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

In this paper we study the existence of bound states for the following class of quasilinear problems, \(\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^p\Delta _pu+V(x)u^{p-1}=f(u)+u^{p^*-1},\ u>0,\ \text {in}\ {\mathbb {R}}^{N},\\&\lim _{|x|\rightarrow \infty }u(x) = 0, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& -{\epsilon }^p{\mathrm{\Delta }}_{p}u+V(x)u^{p-1}=f(u)+u^{p^{\ast }-1},u>0,\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & \underset{|x|\to \infty }{lim}u(x)=0,\hfill \end{array}\right)\end{array}$$ where \(\varepsilon >0\) $\epsilon >0$ is small, \(1<p<N,\) $1<p<N,$ f is a nonlinearity with general subcritical growth in the Sobolev sense, \(p^{*} = pN/(N-p)\) $p^{\ast }=pN/(N-p)$ 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 \(|x|\rightarrow \infty \) $|x|\to \infty$ or \(\varepsilon \rightarrow 0,\) $\epsilon \to 0,$ proving that they are uniformly bounded and concentrate around suitable points of \({\mathbb {R}}^N,\) ${\mathbb{R}}^N,$ that may include local minima of V.