On an elliptic equation ofp-Kirchhoff type via variational methods

Abstract

This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type and where Ω is a bounded smooth domain of ℝN, 1 < p < N, s ≥ p* = (pN)/(N – p) and M and f are continuous functions.

Existence of Positive Solutions for a Class ofpx,qx-Laplacian Elliptic Systems with

“…e study of differential equations and variational problems with nonstandard p(x)-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc (see [44]). Many existence results have been obtained on this kind of problems (see, for example, [44][45][46][47][48][49][50][51][52][53][54][55][56][57]) and in [45] a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term has been investigated.…”

Existence of Solutions for Some <i>p</i>(<i>x</i>)-polyharmonic Elliptic Kirchhoff Equations

“…which becomes the usual p-Laplacian for 1 s = . Kratochvl and Necâs introduced the p-biharmonic operator in [1] [2] [3] to study the physical equations, the p-biharmonic operator for 2 s = and the polyharmonic operator for 2 p = , which reduces to the more appoximate case ( ) ( The existence of positive solutions of non-degenerate Kirchhoff-type problems has been proved in [4] [5] for 1 L = . The novelty of this paper is to treat the degenerate case with allowing Kirchhoff function to take the zero value.…”

Existence and multiplicity of positive solutions for a class of p ( x )-Kirchhoff type equations

“…Moreover, nonlocal boundary value problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density [23][24][25][26]. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [27][28][29]) and p(x)-Laplacian (see [30][31][32][33]). …”