Nontrivial solutions for a quasilinear elliptic system with weight functions

Abstract

In this paper, we consider nontrivial solutions of a quasilinear elliptic system with the weight functions $h(x)$ and $f_i(x)$ $(i = 1, 2)$ by applying the Nehari manifold method along with the fibrering maps and the minimization method. We analyze the effect of the parameters and weight functions on the existence and multiplicity of nontrivial solutions for the quasilinear elliptic system. When $(\lambda_1 f_1)^+ = (\lambda_2 f_2)^+ \equiv 0$, we prove that the system has at least one nontrivial nonnegative solution; and when $(\lambda_1 f_1)^+ \not \equiv 0$ or $(\lambda_2 f_2)^+ \not \equiv 0$, we show that the system has at least two nontrivial nonnegative solutions for any $(\lambda_1, \lambda_2)$ belonging to a certain subset of $\mathbb{R}^2$. These results are novel even for the corresponding scalar quasilinear elliptic equations and semilinear elliptic systems, which improve and extend the existing results in the literature.