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Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on \(\mathbb {R}^{N}\). (English) Zbl 1238.35034
Summary: We study weighted Sobolev type embeddings of radially symmetric functions from \( W_r^{1,p}(\mathbb {R}^N; V)\) into \( L^q(\mathbb {R}^N; Q)\) for \( q<p\) with singular potentials. We then investigate the existence of nontrivial radial solutions of quasilinear elliptic equations with singular potentials and sub-\(p\)-linear nonlinearity. The model equation is of the form \[ \begin{cases} -\text{div}(|\nabla u|^{p-2}\nabla u)+V(| x|) | x|^{p-2} u = Q(| x|) | x|^{q-2} u, \qquad x\in \mathbb R^N, \\ u(x)\rightarrow0, \quad | x|\rightarrow\infty.\end{cases} \]

MSC:
35J62 Quasilinear elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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