TY - JOUR

T1 - Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

AU - Tanaka, Satoshi

PY - 2009/12/1

Y1 - 2009/12/1

N2 - The following Dirichlet problem (1.1){(Δ u + K (| x |) f (u) = 0, in B,; u = 0, on ∂ B,) is considered, where B = {x ∈ RN : | x | < 1}, N ≥ 2, K ∈ C2 [0, 1] and K (r) > 0 for 0 ≤ r ≤ 1, f ∈ C1 (R), s f (s) > 0 for s ≠ 0. Assume moreover that f satisfies the following sublinear condition: f (s) / s > f′ (s) for s ≠ 0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k - 1 nodes, where k ∈ N. It is also shown that there exists K ∈ C∞ [0, 1] such that (1.1) has three radial solutions having exactly one node in the case N = 3.

AB - The following Dirichlet problem (1.1){(Δ u + K (| x |) f (u) = 0, in B,; u = 0, on ∂ B,) is considered, where B = {x ∈ RN : | x | < 1}, N ≥ 2, K ∈ C2 [0, 1] and K (r) > 0 for 0 ≤ r ≤ 1, f ∈ C1 (R), s f (s) > 0 for s ≠ 0. Assume moreover that f satisfies the following sublinear condition: f (s) / s > f′ (s) for s ≠ 0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k - 1 nodes, where k ∈ N. It is also shown that there exists K ∈ C∞ [0, 1] such that (1.1) has three radial solutions having exactly one node in the case N = 3.

KW - Elliptic equation

KW - Nodal solution

KW - Radial solution

KW - Sublinear

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U2 - 10.1016/j.na.2009.04.009

DO - 10.1016/j.na.2009.04.009

M3 - Article

AN - SCOPUS:68349113795

SN - 0362-546X

VL - 71

SP - 5256

EP - 5267

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 11

ER -