The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the \(p\)-Laplacian operator. (English) Zbl 1108.34015
Summary: We study the existence of countable many positive solutions for a class of nonlinear singular boundary value problems with \(p\)-Laplacian operator
\[ \begin{cases} \bigl(\varphi_{p_1}(u')\bigr)'+a_1(t)f(u,v)=0,\quad & 0<t<1,\\ \bigl( \varphi_{p_2}(u')\bigr)'+a_2(t)g(u,v)=0,\quad & 0<t<1, \end{cases} \]
\[ \begin{cases} \alpha_1\varphi_{p_1}\bigl(u(0)\bigr)-\beta_1 \varphi_{p_1}\bigl(u'(0)\bigr)=0, \quad & \gamma_1\varphi_{p_1}\bigl( u(1)\bigr)+\delta_1\varphi_{p_1}\bigl(u'(1) \bigr)=0,\\ \alpha_2 \varphi_{p_2}\bigl(v(0)\bigr)-\beta_2\varpi_{p_2}\bigl(v'(0) \bigr)=0, \quad & \gamma_2\varphi_{p_2}\bigl(v(1)\bigr)+\delta_2\varphi_{p_2} \bigl(v'(1)\bigr)=0,\end{cases} \]
where \(\varphi_{p_1}(s)=|s|^{p_i-2} s\), \(p_i>1\), \(f,g\) are lower semi-continuous functions and \(a_1(t)\) has countable many singularities on \((0,1/2)\), \(i=1,2\). By using the fixed-point theorem of cone expansion and compression of norm type, the existence of countable many positive solutions for the nonlinear singular boundary value problem with \(p\)-Laplacian operator are obtained.
\[ \begin{cases} \bigl(\varphi_{p_1}(u')\bigr)'+a_1(t)f(u,v)=0,\quad & 0<t<1,\\ \bigl( \varphi_{p_2}(u')\bigr)'+a_2(t)g(u,v)=0,\quad & 0<t<1, \end{cases} \]
\[ \begin{cases} \alpha_1\varphi_{p_1}\bigl(u(0)\bigr)-\beta_1 \varphi_{p_1}\bigl(u'(0)\bigr)=0, \quad & \gamma_1\varphi_{p_1}\bigl( u(1)\bigr)+\delta_1\varphi_{p_1}\bigl(u'(1) \bigr)=0,\\ \alpha_2 \varphi_{p_2}\bigl(v(0)\bigr)-\beta_2\varpi_{p_2}\bigl(v'(0) \bigr)=0, \quad & \gamma_2\varphi_{p_2}\bigl(v(1)\bigr)+\delta_2\varphi_{p_2} \bigl(v'(1)\bigr)=0,\end{cases} \]
where \(\varphi_{p_1}(s)=|s|^{p_i-2} s\), \(p_i>1\), \(f,g\) are lower semi-continuous functions and \(a_1(t)\) has countable many singularities on \((0,1/2)\), \(i=1,2\). By using the fixed-point theorem of cone expansion and compression of norm type, the existence of countable many positive solutions for the nonlinear singular boundary value problem with \(p\)-Laplacian operator are obtained.
MSC:
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
Keywords:
fixed-point theoremReferences:
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