Potential well method forp(x)-Laplacian equations with variable exponent sources

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.…”

Singular solutions for the fourth‐order parabolic equation with nonstandard growth conditions and absorption

“…For more results on second-order problems with nonstandard growth conditions, we refer the interested readers to the works. [21][22][23][24][25][26] If the positive source in the equation of (1.2) satisfies a super linear growth condition, then the weak solution could possess the singularity at some finite time, while the absorption term contributes to the existence of extinction or global existence of solutions. Inspired by the works, 1,18,20 we would give a complete and optimal classification on the singularity of solutions.…”

“…Li and Liu 20 solved the blow‐up rates and sets for (). For more results on second‐order problems with nonstandard growth conditions, we refer the interested readers to the works 21–26 …”

Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

“…Hence it is necessary and useful to extend this method to apply for PDEs with non-constant exponents. However, to our best knowledge, there are just a few results for PDEs with nonstandard growth conditions, see [13,27] for example. Also notice that as in the comments of the authors in [13], it is not a trivial extension even in case s = 0 because of the lack of homogeneity.…”