Existence and multiplicity of solutions for fractional \(p\)-Laplacian Schrödinger-Kirchhoff type equations. (English) Zbl 1390.35403
Summary: In this paper, we use variational methods to study existence of solutions for the following fractional \(p\)-Laplacian equations of Schrödinger-Kirchhoff type
\[
\left( a+b \int\limits_{\mathbb R}\int\limits_{2N} \frac{| u(x)-u(y)|^p}{| x-y|^{N+ps}}\mathrm{d}x\, \mathrm{d}y\right)^{p-1} \times (-\Delta)^s_p u+\lambda V(x)| u|^{p-2} u=f(x,u)\quad \text{in }\mathbb R^N,
\]
where \(N>ps\), \(a\), \(b>0\) are constants, \(\lambda\) is a parameter, \((-\Delta)^s_p\) is the fractional \(p\)-Laplace operator with \(0<s<1<p<\infty\), \(f\in C(\mathbb R^N\times\mathbb R,\mathbb R)\) and \(V:\mathbb R^N\to\mathbb R^+\) is a potential function.
MSC:
| 35R11 | Fractional partial differential equations |
| 35A15 | Variational methods applied to PDEs |
| 35J60 | Nonlinear elliptic equations |
| 47G20 | Integro-differential operators |
Keywords:
fractional \(p\)-Laplacian; fractional Schrödinger equation; mountain pass theorem; variational methodsReferences:
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