Multiple solutions for (p, q)-Laplacian equations in $$\mathbb {R}^N$$ with critical or subcritical exponents

Abstract

In this paper we study the following \(\left( p,q\right) \)-Laplacian equation with critical exponent

$$\begin{aligned} -\Delta _{p}u-\Delta _{q}u=\lambda h(x)|u|^{r-2}u+g(x)|u|^{p^{*} -2}u \quad \text {in }\mathbb {R}^{N} , \end{aligned}$$

where \(1<q\le p<r<p^{*}\). After establishing \((PS)_c\) condition for \(c\in (0,c^*)\) for a certain constant \(c^*\) by employing the concentration compactness principle of Lions, multiple solutions for \(\lambda \gg 1\) are obtained by applying a critical point theorem due to Perera (J Anal Math, 2023. arxiv:2308.07901). A similar problem with subcritical exponents is also considered.