Multiplicity of solutions for a class of fractional $p(x,\cdot)$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition
Abstract
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-\Delta)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in \Omega, \\ \\ u= 0, \quad x\in \partial\Omega, \end{array}\right.$$ where $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overline{\Omega}\times \overline{\Omega} \rightarrow (1, \infty)$, $(-\Delta)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,\infty) \to [0, \infty),$ and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo-Benci-Fortunato (Nonlinear Anal. 7:9 (1983), 981-1012), 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.07493
- arXiv:
- arXiv:2009.07493
- Bibcode:
- 2020arXiv200907493H
- Keywords:
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- Mathematics - Analysis of PDEs;
- Primary: 35R11;
- Secondary: 35J20;
- 35J60
- E-Print:
- Bound. Value Probl. 2020 (2020), art. 150, 16 pp