Uniqueness of entire solutions to quasilinear equations of $ p $-Laplace type

Abstract

<abstract><p>We prove the uniqueness property for a class of entire solutions to the equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 &lt; p &lt; \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>in the sub-natural growth case $ 0 &lt; q &lt; p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.</p></abstract>