Multiplicity of positive solutions for a class of concave-convex elliptic equations with critical growth

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,

$\{\begin{array}{ll}-\Delta u=g\left(x\right){\left|u\right|}^{2*-2}u+\lambda f\left(x\right){\left|u\right|}^{q-2}u,\hfill & x\in \Omega \hfill \\ u=0,\hfill & x\in ?\Omega ,\hfill \end{array}$

where Ω ? ${\mathbb{R}}^N$ (N ≥ 3) is an open bounded domain with smooth boundary, 1 < q < 2,λ > 0. $2*=\frac{2N}{N-2}$ is the critical Sobolev exponent, $f\in L^{\frac{2*}{2*-q}}\left(\Omega \right)$ is nonzero and nonnegative, and g ∈ C $\left(\overline{\Omega }\right)$ is a positive function with k local maximum points. By the Nehari method and variational method, k + 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin MR2870946, Nonlinear Anal. 75(2012) 2660-2671].