Existence of positive solutions for the $p(x)$-Laplacian equation

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

“…Up to now, a large number of results have been obtained for solutions to equations related to this operator. For instance, we must mention the works of Kefi-Radulescu [9], Saoudi [10], Kefi-Saoudi [11], Xie-Chen [12], Nhan-Chuong-Truong [13], Zhang [14], Zhang-Motreanu [15], Yucedag [16], Yin-Li-Ke [17], Ait Hammou-Azroul-Lahmi [18], Ayazoglu-Ekincioglu [19], Ge-Wang [20], Ge-Lv [21], Ge-Radulescu [22], Heidarkhani-Afrouzi-Moradi-Caristi-Ge [23], Ge-Zhou-Wu [24], Fan [25] and the references therein. In relation to the regularity for solutions of differential equations with p(x)-Laplacian operator, we refer the readers to to [26][27][28][29][30][31][32][33][34][35][36], respectively, and the references therein.…”

“…Since the nonlinearity f depends on the gradient ∇u, problem (1) does not have a variational structure, so the variational methods cannot be applied directly. In order to overcome this difficulty, Yin, Li and Ke in [17], by using Krasnoselskii's fixed point theorem on the cone, proved the existence of positive solutions for problem (1) under certain assumptions. Very recently, the authors in [18] used the topological degree theory for a class of demicontinuous operators of generalized (S + ) type to prove the existence of at least one weak solution for problem (1) under the following assumption on f (x, t, ξ):…”

“…To the best of our knowledge, our setting is more general than those of [17,18] and our method contrasts with other treatments of (1). The functional framework contains the generalized Lebesgue and Sobolev spaces with variable exponents and our technique is based on the surjectivity result for pseudomonotone operators.…”

Solutions for the P(x)-Laplacian With Dependence on the Gradient

“…Motivated from the above mentioned papers, especially [12,13], we consider problem (P). Further, as far as we know, there is only one paper which deals with an elliptic equation with variable exponent with dependence on the gradient of the solutions [25], and the present paper is the second.…”

“…Problems with variable exponent growth conditions also appear in the modelling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the p(x)-Laplacian can be found in [3,6,22,25] and references therein.…”