Existence and multiplicity of the solutions of the p (x )-Kirchhoff type equation via genus theory

Abstract

Theorem 3.7Let J 2 C 1 .X, R/ be functional satisfying the .PS/ condition. Furthermore, let us suppose that (i) J is bounded from below and even; (ii) There is a compact set K 2 R such that .K/ D k and sup x2K J .x/ < J .0/.Then J possesses at least k pairs of distinct critical points, and their corresponding critical values are less than J .0/. Lemma 3.8Suppose .M 1 /, .f 1 /, and q C <ˇp hold. Then I is bounded from below.

On a bi‐nonlocal p(x)‐Kirchhoff equation via Krasnoselskii's genus

“…Because k is arbitrary, we obtain infinitely many critical points of J .□ Remark Following the same steps in this article, we obtain the main result, changing the condition by: A 0 ≤ M ( t ) ≤ B 0 , At α ( x ) ≤ M ( t ) ≤ Bt β ( x ) , A 0 ≤ M ( t ) ≤ Bt β ( x ) , A 0 ≤ M ( t ) ≤ B 0 + Bt β ( x ) , At α ( x ) ≤ M ( t ) ≤ B 0 + Bt β ( x ) , A 0 + At α ( x ) ≤ M ( t ) ≤ Bt β ( x ) , where A 0 , B 0 , A , and B are positive constants. Note that the result obtained in this paper is more general than the result obtained in and , because we introduced the additional nonlocal term, we varied the exponents of functions limits, we did translation of these functions, and we conclude that the result also holds true for bounded functions M . Remark This work partially complements a previous paper by the authors. See Corrêa‐Costa .…”

“…and [2]. However, after concluding this paper, we have learned that Avci-Cekic-Mashyiev [3] have obtained a similar result using Genus Theory. Because of this, we should point out that the novelty in the present paper is the appearance of another nonlocal term This paper is organized as follows: In section 2, we present some preliminaries on the variable exponent spaces.…”

“…Problem was initially motivated by Corrêa‐Figueiredo . However, after concluding this paper, we have learned that Avci‐Cekic‐Mashyiev have obtained a similar result using Genus Theory. Because of this, we should point out that the novelty in the present paper is the appearance of another nonlocal term and the variation of the function M .…”

Solutions of an anisotropic nonlocal problem involving variable exponent

“…To be more precise, R L 0 j @u @x j 2 dx of the kinetic energy 1 2 j @u @x j 2 on OE0; L, and hence the equation is no longer a pointwise identity. For Kirchhoff-type equations involving the p. /-Laplacian operator see, e.g., [3,6,8,9,11].…”

EXISTENCE OF WEAK SOLUTIONS FOR $p(x)$-KIRCHHOFF-TYPE EQUATION

“…In recent years, elliptic problems involving p−Kirchhoff and p (x) −Kirchhoff type operators have been studied in many papers, we refer to [2,5,7,8,9,10,18,23,25].…”