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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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##### References:
 [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030 [2] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063 [3] Thomas Bartsch, Alexander Pankov, and Zhi-Qiang Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), no. 4, 549 – 569. · Zbl 1076.35037 · doi:10.1142/S0219199701000494 · doi.org [4] Thomas Bartsch and Zhi Qiang Wang, Existence and multiplicity results for some superlinear elliptic problems on \?^\?, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725 – 1741. · Zbl 0837.35043 · doi:10.1080/03605309508821149 · doi.org [5] Zhaoli Liu and Zhi-Qiang Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys. 56 (2005), no. 4, 609 – 629. · Zbl 1113.35071 · doi:10.1007/s00033-005-3115-6 · doi.org [6] Pablo De Nápoli and María Cristina Mariani, Mountain pass solutions to equations of \?-Laplacian type, Nonlinear Anal. 54 (2003), no. 7, 1205 – 1219. · Zbl 1274.35114 · doi:10.1016/S0362-546X(03)00105-6 · doi.org [7] Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002 [8] Paul Sintzoff, Symmetry of solutions of a semilinear elliptic equation with unbounded coefficients, Differential Integral Equations 16 (2003), no. 7, 769 – 786. · Zbl 1161.35398 [9] P. Sintzoff and M. Willem, A semilinear elliptic equation on \?^\? with unbounded coefficients, Variational and topological methods in the study of nonlinear phenomena (Pisa, 2000) Progr. Nonlinear Differential Equations Appl., vol. 49, Birkhäuser Boston, Boston, MA, 2002, pp. 105 – 113. · Zbl 1094.35054 [10] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149 – 162. · Zbl 0356.35028 [11] Jiabao Su and Rushun Tian, Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations, Commun. Pure Appl. Anal. 9 (2010), no. 4, 885 – 904. · Zbl 1200.35143 · doi:10.3934/cpaa.2010.9.885 · doi.org [12] Jiabao Su, Zhi-Qiang Wang, and Michel Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), no. 4, 571 – 583. · Zbl 1141.35056 · doi:10.1142/S021919970700254X · doi.org [13] Jiabao Su, Zhi-Qiang Wang, and Michel Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), no. 1, 201 – 219. · Zbl 1220.35026 · doi:10.1016/j.jde.2007.03.018 · doi.org [14] Zhi-Qiang Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 1, 15 – 33. · Zbl 0983.35052 · doi:10.1007/PL00001436 · doi.org [15] Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. · Zbl 0856.49001
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