Positive solution for a quasilinear equation with critical growth in \(\mathbb {R}^N\) (Q2809373)

The authors examine and extend previous existence results for positive solutions to the quasilinear problem NEWLINE\[NEWLINE -\Delta_{p}u + V(x)|u|^{p-2}u = f(u,|\nabla u|^{p-2}\nabla u), ~ x \text{ in } \mathbb{R}^N, \tag{1}NEWLINE\]NEWLINE where \(N \geq 2\) is an integer, \(1 < p \leq N\), \(-\Delta_p u := \mathrm{div}(|\nabla u|^{p-2}\nabla u)\) is the usual \(p\)-Laplace operator, the source term \(f:\mathbb{R}\times \mathbb{R}^N \mapsto \mathbb{R}\) is a continuous function exhibiting critical growth at the origin and infinity, and the potential \(V: \mathbb{R}^N \mapsto (0,\infty)\) is continuous and periodic, with \(f\) and \(V\) satisfying various other natural conditions. In particular, the authors focus on the borderline case \(p = N\) with the \(N\)-Laplace operator. They adopt an iterative approach based on Mountain Pass variational techniques to establish the existence of a positive weak solution \(u\) of equation (1) belonging to \(W^{1,N}(\mathbb{R}^N)\). This extends the previous existence results of \textit{G. M. Figueiredo} [Appl. Math. Comput. 203, No. 1, 14--18 (2008; Zbl 1161.35389)] for equation (1) in the range \(1 < p < N\).