In particular, we prove that if\n
frac12lt;
beta-:=
liminft
rightarrow
infty\n
fracf'(t)F(t)f(t)2
leq
beta+:=
limsupt
rightarrow
infty\n
fracf'(t)F(t)f(t)2lt;
infty where F(t)=\∫0tf(s)ds, then\nu*\∈ L\∞(\Ω), for n\≤ 6. Also, if\n\β-=\β+>\(1)/(2) or\n\(1)/(2)<\β-\≤\β+<\(7)/(10), then u*\∈\nL\∞(\Ω), for n\≤ 9. Moreover, if \β->\(1)/(2) then\nu*\∈ H10(\Ω) for n\≥ 2.\n