Concentrating Bound States for Kirchhoff type problems in ${\mathbb{R}}^3$ involving critical Sobolev exponents

Abstract

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{\R^3} {{{\left| {\nabla u} \right|}^2}} \Bigr)Δu + V(z)u = f(u) + {u^5}{\text{in}}{\R^3}, \hfill u \in {H^1}({\R^3}),{\text{}}u > 0{\text{in}}{\R^3}, \hfill \\ \end{gathered} \right.\] where $\epsilon$ is a small positive parameter and $a,b>0$ are constants, $f\in C^1({\mathbb{R}}^{+},\mathbb{R})$ is subcritical, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a locally Hölder continuous function. We first prove that for ${\epsilon }_0>0$ sufficiently small, the above problem has a weak solution $u_{\epsilon }$ with exponential decay at infinity. Moreover, $u_{\epsilon }$ concentrates around a local minimum point of $V$ in $\Lambda$ as $\epsilon \to 0$. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topology construct of the set where the potential $V\left(z\right)$ attains its minimum.

Cited by in corpus (4)

• Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems
• Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in ${\mathbb{R}}^3$
• Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
• Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth