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Existence of bound states for quasilinear elliptic problems involving critical growth and frequency

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Abstract

In this paper we study the existence of bound states for the following class of quasilinear problems,

$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^p\Delta _pu+V(x)u^{p-1}=f(u)+u^{p^*-1},\ u>0,\ \text {in}\ {\mathbb {R}}^{N},\\&\lim _{|x|\rightarrow \infty }u(x) = 0, \end{aligned} \right. \end{aligned}$$

where \(\varepsilon >0\) is small, \(1<p<N,\) f is a nonlinearity with general subcritical growth in the Sobolev sense, \(p^{*} = pN/(N-p)\) and V is a continuous nonnegative potential. By introducing a new set of hypotheses, our analysis includes the critical frequency case which allows the potential V to not be necessarily bounded below away from zero. We also study the regularity and behavior of positive solutions as \(|x|\rightarrow \infty \) or \(\varepsilon \rightarrow 0,\) proving that they are uniformly bounded and concentrate around suitable points of \({\mathbb {R}}^N,\) that may include local minima of V.

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Acknowledgements

The author would like to thank Ailton Rodrigues da Silva (DMAT/UFRN) for several discussions about this subject.

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  1. Departamento de Matemática, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-970, Brazil

    Diego Ferraz

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Ferraz, D. Existence of bound states for quasilinear elliptic problems involving critical growth and frequency. Nonlinear Differ. Equ. Appl. 31, 46 (2024). https://doi.org/10.1007/s00030-024-00932-9

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