Existence of solutions and multiple solutions for a class of weighted \(p(r)\)-Laplacian system (Q1022962)

Existence of solutions is considered for the following weighted \(p(r)\)-Laplacian system \[ -\left( w(r)|u'|^{p(r)-2}u'\right)' + f(r,u,\left(w(r)\right)^{\frac 1{p(r)-1}}u' =0\qquad r\in (T_1,T_2) \] under one of the following boundary conditions of periodic, Neumann, Dirichlet or mixed type: \[ u(T_1)=u(T_2); \qquad \lim_{r\to T_1^+}w(r)|u'|^{p(r)-2}u'(r) = \lim_{r\to T_2^-}w(r)|u'|^{p(r)-2}u'(r); \] \[ \lim_{r\to T_1^+}w(r)|u'|^{p(r)-2}u'(r) = \lim_{r\to T_2^-}w(r)|u'|^{p(r)-2}u'(r) = 0; \] \[ u(T_1)=u(T_2)=0; \] \[ \lim_{r\to T_1^+}w(r)|u'|^{p(r)-2}u'(r) = u(T_2)=0. \] Here, \(p:[T_1,T_2]\to \mathbb R_{>1}\) and \(w:[T_1,T_2]\to \mathbb R_{\geq 0}\) are continuous functions with \(w(r)>0\) for \(r\in (T_1,T_2)\). Existence of solutions is established by the use of the Leray-Schauder's degree, and multiplicity of solutions is discussed via critical point theory.