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

TL;DR

This paper proves the existence of bound states for a class of quasilinear elliptic equations with critical growth, including cases where the potential is not bounded below, and analyzes their regularity and concentration behavior.

Contribution

It introduces new hypotheses that include critical frequency cases, allowing the potential to be non-bounded below, and studies solution regularity and concentration as parameters vary.

Findings

Solutions are uniformly bounded.

Solutions concentrate around potential minima.

Results include cases with unbounded below potential.

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 $V$.