Critical point approaches for a class of second-order boundary value problems with variable exponents

The first important discovery on electrorheological fluids was contributed by Willis Winslow in ۱۹۴۹. The viscosity of these fluids depends upon the electric field of the fluids. He discovered that the viscosity of such fluids as instance lithium polymetachrylate in an electrical field is an inverse relation to the strength of the field. The field causes string-like formations in the fluid, parallel to the field. They can increase the viscosity five orders of magnitude. This event is called the Winslow effect. Electrorheological fluids also have functions in robotics and space technology. In this paper, using variational methods and critical point theory, we establish the existence of two and infinitely many solutions for the following boundary value problem with variable exponent\begin{equation*}\begin{cases}-(\vert u'(x)\vert^{p(x)-۲}u'(x))'+\alpha(x)\vert u(x)\vert^{p(x)-۲}u(x)= f(x,u), \,\ x\in ]۰,۱[,\\\vert u'(۰)\vert^{p(۰)-۲}u'(۰)=- g(u(۰)), \nonumber\\\vert u'(۱)\vert^{p(۱)-۲}u'(۱)=- h(u(۱)).\nonumber\end{cases}\end{equation*}

کلیدواژه ها:

Palais-Smale condition ، p(x)-Laplacian equation ، nonhomogeneous Neumann condition ، Variational methods