## Some existence and nonexistence principles for a class of singular boundary value problems.(English)Zbl 0860.34010

The author presents existence and nonexistence results to the following singular boundary value problem \begin{aligned} (p(t)x'(t))'+ p(t)f(t,x(t)) &=0, \qquad 0<t<1,\\ \alpha x(0)- \beta \lim_{t\to 0}p(t)x'(t) &=0, \tag{1}\\ \gamma x(1)+ \delta \lim_{t\to1} p(t)x'(t) &=0, \end{aligned} where $$f:(0,1)\times (0,\infty)\to (0,\infty)$$ is continuous, $$p\in C^1[0,1]$$, and $$p(t)>0$$ for $$t\in(0,1)$$, and $$\alpha,\beta, \gamma,\delta\geq 0$$, $$\beta\gamma+ \alpha\delta+ \alpha\gamma>0$$. The nonlinear term $$f(t,u)$$ may be singular at $$u=0$$ and $$t=0,1$$.
An example is $$f(t,u)= t^{-m} (1-t)^{-m} u^{-n}$$, where $$m,n>0$$. The author shows that, when $$m<2$$, $$n>0$$, problem (1) has a solution for general boundary conditions. When $$m>2$$, no solutions exist.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

nonexistence; singular boundary value problem
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### References:

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