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Ground State Solutions for a Class of Periodic Kirchhoff-Type Equation in \({\mathbb {R}}^3\) Involving Critical Sobolev Exponent

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Abstract

This paper is concerned with the following Kirchhoff-type equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2dx\right) \varDelta u+V(x)u=Q(x)u^5+ f(x,u), \quad x \in {\mathbb {R}}^{3}, \end{aligned}$$

where \(a,b>0\) are constants, V(x), Q(x) and f(x, u) are periodic in x, and the nonlinear growth of \(u^5\) reaches the Sobolev critical exponent since \(2^*=6\) in dimension 3. Under some suitable assumptions on V, Q and f, using variational methods, we establish the existence of nontrivial ground state solution for the above equation. Recent results from the literature are improved and extended.

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Acknowledgements

The author would like to thank the handling editor and the anonymous referee for careful reading the manuscript and suggesting many valuable comments.

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  1. Faculty of Mathematics, University of Science and Technology Houari Boumediene, PB 32 El-Alia, Bab Ezzouar, 16111, Algiers, Algeria

    Sofiane Khoutir

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Khoutir, S. Ground State Solutions for a Class of Periodic Kirchhoff-Type Equation in \({\mathbb {R}}^3\) Involving Critical Sobolev Exponent. Differ Equ Dyn Syst 30, 535–548 (2022). https://doi.org/10.1007/s12591-019-00496-6

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