Existence of solutions for weighted \(p(t)\)-Laplacian system multi-point boundary value problems
The authors study the existence of solutions for weighted \(p(t)\)-Laplacian system: \[ -\triangle_{p(t)}u+\delta f(t,u,(w(t))^{\frac{1}{p(t)-1}}u')=0,\quad t\in(0,1), \] with one of the following multi-point boundary value conditions: \[ \begin{aligned} &u(0)=0,\quad u(1)=\sum^{m-2}_{i=1}a_iu(\eta_i)+e; \\ &\lim_{t\to 0^+}w(t)|u'|^{p(t)-2}u'(t)=0,\quad u(1)=\sum^{m-2}_{i=1}a_iu(\eta_i)+e; \end{aligned} \] where \(p\in C([0,1],\mathbb{R})\) and \(p(t)>1\), \(-\triangle_{p(t)}u=-(w(t)|u'|^{p(t)-2}u')'\), \(w\in C([0,1])\) satisfies \(0<w(t)\) for \(t\in (0,1)\) and \((w(t))^{\frac{-1}{p(t)-1}}\in L^1(0,1)\); \(0<\eta_1<\dots<\eta_{m-2}<1\); \(a_i\geq 0\;(i=1,\dots,m-2)\) and \(0<\sum_{i=1}^{m-2}a_i<1\). The authors prove the existence of solutions when the nonlinearity \(f(t,\cdot,\cdot)\) satisfies some growth conditions. The method is based on Leray-Schauder's degree.
scientific article; zbMATH DE number 5577440