Multiplicity of asymmetric solutions for nonlinear elliptic problems

In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem: \[ \begin {cases} -\Delta u+(\lambda -h(x))u=(1-f(x))u^p, & x\in \mathbb {R}^N,\\ u(x)>0, &\quad x\in \mathbb {R}^N,\\ u\in H^1(\mathbb {R}^N), \end {cases} \] where $\lambda >0$ is a parameter, $h(x)$ and $f(x)$ are nonnegative radially symmetric functions in $L^\infty (\mathbb {R}^N)$, $h(x)$ and $f(x)$ have compact support in $\mathbb {R}^N$, $f(x)\leq 1$ for all $x\in \mathbb {R}^N$, $1<p<+\infty$ for $N=1,2$, $1<p<\frac {N+2}{N-2}$ for $N\geq 3$. We prove that for any $k=1,2, \ldots$, if $\lambda$ is large enough the above problem has positive solutions $u_\lambda$ concentrating at $k$ distinct points away from the origin as $\lambda$ goes to $\infty$.