Summary: We use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions of the following scalar field equations with strong nonlinearity \[ - \sum^{N}_{i=1}\partial /\partial x_ i(| \nabla u|^{p- 2}(\partial u/\partial x_ i))+a(x)| u|^{q-2} u=f(x,u),\quad x\in R^ N,\quad N\geq 2, \]
\[ u\in E\equiv \{u\in L^ q(R^ N)| \quad u\quad real,\quad \partial u/\partial x_ i\in L^ p(R^ N),\quad 1\leq i\leq N\}, \] where \(2\leq p\leq q\), \(p=N\).