A Caffarelli-Kohn-Nirenberg-type inequality with variable exponent and applications to PDEs

Given Omega subset of R-N (N >= 2) a bounded smooth domain we establish a Caffarelli-Kohn-Nirenberg type inequality on Omega in the case when a variable exponent p(x), of class C-1, is involved. Our main result is proved under the assumption that there exists a smooth vector function (a) over right arrow : Omega -> R-N, satisfying div (a) over right arrow (x) > 0 and (a) over right arrow (x) . del p(x) = 0 for any x is an element of Omega. Particularly, we supplement a result by Fan et al. [X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), pp. 306-317] regarding the positivity of the first eigenvalue of the p(x)-Laplace operator. Moreover, we provide an application of our result to the study of degenerate PDEs involving variable exponent growth conditions.