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We prove a weak comparison principle in narrow unbounded domains for solutions to $$-\Delta _p u=f(u)$$ in the case $$2<p< 3$$ and $$f(\cdot )$$ is a power-type nonlinearity, or in the case $$p>2$$ and $$f(\cdot )$$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $$-\Delta _p u=f(u)$$ in half spaces (with zero Dirichlet assumption)...
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