Infinitely many non-radial positive solutions to the double-power nonlinear Schrödinger equations

Abstract

We consider the following problem with combined nonlinearities: $\{\begin{array}{l}-\mathrm{\Delta }u+a(y)u+b(y)u^q-u^p=0,u>0,\text{in}{\mathbb{R}}^N,\\ u\in H^1({\mathbb{R}}^N),\end{array}$where $1<p<2^{\ast }-1,q>1,$ $2^{\ast }=\frac{2N}{N-2}$ when $N\ge 3$; $2^{\ast }=+\mathrm{\infty }$ when N = 2. By use of the Lyapunov-Schmidt reduction argument, under some expansion conditions of the potentials $a(y)$ and $b(y)$, we establish infinitely many non-radial solutions. Especially, the nonlinearity $b(y)u^q$ is allowed to be (super-)critical, that is, $q\ge 2^{\ast }-1$.

Keywords:

• Combined nonlinearities
• infinitely many non-radial solutions
• Lyapunov-Schmidt reduction

2010 Mathematics Subject Classifications:

• 35J15
• 35B08
• 35B09

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Qing Guo was supported by NSFC grants [grant number 11771469].