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Abstract

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: $\begin{matrix} \{\begin{matrix} - (ε^{2} a + ε b \int_{R^{3}} | \nabla u |^{2} d x) Δ u + V (x) u = f (u) + γ u^{5} in R^{3}, \\ u \in H^{1} (R^{3}), u > 0 in R^{3}, \end{matrix} \end{matrix}$ where $ε > 0$ is a small parameter, $a, b > 0$ are constants, $γ \in {0, 1}$ , V is a continuous positive potential with a local minimum, and f is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud. 14 (2014), 483–510; J. Differ. Equ. 252 (2012), 1813–1834; J. Differ. Equ. 253 (2012), 2314–2351).

Keywords

Kirchhoff problems penalization method Ljusternik–Schnirelmann theory critical growth supercritical exponent

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On a class of Kirchhoff problems via local mountain pass

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