Existence and multiplicity of solutions for Kirchhoff type equations involving fractional \(p\)-Laplacian without compact condition. (English) Zbl 1427.35065
Summary: The purpose of this paper is mainly to investigate the following fractional Kirchhoff equation in \(\mathbb{R}^N\) \[ \left(a+b\iint _{\mathbb{R}}^{2N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\text{d}x\text{d}y\right) ^{p-1} (-\Delta )^s_p u+\lambda V(x)|u|^{p-2}u=f(x,u),\] where \(0<s<1\), \(2\le p<\infty\), \( a,b>0\) are constants, \(\lambda\) is a parameter, \(V\) is sign-changing potential function satisfying some assumptions which may not guarantee the compactness of the corresponding Sobolev embedding. Under some suitable conditions, we prove the existence and multiplicity of nontrivial solutions by applying some new tricks for the above equation.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35R11 | Fractional partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
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