Blow-up rates of radially symmetric large solutions

Abstract

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f ∈ C[0, ∞) is positive and non-decreasing, V ∈ C[0, ∞) ∩ C 2 (0, ∞) satisfies V (0) = 0, V (u) > 0, V (u) 0, for every u > 0, and V (u) ∼ H u p−1 as u ↑ ∞, for some H > 0 and p > 1, then, for each λ 0, −Δu = λu − f dist(x, ∂Ω) V (u)u possesses a unique positive large solution in Ω, L, which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of L(x) at ∂Ω in terms of p, H and f (see Theorem 1.1).

Blow-up rates of large solutions for elliptic equations

“…The problem (1.1) arises from many branches of mathematics and has been discussed by many authors, see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24][25][26][27]29,[32][33][34][35][36][37] and the references therein.…”

“…Moreover, López-Gómez [25], and Cano-Casanova and López-Gómez [7] established the following optimal uniqueness result without the first expansion of solutions and the first expansion of solutions for more general weight b.…”

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

“…López-Gómez [27] established the blow-up rate where b(x) is more general without assuming the decay rate of b(x) to be approximated by a power of the distance function near the boundary. Some other related work can be found in [6,8,9,12,28,30], etc. when m > 1, the existence, uniqueness (for the case a(x) ≥ 0) and asymptotic behavior had been studied by Delgado et al [14,15], Peng [32] when f (u) ∼ Ku p/m with p > m, and by Li et al [23] when the variation of f is regular, and [24] for some kinds of functions with non-regular variation at infinity.…”

Uniqueness and blow-up rate of large solutions for elliptic equation−Δu=λu−b(x)h(u)<

“…If the results in [6] and [20] are combined, they would only require the monotonicity of f and the concavity of V (u) and would not require that V (u) ∼ Hu p−1 as u → ∞.…”