一类半线性椭圆型方程的爆破解

Explosive solutions of a class of semilinear elliptic equation

讨论半线性椭圆型方程Δu =p(x)f(u) ,其中f(s)是 (0 ,+∞ )中非负连续可微的单调递增函数 ,且lims→ 0 f(s) =0 ,lims→∞f(s)s =k(k <∞ ) ,p(x)是RN(N≥ 3)中局部H lder连续的非负函数 .当p(x)=p(x )时 ,方程存在整体爆破解的充要条件是∫∞0 tp(t)dt=∞ ;而当p(x)满足∫∞0 tφ(t)dt<∞ ,其中 φ(t) =maxx =tp(x)时 ,方程存在整体有界解 .

The semilinear elliptic equationΔu=p(x)f(u)is considered,where f(s)is a nonnegative continuous differ entiable function in(0,∞)which is monotone increasing,and lim s→∞f(s)=0, lim s→∞f(s)s=k(k<∞) ,p(x) is a nonnegative locally Hlder continuous function in R N(N≥3).When p(x) =p(x),the equation has a entire explosive so lution if and only if∫ ∞ 0tp(t)dt=∞.However,when p(x) satisfies ∫ ∞ 0tφ(t)dt<∞,the equation has a entire bo unded solution,where φ(t)= max x=tp(x).