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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.

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
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