论文标题
一类分数$ p（x，\ cdot）$的解决方案的多样性 - 没有Ambrosetti-Rabinowitz条件的Kirchhoff类型问题
Multiplicity of solutions for a class of fractional $p(x,\cdot)$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition
论文作者
论文摘要
我们对以下分数$ p（x，\ cdot）$ - kirchhoff类型问题$$ \ left \ left \ oken {array} {ll} {ll} m \，\ left（\ displaystyle \ int_ {ω{ω\ timesω}}} \ displayStyle {\ frac {| u（x）-U（y）|^{p（x，y）}}} {p（x，x，y）\ | x-y |^{n+p（x，x，x，y）s}}}}}}}} \ dx \ dx \ dx \ dx \ d \ right）（dy） \ quad x \ inω，\\\\ u = 0，\ quad x \ in \partialΩ，\ end \ end {array} \ right。 \edlineΩ\ rightarrow（1，\ infty）$，$（δ）^{s} _ { \ to \ mathbb {r} $是连续函数。使用变异方法，尤其是由于Bartolo-Benci-Fortunato（非线性肛门7：9（1983），981-1012）引起的对称山间定理，我们在没有Ambrosetti-Rabinowitz条件的情况下建立了无限的解决方案。我们在几个方向上的主要结果扩展了以前出现在文献中的结果。
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{Ω\times Ω} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-Δ)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in Ω, \\ \\ u= 0, \quad x\in \partialΩ, \end{array}\right.$$ where $Ω\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overlineΩ\times \overlineΩ \rightarrow (1, \infty)$, $(-Δ)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,\infty) \to [0, \infty),$ and $f: Ω\times \mathbb{R} \to \mathbb{R}$ 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.
