Abstract
We consider the existence and multiplicity of solutions for thepx-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.
Highlights
The Kirchhoff equation ρ ∂2u ∂t2 − ( ρ0 h + E 2L L ∫ ∂u ∂x dx)
Motivated by the above works, the purpose of this paper is to study the p(x)-Kirchhoff-type equation
By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the p(x)-Kirchhoff-type equation (6) under no (AR) condition
Summary
Was introduced by Kirchhoff [1] in the study of oscillations of stretched strings and plates, where ρ, ρ0, h, E, and L are constants. We refer the reader to [15,16,17,18,19] for an overview on the variable exponent Sobo-lev space, and to [20,21,22,23,24,25,26,27,28,29] for the study of the p(x)Laplacian-type equations. There has been an increasing interest in studying the Kirchhoff equation involving the p(x)-Laplace operator. Motivated by the above works, the purpose of this paper is to study the p(x)-Kirchhoff-type equation. By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the p(x)-Kirchhoff-type equation (6) under no (AR) condition
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