Existence of solutions for a class of p(x)-biharmonic problems without (A-R) type conditions

Abstract

In this paper, we study the existence and multiplicity of nontrivial solutions for a class of p(x)-biharmonic problems. The interesting point lines in the fact that we do not need the usual Ambrosetti-Rabinowitz type condition for the nonlinear term f. The proofs are essentially based on the mountain pass theorem and its Z 2 symmetric version.

On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

“…Similar problems have been studied before by various authors, see e.g. recent papers of Afrouzi-Chung-Mirzapour [1], Kefi-Rȃdulescu [14], Kong [15,16], and Chung-Ho [6]. In particular, Kefi [13] studied the following problem…”

Existence and multiplicity of solutions for $m(x)-$polyharmonic elliptic Kirchhoff type equations without Ambrosetti-Rabinowitz conditions

“…For this reason, in recent years there were some authors studied the problem (4.1) trying to drop the condition (AR). For instance, we refer the interested readers to [29,50] for the m(x)-Laplacian equation, [5,14,68] for the m(x)-biharmonic equation. However, in literature the only results involving the m(x)-polyharmonic without assuming the (AR) condition can be found in [12] which the authors extended the condition (H 5 ) of [51] to the following assumption:…”

On a m(x)$$ m(x) $$‐polyharmonic Kirchhoff problem without any growth near 0 and Ambrosetti–Rabinowitz conditions

“…For this reason, more recently, some authors studied problem (1.1) trying to drop the (AR) condition. For instance, we refer the interested readers to [18,19] for the m(x)-Laplacian equation, [20][21][22] for the m(x)-biharmonic equation, and [23] for the m(x)-triharmonic equation. However, in literature, the only results involving the m(x)-polyharmonic without assuming the (AR) condition can be found in [24], where the authors extended condition (H 5 ) of [25] to the following assumption:…”

“…For this reason, more recently, some authors studied problem () trying to drop the $$$ (AR) $$$ condition. For instance, we refer the interested readers to [18, 19] for the $$$ m(x) $$$‐Laplacian equation, [20–22] for the $$$ m(x) $$$‐biharmonic equation, and [23] for the $$$ m(x) $$$‐triharmonic equation. However, in literature, the only results involving the $$$ m(x) $$$‐polyharmonic without assuming the $$$ (AR) $$$ condition can be found in [24], where the authors extended condition $$$ \left({H}_5\right) $$$ of [25] to the following assumption: $$$ \left({F}_4\right) $$$:there exist $$$ {c}_0\ge 0,{r}_0\ge 0 $$$, and $$$ k>\max \left\{1,\frac{N}{L{p}_{-}}\right\} $$$ such that …”