The existence of infinitely many weak solutions for a Navier doubly eigenvalue boundary value problem involving the $p(x)$-biharmonic operator is established. In our main result, under an appropriate oscillating behavior of the nonlinearity and suitable assumptions on the variable exponent, a sequence of pairwise distinct solutions is obtained. Furthermore, some applications are pointed out.

This paper studies variable-length coding with noiseless feedback for discrete memoryless channels. Yamamoto and Itoh’s two-phase scheme achieves the optimal error-exponent, but due to the block-coding nature it is not optimal in the expansion of the message size logM . Polyanskiy et al. showed that with feedback, the back-off from capacity is logarithmic in the expected latency l. The O(log l)...

‎By considering a degenerate $p(x)-$Laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent Sobolev space is presented‎. ‎Multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding Nehari manifold‎. 

We study superharmonic functions related to elliptic equations with structural conditions involving a variable growth exponent. We establish pointwise estimates for such functions in terms of a Wolff type potential. We apply these estimates to prove a variable exponent version of the Hedberg–Wolff theorem on the dual of Sobolev spaces with zero boundary values.

We consider the regularity of solutions for some parabolic equations. We show Hölder continuity with exponent (1 + α)/2, with respect to the time variable, when the gradient in the space variable of the solution has the Hölder continuity with exponent α.

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ *-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.