Concentrating solutions for a fractional Kirchhoff equation with critical growth

Abstract

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\epsilon >0$ is a small parameter, $a,b>0$ are constants, $s\in (\frac{3}{4},1)$, $2_s^{\ast }=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta {)}^s$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{\epsilon }$ which concentrates around a local minimum of $V$ as $\epsilon \to 0$.

arXiv admin note: text overlap with arXiv:1810.04561

References in corpus (3)

• A multiplicity result for a fractional Kirchhoff equation in ${\mathbb{R}}^N$ with a general nonlinearity
• Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation
• Concentration phenomena for a class of fractional Kirchhoff equations in ${\mathbb{R}}^N$ with general nonlinearities

Cited by in corpus (2)

• Concentration phenomena for a class of fractional Kirchhoff equations in ${\mathbb{R}}^N$ with general nonlinearities
• Nodal solutions for double phase Kirchhoff problems with vanishing potentials