Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent

Abstract

In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type −Δu+u2∗1∣4πx∣u=μf(x)∣u∣p−2u+g(x)∣u∣4uinR3,-\Delta u+\left({u}^{2}\ast \frac{1}{| 4\pi x| }\right)u=\mu f\left(x){| u| }^{p-2}u+g\left(x){| u| }^{4}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where μ>0\mu \gt 0, 10\mu \gt 0 small; while gg has kk strict local maximum points, we prove that the equation has at least k+1k+1 distinct positive solutions for μ>0\mu \gt 0 small by the Nehari manifold. Moreover, we show that one of the solutions is a ground state solution.

Keywords

• schrödinger-poisson-slater equations
• critical exponent
• multiple solutions
• nehari manifold
• concentration-compactness principle
• 35b33
• 35j20
• 35q55