Sum of weighted Lebesgue spaces and nonlinear elliptic equations

Abstract

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space \({(\Omega ,\mathcal{A},\mu)}\) and investigating the main properties of both the Banach space $$L\left( \Omega \right) =\left\{u_{1}+u_{2}:u_{1} \in L^{q_{1}} \left(\Omega \right),u_{2} \in L^{q_{2}} \left( \Omega \right) \right\}, L^{q_{i}} \left( \Omega \right) :=L^{q_{i}} \left( \Omega ,d\mu \right),$$and the Nemytskiĭ operator defined on it. Then we apply our general results to prove existence and multiplicity of solutions to a class of nonlinear p-Laplacian equations of the form $$-\triangle _{p}u+V\left( \left| x\right| \right) \left| u\right| ^{p-2}u=f\left( \left| x\right| ,u\right) \quad {\rm in} \mathbb{R}^{N}$$where V is a nonnegative measurable potential, possibly singular and vanishing at infinity, and f is a Carathéodory function satisfying a double-power growth condition in u.