Solutions of perturbed Schrödinger equations with critical nonlinearity

Abstract

We consider the perturbed Schrödinger equation

$$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} & {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} & \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$$

where $N\geq 3, \ 2^*=2N/(N-2)$ is the Sobolev critical exponent, $p\in (2, 2^*)$ , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that $\varepsilon\leq{\mathcal{E}}$ ; for any $m\in{\mathbb{N}}$ , it has m pairs of solutions if $\varepsilon\leq{\mathcal{E}}_{m}$ ; and suppose there exists an orthogonal involution $\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that $\varepsilon\leq{\mathcal{E}}$ , where ${\mathcal{E}}$ and ${\mathcal{E}}_{m}$ are sufficiently small positive numbers. Moreover, these solutions $u_\varepsilon\to 0$ in $H^1({\mathbb{R}}^N)$ as $\varepsilon\to 0$ .