### Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta}_{p}u+{\left|u\right|}^{p-2}u={f}_{1\lambda \u2081}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta ){g}_{\mu}{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta}$, x ∈ Ω, ⎨ $-{\Delta}_{p}v+{\left|v\right|}^{p-2}v={f}_{2\lambda \u2082}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta ){g}_{\mu}{\left|u\right|}^{\alpha}{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {\mathbb{R}}^{N}$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and ${f}_{i{\lambda}_{i}}\left(x\right)={\lambda}_{i}{f}_{i+}\left(x\right)+{f}_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where ${f}_{i\pm}\left(x\right)=max\pm {f}_{i}\left(x\right),0$, ${g}_{\mu}\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta}_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.