Infinitely Many Solutions for Schrodinger-Choquard-Kirchhoff Equations Involving the Fractional p-Laplacian

In this article, we show the existence of infinitely many solutions for the fractional p-Laplacian equations of Schroodinger-Kirchhoff type equation M([u]s,pp)(-Delta)psu+V(x)|u|p-u=lambda(I alpha*|u|ps,alpha*)|u|ps,alpha*-u+beta k(x)|u|q-u,x is an element of RN,$$M\left( {\left[ u \right]_{s,p}<^>p} \right)( - \Delta )_p<^>su + V(x){\left| u \right|<^>{p - 2}}u = \lambda ({I_\alpha }*{\left| u \right|<^>{p_{s,\alpha }<^>*}}){\left| u \right|<^>{p_{s,\alpha }<^>* - 2}}u + \beta k(x){\left| u \right|<^>{q - 2}}u,x \in {\mathbb{R}<^>N},$$\end{document} where (-Delta)ps is the fractional p-Laplacian operator, [u](s,p) is the Gagliardo p-seminorm, 0 < s < 1 < q < p < N/s, alpha is an element of (0,N), M and V are continuous and positive functions, and k(x) is a non-negative function in an appropriate Lebesgue space. Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters lambda and beta.

Automated summary

The article demonstrates the existence of infinitely many solutions for the fractional p-Laplacian equations of Schroodinger-Kirchhoff type by combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma. These solutions tend to zero for suitable positive parameters lambda and beta.