Compactness results for the $p$-Laplace equation

Abstract

Given $1<p<N$ and two measurable functions $V(r)\ge 0$ and $K(r)>0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^N):\int_{\mathbb{R}^N}V\left(\left|x\right|\right) \left|u\right|^p dx<\infty \right\} , \quad L_{K}^q =L^q(\mathbb{R}^N,K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_K^{q_1}+L_K^{q_2}$, and thus into $L_K^q$ ($=L_K^q+L_K^q$) as a particular case. Both exponents $q_1,q_2,q$ greater and lower than $p$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately.

arXiv admin note: substantial text overlap with arXiv:1403.3803

References in corpus (3)

• Compactness and existence results for the $p$-Laplace equation
• Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence
• Nonlinear Schrödinger equations without compatibility conditions on the potentials

Cited by in corpus (1)

• Compactness and existence results for the $p$-Laplace equation