##
**Multiple positive solutions of singular Dirichlet problems on time scales via variational methods.**
*(English)*
Zbl 1129.34015

In this work, the existence and multiplicity of positive solutions are studied for the Dirichlet boundary value problem
\[
-u^{\triangle \triangle}(t) = f(\sigma(t),u(\sigma(t))), \quad t \in (a,\rho(b)) \cap \mathbb{T},
\]

\[ u(0) = u(b) = 0, \] where \(\mathbb{T}\) is a time scale with \(a = \min \mathbb{T}\) and \(b = \max \mathbb{T}\). The inhomogeneous term \(f(\sigma(t),z)\) (\(\triangle\)-measurable in \(\mathbb{T}\)) possesses a singularity in the dependent variable at \(z=0\) and also is bounded away from zero by a constant proportional to the the size of every interval near zero, where such lower bound is assumed to be valid. It also is dominated by a \(\triangle\)-Lebesgue integrable function on an interval of \(z\) bounded away from zero. In addition, the \(\triangle\)-antiderivative, \(F(\sigma(t),z)\), of \(f\), with respect to the dependent variable, is a \(\triangle\)-measurable function of the independent variable in all of \(\mathbb{T}\) for each nonnegative value of \(z\). There is a crucial condition imposed on the time scale, namely, that there exists a \(\xi \in (0,1)\) such that

\[ \int_{[a,\rho(b)) \cap\mathbb{T}}(b-\sigma(s))^{-\xi} \, \triangle s < +\infty. \]

The constant \(\xi\) plays an additional role by involvement in control of the singular behavior of \(f\) near \(z=0\). The above assumptions allow the authors to introduce a suitable regularization of \(f\). Using variational methods and under additional (growth-type) assumptions on \(f\), several existence results are obtained for at least one solution in the resonant and non-resonant scenarios. Multiplicity results are also obtained. This work will definitely aid those interested in applications of the critical point theory to boundary value problems on time scales.

\[ u(0) = u(b) = 0, \] where \(\mathbb{T}\) is a time scale with \(a = \min \mathbb{T}\) and \(b = \max \mathbb{T}\). The inhomogeneous term \(f(\sigma(t),z)\) (\(\triangle\)-measurable in \(\mathbb{T}\)) possesses a singularity in the dependent variable at \(z=0\) and also is bounded away from zero by a constant proportional to the the size of every interval near zero, where such lower bound is assumed to be valid. It also is dominated by a \(\triangle\)-Lebesgue integrable function on an interval of \(z\) bounded away from zero. In addition, the \(\triangle\)-antiderivative, \(F(\sigma(t),z)\), of \(f\), with respect to the dependent variable, is a \(\triangle\)-measurable function of the independent variable in all of \(\mathbb{T}\) for each nonnegative value of \(z\). There is a crucial condition imposed on the time scale, namely, that there exists a \(\xi \in (0,1)\) such that

\[ \int_{[a,\rho(b)) \cap\mathbb{T}}(b-\sigma(s))^{-\xi} \, \triangle s < +\infty. \]

The constant \(\xi\) plays an additional role by involvement in control of the singular behavior of \(f\) near \(z=0\). The above assumptions allow the authors to introduce a suitable regularization of \(f\). Using variational methods and under additional (growth-type) assumptions on \(f\), several existence results are obtained for at least one solution in the resonant and non-resonant scenarios. Multiplicity results are also obtained. This work will definitely aid those interested in applications of the critical point theory to boundary value problems on time scales.

Reviewer: Nickolai Kosmatov (Little Rock)

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

39A10 | Additive difference equations |

34B16 | Singular nonlinear boundary value problems for ordinary differential equations |

### Keywords:

singular; dynamic boundary value problem; multiple positive solutions; variational methods; critical point theory
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\textit{R. P. Agarwal} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 2, 368--381 (2007; Zbl 1129.34015)

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### References:

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