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

Abstract: In this paper we study the existence of bound states of the following class of quasilinear problems, \begin{equation*} \left{ \begin{aligned} &-\varepsilon ^p\Delta_pu+V(x)u^{p-1}=f(u)+u^{p^\ast -1},\ u>0,\ \text{in}\ \mathbb{R}^{N}, &\lim _{|x|\rightarrow \infty }u(x) = 0 , \end{aligned} \right. \end{equation*} where $\varepsilon>0$ is small, $1<p<N,$ $f$ is a nonlinearity with general subcritical growth in the Sobolev sense, $p^{\ast } = pN/(N-p)$ 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 $|x|\rightarrow \infty$ 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 $V$.