Existence results for second-order boundary-value problems with variable exponents

Differential equations and variational problems with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. This paper presents several sufficient conditions for the existence of at least one weak solution for the following boundary value problem involving an ordinary differential equation with p(x)-Laplacian operator, and nonhomogeneous Neumann conditions −|u′(x)|p(x)−2u′(x)′+α(x)|u(x)|p(x)−2u(x)=λf(x,u(x))in(0,1),|u′(0)|p(0)−2u′(0)=−λg(u(0)),|u′(1)|p(1)−2u′(1)=λh(u(1))where p∈C([0,1],R), f:[0,1]×R→R is a Carathéodory function, g,h:R→R are nonnegative continuous functions, λ>0, α∈L1([0,1]), with essinf[0,1]α>0. Our technical approach is based on variational methods. Some recent results are extended and improved. Moreover, a concrete example of an application is presented.