Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent - Zheng Tian-Tian; Lei Chun-Yu; Liao Jia-Feng (Advances in Nonlinear Analysis, 2024)
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,-Δ u+\left({u}^{2}* \frac{1}{| 4π x| }\right)u=μ 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>0μ \gt 0, 1<p<21\lt p\lt 2, f∈L66−p(R3)f\in {L}^{\tfrac{6}{6-p}}\left({{\mathbb{R}}}^{3}), and f,g∈C(R3,R+)f,g\in C\left({{\mathbb{R}}}^{3},{{\mathbb{R}}}^{+}). Using Ekeland’s variational principle and a measure representation concentration-compactness of Lions, when gg has one local maximum point, we obtain two positive solutions for μ>0μ \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>0μ \gt 0 small by the Nehari manifold. Moreover, we show that one of the solutions is a ground state solution.