Existence results for zero mass polyharmonic system

In this paper, we study the following p(x)-curl systems: ∇ × (|∇ × u| p(x)−2 ∇ × u) + a(x)|u| p(x)−2 u = λf (x, u) + µg(x, u), ∇ • u = 0, in Ω, |∇ × u| p(x)−2 ∇ × u × n = 0, u • n = 0, on ∂Ω, where Ω ⊂ R 3 is a bounded simply connected domain with a C 1,1-boundary, denoted by ∂Ω, p : Ω → (1, +∞) is a continuous function, a ∈ L ∞ (Ω), f, g : Ω × R 3 → R 3 are Carathéodory functions, and λ, µ are two parameters. Using variational arguments based on Fountain theorem and Dual Fountain theorem, we establish some existence and non-existence results for solutions of this problem. Our main results generalize the results of Xiang et al.

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“…Since 1 We refer the readers to [24,25,26,28] for further information on the Palais-Smale condition with assumption weaker than (f 3 ).…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Existence of solutions for systems arising in electromagnetism

Hamdani

Repovš

2020

Journal of Mathematical Analysis and Applications

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“…Clearly, our condition (H exp 1 ) is weaker also than (F 4 ). For more details we refer the reader to point 5 of Remark 1.1 in the recent work of Harrabi-Hamdani-Selmi [34] for the polyharmonic case.…”

Section: A M(x)−polyharmonic Kirchhoff Equationmentioning

confidence: 99%

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

Hamdani,

Harrabi

2021

Preprint

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In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m(x)polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case m(x) ≡ Const, we introduce a positive quantity λ M similar to the first eigenvalue of the mpolyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020).

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“…where l is a constant. For this reason, in recent years some authors have studied problem (1.1) by trying to omit the condition (AR); see for example [18,22,27]. Not having the (AR) condition brings great difficulties, so it is natural to consider if this kind of fractional problems have corresponding results even if the nonlinearity does not satisfy the (AR) condition.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition

et al. 2020

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We are interested in the existence of solutions for the following fractional $p(x,\cdot )$ p ( x , ⋅ ) -Kirchhoff-type problem: $$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $$ { M ( ∫ Ω × Ω | u ( x ) − u ( y ) | p ( x , y ) p ( x , y ) | x − y | N + p ( x , y ) s d x d y ) ( − Δ ) p ( x , ⋅ ) s u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N , $N\geq 2$ N ≥ 2 is a bounded smooth domain, $s\in (0,1)$ s ∈ ( 0 , 1 ) , $p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$ p : Ω ‾ × Ω ‾ → ( 1 , ∞ ) , $(-\Delta )^{s}_{p(x,\cdot )}$ ( − Δ ) p ( x , ⋅ ) s denotes the $p(x,\cdot )$ p ( x , ⋅ ) -fractional Laplace operator, $M: [0,\infty ) \to [0, \infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) , and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ f : Ω × R → R are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo–Benci–Fortunato (Nonlinear Anal. 7(9):981–1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti–Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

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Existence results for zero mass polyharmonic system

Cited by 10 publications

References 20 publications

Existence of solutions for systems arising in electromagnetism

Existence of solutions for systems arising in electromagnetism

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

Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition

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