Nontrivial solutions of \(m\)-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term. (English) Zbl 1163.34011
Summary: This paper concerns the existence of nontrivial solutions for the following singular m-point boundary value problem with a sign-changing nonlinear term
\[ \begin{cases} (Lu)(t)+h(t)f(t,u)=0,\quad 0<t<1,\\ u(0)=0,\quad u(1)=\sum^{m-2}_{i=1} a_iu(\xi_i),\end{cases} \]
where \( (Lu)(t)=(\widetilde p(t)u'(t))'+q(t)u(t),\quad 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\), \(a_i\in[0,+\infty)\), \(h(t)\) is allowed to be singular at \(t=0,1\), and \(f:[0,1]\times (-\infty,+\infty)\to(-\infty,+\infty)\) is a sign-changing continuous function and may be unbounded from below. By applying the topological degree of a completely continuous field and the first eigenvalue and its corresponding eigenfunction of a special linear operator, some new results on the existence of nontrivial solutions for the above singular \(m\)-point boundary value problem are obtained. An example is then given to demonstrate the application of the main results.
\[ \begin{cases} (Lu)(t)+h(t)f(t,u)=0,\quad 0<t<1,\\ u(0)=0,\quad u(1)=\sum^{m-2}_{i=1} a_iu(\xi_i),\end{cases} \]
where \( (Lu)(t)=(\widetilde p(t)u'(t))'+q(t)u(t),\quad 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\), \(a_i\in[0,+\infty)\), \(h(t)\) is allowed to be singular at \(t=0,1\), and \(f:[0,1]\times (-\infty,+\infty)\to(-\infty,+\infty)\) is a sign-changing continuous function and may be unbounded from below. By applying the topological degree of a completely continuous field and the first eigenvalue and its corresponding eigenfunction of a special linear operator, some new results on the existence of nontrivial solutions for the above singular \(m\)-point boundary value problem are obtained. An example is then given to demonstrate the application of the main results.
MSC:
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
singular second-order differential equations; \(m\)-point BVP; sign-changing; eigenvalue; topological degreeReferences:
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