Journal of the
Korean Mathematical Society JKMS

We study existence of positive solutions of the classical non-linear Schr\"odinger equation $$ \begin{array}{rcl} -\triangle u +V(x)u-f(x,u)-H(x)u^{2^*-1}&=&0,\quad u>0\quad \mbox{ in } \mathbb R^n \\ u & \to & 0 \quad \mbox{ as } |x|\to\infty. \end{array}$$ In fact, we consider the following more general quasi-linear Schr\"odinger equation $$ \begin{array}{rcl} -{\rm div}(|\nabla u|^{m-2}\nabla u) + V(x)u^{m-1}&&\\-f(x,u)-H(x)u^{m^*-1}&=&0, \quad u>0\quad \mbox{ in } \mathbb R^n \\ u & \to & 0 \quad \mbox{ as } |x|\to\infty, \end{array}$$ where $m\in(1,n)$ is a positive number and $$ m^*:=\frac{mn}{n-m}>0,$$ is the corresponding critical Sobolev embedding number in $\mathbb R^n$. Under appropriate conditions on the functions $V(x)$, $f(x,u)$ and $H(x)$, existence and non-existence results of positive solutions have been established.

Keywords: concentration-compactness, critical Sobolev exponent, existence, $m$-Laplacian, minimax methods, mountain-pass lemma, Schr\"odinger equations

MSC numbers: Primary 35J20; Secondary 35J10