Finite energy solutions of quasilinear elliptic equations with Orlicz growth
Estevan Luiz da Silva, João Marcos do Ó
TL;DR
This work studies finite energy solutions to quasilinear elliptic equations with Orlicz growth driven by measure data, focusing on sublinear growth in the form $f(u)=g(u^{\gamma})$ with $g(t)=t^{p-1}+t^{q-1}$. The authors develop a Wolff-potential framework, proving a sufficient condition expressed through integrability of generalized Wolff potentials of the data measure $\sigma$ that guarantees the existence of a finite-energy solution in $\mathcal{D}^{1,G}(\mathbb{R}^n)\cap L^F(\mathbb{R}^n,\mathrm{d}\sigma)$, and they establish a minimality property for such a solution. They also derive a necessary condition via a lower bound in terms of the generalized Wolff potential and provide a comprehensive potential-estimate theory (Maly-type) linking solutions to $\mathbf{W}_G$ and $\mathbf{W}_G^R$ bounds. The results extend classical $p$-Laplacian potential theory to the Orlicz-growth setting and apply to a broader class of quasilinear operators, offering a robust approach for existence, minimality, and sharp potential estimates in nonlinear elliptic problems with measure data.
Abstract
We present a sufficient condition, expressed in terms of Wolff potentials, for the existence of a finite energy solution to the measure data $(p,q)$-Laplacian equation with a "sublinear growth" rate. Furthermore, we prove that such a solution is minimal. Additionally, a lower bound of a suitably generalized Wolff-type potential is necessary for the existence of a solution, even if the energy is not finite. Our main tools include integral inequalities closely associated with $(p,q)$-Laplacian equations with measure data and pointwise potential estimates, which are crucial for establishing the existence of solutions to this type of problem. This method also enables us to address other nonlinear elliptic problems involving a general class of quasilinear operators.