In this paper, we study existence of nontrivial solutions to the elliptic equation
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\begin{document}$$-\Delta u = f(x, u) \ \ {\rm in} \ \Omega, \quad \ u=0 \ \ {\rm on} \ \ \partial \Omega$$\end{document}and to the elliptic system
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\begin{document}$$-\Delta u = \nabla_{u}V(x, u) \ \ {\rm in} \ \Omega, \quad \ u=0 \ \ {\rm on} \ \ \partial \Omega,$$\end{document}where Ω is a bounded domain in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb R^{N}}$$\end{document} with smooth boundary ∂Ω, \documentclass[12pt]{minimal}
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\begin{document}$${f\in C^{1}(\bar\Omega\times\mathbb{R}, \mathbb{R})}$$\end{document}, f (x, 0) = 0, \documentclass[12pt]{minimal}
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\begin{document}$${V\in C^{2}(\bar\Omega\times\mathbb{R}^{m}, \mathbb{R})}$$\end{document} with m ≥ 2 and \documentclass[12pt]{minimal}
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\begin{document}$${\nabla_{u} V(x,0)=0}$$\end{document}. Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, \documentclass[12pt]{minimal}
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\begin{document}$${|f(x,u)|\leqslant c|u|}$$\end{document} for \documentclass[12pt]{minimal}
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\begin{document}$${x\in\Omega}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${u\in\mathbb R}$$\end{document}, and \documentclass[12pt]{minimal}
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\begin{document}$${-cI_{m}\,\leqslant\,\nabla_{u}^{2}V(x,u)\,\leqslant\,cI_{m}}$$\end{document} for \documentclass[12pt]{minimal}
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\begin{document}$${x\in\Omega}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${u\in\mathbb R^{m}}$$\end{document}, where Im is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity on the asymptotic behaviors of the nonlinearity f and \documentclass[12pt]{minimal}
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\begin{document}$${\nabla_{u} V}$$\end{document}.
