Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

Abstract

Motivated by the idea which has been introduced by Boulaaras and Guefaifia (Math. Meth. Appl. Sci. 41 (2018), 5203–5210 and, by Afrouzi and Shakeri (Afr. Mat. (2015) 26:159–168) combined with some properties of Kirchhoff type operators, we prove the existence of positive solutions for a class of nonlocal p x -Kirchhoff evolutionary systems by using the sub and super solutions concept.

Subsuper solutions method for elliptic systems involving p1,...,pm Laplacian operator

“…In this work, the existence and nonexistence of positive weak solution are proved for the elliptic systems involving Laplacian operator with zero Dirichlet boundary condition in bounded domain by using subsuper solutions method, which has been widely applied in many work (see, for example, previous studies) . The obtained results are natural generalization and extension of previous work .…”

Existence of positive solutions of (p(x),q(x))‐Laplacian parabolic systems with right hand side defined as a multiplication of two separate functions

“…It arises from nonlinear elasticity theory, electrorheological fluids, etc (see ). Many existing results have been obtained on this kind of problems, see for example Alves and Correa [5][6][7][8][9] and Mezouar and Boulaaras. 10 In addition, in Alves and Correa, 5 a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term has been investigated.…”

Further results of existence of positive solutions of elliptic Kirchhoff equation with general nonlinearity of source terms

“…where we used the subsolution-supersolution method (see previous studies [7][8][9][10][11][12][13][14][15][16][17][18] ), Galerkin approach, and a variational method to prove the existence of positive solutions in different cases depending on the parameters p and . We chose the logarithmic nonlinearity of source terms, because it appears in several branches of physics such as inflationary cosmology, nuclear physics, optics, and geophysics (see previous studies 2,4,12,[19][20][21][22][23] ).…”