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Abstract

In this work, we consider a functional $I : W_{0}^{1, p} (Ω) \times W_{0}^{1, p} (Ω) \to R$ of the form $\begin{matrix} I (u, v) = \frac{1}{p} \int_{Ω} (| \nabla u |^{p} + | \nabla v |^{p}) d x - \int_{Ω} H (x, u (x), v (x)) d x \end{matrix}$ where $Ω \subset R^{N}$ is a smooth bounded domain, $max {| \partial_{s} H (x, s, t) |, | \partial_{t} H (x, s, t) |} ⩽ C (1 + | s |^{σ_{1} - 1} + | t |^{σ_{2} - 1})$ a.e. $x \in Ω$ , for some $C > 0$ , $\forall t, s \in R$ , $p < σ_{i} ⩽ p^{*} : = N p / (N - p)$ , $i = 1, 2$ , and $1 < p < N$ . We prove that a local minimum in the topology of $C_{0}^{1} (Ω) \times C_{0}^{1} (Ω)$ is a local minimum in the topology of $W_{0}^{1, p} (Ω) \times W_{0}^{1, p} (Ω)$ . An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.

Keywords

Local minimization p-Laplacian system Sub-super solution method

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W 0 1,p ( Ω ) × W 0 1,p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers

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