TY - JOUR

T1 - Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations

AU - Tanaka, Satoshi

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/10/1

Y1 - 2013/10/1

N2 - In this paper, the Morse index and the symmetry-breaking for positive solutions of the following two-point boundary value problem{u″+h(x)f(u)=0,x∈(-1,1),u(-1)=u(1)=0, are studied, where h∈C[-1, 1]∩C1([-1, 1]\{0}), h(x)>0, h(-x)=h(x) on [-1, 1]\{0}, f∈C1[0, ∞), f(s)>0 for s>0, and f(0)=0. The problem for the one-dimensional Hénon equation{u″+|x|lup=0,x∈(-1,1),u(-1)=u(1)=0 is a typical example, where l≥0 and p>1. This problem always has the unique positive even solution. It is well-known that if l=0, then there is no positive non-even solution, and if l>0 is sufficiently large, then there exist positive non-even solutions. The result in this paper shows that if l(p-1)≥4, then the Morse index of the positive least energy solution equals 1 and the Morse index of the positive even solution equals 2, and hence the positive least energy solution is non-even and symmetry-breaking phenomena occur. It is also shown that if l≥0 and p>1 are sufficiently small, then there is no positive non-even solution and the Morse index of the even positive solution equals 1.

AB - In this paper, the Morse index and the symmetry-breaking for positive solutions of the following two-point boundary value problem{u″+h(x)f(u)=0,x∈(-1,1),u(-1)=u(1)=0, are studied, where h∈C[-1, 1]∩C1([-1, 1]\{0}), h(x)>0, h(-x)=h(x) on [-1, 1]\{0}, f∈C1[0, ∞), f(s)>0 for s>0, and f(0)=0. The problem for the one-dimensional Hénon equation{u″+|x|lup=0,x∈(-1,1),u(-1)=u(1)=0 is a typical example, where l≥0 and p>1. This problem always has the unique positive even solution. It is well-known that if l=0, then there is no positive non-even solution, and if l>0 is sufficiently large, then there exist positive non-even solutions. The result in this paper shows that if l(p-1)≥4, then the Morse index of the positive least energy solution equals 1 and the Morse index of the positive even solution equals 2, and hence the positive least energy solution is non-even and symmetry-breaking phenomena occur. It is also shown that if l≥0 and p>1 are sufficiently small, then there is no positive non-even solution and the Morse index of the even positive solution equals 1.

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U2 - 10.1016/j.jde.2013.05.029

DO - 10.1016/j.jde.2013.05.029

M3 - Article

AN - SCOPUS:84880507485

VL - 255

SP - 1709

EP - 1733

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -