Global multiplicity of solutions for a quasilinear elliptic equation with concave and convex nonlinearities

We consider the modified elliptic problem $$ -\Delta u-u\Delta u^2= a(x) u^\alpha+ \lambda b(x)u^{\beta}~ \mbox{in}~ \Omega, $$ with $u(x)=0$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^N$ is a regular domain and $N\geq3$, $0 0$ is a parameter. By using sub- and super-solutions methods and variational methods, we establish the existence of two nontrivial solutions for the modified equation with appropriate exponents $\alpha,\beta$ and potentials $a(x), b(x)$.