Multiple solutions to multi-critical Schrödinger equations

Abstract

In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1)−Δu+λV(x)u=μ∣u∣p−2u+∑i=1k(∣x∣−(N−αi)∗∣u∣2i∗)∣u∣2i∗−2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\Delta u+\lambda V\left(x)u=\mu | u{| }^{p-2}u+\mathop{\displaystyle \sum }\limits_{i=1}^{k}\left(| x{| }^{-\left(N-{\alpha }_{i})}\ast | u{| }^{{2}_{i}^{\ast }})| u{| }^{{2}_{i}^{\ast }-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\hspace{0.33em}\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where λ,μ∈R+,N≥4\lambda ,\mu \in {{\mathbb{R}}}^{+},N\ge 4, and 2i∗=N+αiN−2{2}_{i}^{\ast }=\frac{N+{\alpha }_{i}}{N-2} with N−4<αi<NN-4\lt {\alpha }_{i}\lt N, i=1,2,…,ki=1,2,\ldots ,k are critical exponents and 2<p<2min∗=min{2i∗:i=1,2,…,k}2\lt p\lt {2}_{\min }^{\ast }={\rm{\min }}\left\{{2}_{i}^{\ast }:i=1,2,\ldots ,k\right\}. Suppose that Ω=intV−1(0)⊂RN\Omega ={\rm{int}}\hspace{0.33em}{V}^{-1}\left(0)\subset {{\mathbb{R}}}^{N} is a bounded domain, we show that for λ\lambda large, problem (0.1) possesses at least catΩ(Ω){{\rm{cat}}}_{\Omega }\left(\Omega ) positive solutions.

Keywords

• multi-critical schrödinger equation
• multiple solutions
• lusternik-schnirelman theory
• 35j10
• 35j20
• 35j61