# American Institute of Mathematical Sciences

January  2022, 21(1): 23-46. doi: 10.3934/cpaa.2021165

## Radial quasilinear elliptic problems with singular or vanishing potentials

 1 Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy 2 Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Roberto Cozzi 53, 20125 Milano, Italy

* Corresponding author

Received  February 2021 Revised  August 2021 Published  January 2022 Early access  September 2021

Fund Project: The first author is partially supported by the PRIN2012 grant "Aspetti variazionali e perturbativi nei problemi differenziali nonlineari"

In this paper we continue the work that we began in [6]. Given
 $1 , two measurable functions $ V\left(r \right)\geq 0 $and $ K\left(r\right)> 0 $, and a continuous function $ A(r) >0 $( $ r>0 $), we consider the quasilinear elliptic equation $ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $where all the potentials $ A,V,K $may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space $ X $into the sum of Lebesgue spaces $ L_{K}^{q_{1}}+L_{K}^{q_{2}} $. The nonlinearity has a double-power super $ p $-linear behavior, as $ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $with $ q_1,q_2>p $(recovering the power case if $ q_1 = q_2 $). With respect to [6], in the present paper we assume some more hypotheses on $ V $, and we are able to enlarge the set of values $ q_1 , q_2 $for which we get existence results. Citation: Marino Badiale, Michela Guida, Sergio Rolando. Radial quasilinear elliptic problems with singular or vanishing potentials. Communications on Pure & Applied Analysis, 2022, 21 (1) : 23-46. doi: 10.3934/cpaa.2021165 ##### References:  [1] T. V. Anoop, P. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975. doi: 10.1007/s00526-014-0773-2. Google Scholar [2] M. Badiale, S. Greco and S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potential, Nonlinear Anal., 185 (2019), 97-122. doi: 10.1016/j.na.2019.01.011. Google Scholar [3] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the$p$-Laplace equation, J. Math. Anal. Appl., 451 (2017), 345-370. doi: 10.1016/j.jmaa.2017.02.011. Google Scholar [4] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence, Nonlinear Differ. Equ. Appl., 23 (2017), 34 pp. doi: 10.1007/s00030-016-0411-0. Google Scholar [5] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: Compactness, Calc. Var. Partial Differ. Equ., 54 (2015), 1061-1090. doi: 10.1007/s00526-015-0817-2. Google Scholar [6] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials, Anal. Appl., 19 (2021), 751-777. doi: 10.1142/S0219530521500020. Google Scholar [7] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y. Google Scholar [8] M. Badiale and F. Zaccagni, Radial nonlinear elliptic problems with singular or vanishing potentials, Adv.Nonlinear Stud., 18 (2018), 409-428. doi: 10.1515/ans-2018-0007. Google Scholar [9] H. Cai, J. Su and Y. Sun, Sobolev type embeddings and an inhomogeneous quasilinear elliptic equation on$\mathbb{R}^N$with singular weights, Nonlinear Anal., 96 (2014), 59-67. doi: 10.1016/j.na.2013.11.002. Google Scholar [10] M. Guida and S. Rolando, Nonlinear Schrödinger equations without compatibility conditions on the potentials, J. Math. Anal. Appl., 439 (2016), 347-363. doi: 10.1016/j.jmaa.2016.02.061. Google Scholar [11] J. Su, Quasilinear elliptic equations on$\mathbb{R}^{N}$with singular potentials and bounded nonlinearity, Z. Angew. Math. Phys., 63 (2012), 51-62. doi: 10.1007/s00033-011-0138-z. Google Scholar [12] J. Su and R. Tian, Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on$\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 140 (2012), 891-903. doi: 10.1090/S0002-9939-2011-11289-9. Google Scholar [13] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [14] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018. Google Scholar [15] J. Su and Z.-Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differ. Equ., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025. Google Scholar [16] Y. Yang and J. Zhang, A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, Bound. Value Probl., 2012 (2012), 7 pp. doi: 10.1186/1687-2770-2012-136. Google Scholar [17] G. Zhang, Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials, Bound. Value Probl., 2013 (2013), 15 pp. doi: 10.1186/1687-2770-2013-189. Google Scholar show all references ##### References:  [1] T. V. Anoop, P. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975. doi: 10.1007/s00526-014-0773-2. Google Scholar [2] M. Badiale, S. Greco and S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potential, Nonlinear Anal., 185 (2019), 97-122. doi: 10.1016/j.na.2019.01.011. Google Scholar [3] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the$p$-Laplace equation, J. Math. Anal. Appl., 451 (2017), 345-370. doi: 10.1016/j.jmaa.2017.02.011. Google Scholar [4] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence, Nonlinear Differ. Equ. Appl., 23 (2017), 34 pp. doi: 10.1007/s00030-016-0411-0. Google Scholar [5] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: Compactness, Calc. Var. Partial Differ. Equ., 54 (2015), 1061-1090. doi: 10.1007/s00526-015-0817-2. Google Scholar [6] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials, Anal. Appl., 19 (2021), 751-777. doi: 10.1142/S0219530521500020. Google Scholar [7] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y. Google Scholar [8] M. Badiale and F. Zaccagni, Radial nonlinear elliptic problems with singular or vanishing potentials, Adv.Nonlinear Stud., 18 (2018), 409-428. doi: 10.1515/ans-2018-0007. Google Scholar [9] H. Cai, J. Su and Y. Sun, Sobolev type embeddings and an inhomogeneous quasilinear elliptic equation on$\mathbb{R}^N$with singular weights, Nonlinear Anal., 96 (2014), 59-67. doi: 10.1016/j.na.2013.11.002. Google Scholar [10] M. Guida and S. Rolando, Nonlinear Schrödinger equations without compatibility conditions on the potentials, J. Math. Anal. Appl., 439 (2016), 347-363. doi: 10.1016/j.jmaa.2016.02.061. Google Scholar [11] J. Su, Quasilinear elliptic equations on$\mathbb{R}^{N}$with singular potentials and bounded nonlinearity, Z. Angew. Math. Phys., 63 (2012), 51-62. doi: 10.1007/s00033-011-0138-z. Google Scholar [12] J. Su and R. Tian, Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on$\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 140 (2012), 891-903. doi: 10.1090/S0002-9939-2011-11289-9. Google Scholar [13] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [14] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018. Google Scholar [15] J. Su and Z.-Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differ. Equ., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025. Google Scholar [16] Y. Yang and J. Zhang, A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, Bound. Value Probl., 2012 (2012), 7 pp. doi: 10.1186/1687-2770-2012-136. Google Scholar [17] G. Zhang, Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials, Bound. Value Probl., 2013 (2013), 15 pp. doi: 10.1186/1687-2770-2013-189. Google Scholar  [1] Jiabao Su, Rushun Tian. 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