# 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 $for which we get existence results. Citation: Marino Badiale, Michela Guida, Sergio Rolando. Radial quasilinear elliptic problems with singular or vanishing potentials. Communications on Pure and Applied Analysis, 2022, 21 (1) : 23-46. doi: 10.3934/cpaa.2021165 ##### References:  [1] T. V. Anoop, P. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975. doi: 10.1007/s00526-014-0773-2. [2] M. Badiale, S. Greco and S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potential, Nonlinear Anal., 185 (2019), 97-122. doi: 10.1016/j.na.2019.01.011. [3] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the$p$-Laplace equation, J. Math. Anal. Appl., 451 (2017), 345-370. doi: 10.1016/j.jmaa.2017.02.011. [4] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence, Nonlinear Differ. Equ. Appl., 23 (2017), 34 pp. doi: 10.1007/s00030-016-0411-0. [5] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: Compactness, Calc. Var. Partial Differ. Equ., 54 (2015), 1061-1090. doi: 10.1007/s00526-015-0817-2. [6] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials, Anal. Appl., 19 (2021), 751-777. doi: 10.1142/S0219530521500020. [7] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y. [8] M. Badiale and F. Zaccagni, Radial nonlinear elliptic problems with singular or vanishing potentials, Adv.Nonlinear Stud., 18 (2018), 409-428. doi: 10.1515/ans-2018-0007. [9] H. Cai, J. Su and Y. Sun, Sobolev type embeddings and an inhomogeneous quasilinear elliptic equation on$\mathbb{R}^N$with singular weights, Nonlinear Anal., 96 (2014), 59-67. doi: 10.1016/j.na.2013.11.002. [10] M. Guida and S. Rolando, Nonlinear Schrödinger equations without compatibility conditions on the potentials, J. Math. Anal. Appl., 439 (2016), 347-363. doi: 10.1016/j.jmaa.2016.02.061. [11] J. Su, Quasilinear elliptic equations on$\mathbb{R}^{N}$with singular potentials and bounded nonlinearity, Z. Angew. Math. Phys., 63 (2012), 51-62. doi: 10.1007/s00033-011-0138-z. [12] J. Su and R. Tian, Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on$\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 140 (2012), 891-903. doi: 10.1090/S0002-9939-2011-11289-9. [13] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. [14] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018. [15] J. Su and Z.-Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differ. Equ., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025. [16] Y. Yang and J. Zhang, A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, Bound. Value Probl., 2012 (2012), 7 pp. doi: 10.1186/1687-2770-2012-136. [17] G. Zhang, Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials, Bound. Value Probl., 2013 (2013), 15 pp. doi: 10.1186/1687-2770-2013-189. show all references ##### References:  [1] T. V. Anoop, P. Drábek and S. Sasi, Weighted quasilinear eigenvalue problems in exterior domains, Calc. Var. Partial Differ. Equ., 53 (2015), 961-975. doi: 10.1007/s00526-014-0773-2. [2] M. Badiale, S. Greco and S. Rolando, Radial solutions of a biharmonic equation with vanishing or singular radial potential, Nonlinear Anal., 185 (2019), 97-122. doi: 10.1016/j.na.2019.01.011. [3] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the$p$-Laplace equation, J. Math. Anal. Appl., 451 (2017), 345-370. doi: 10.1016/j.jmaa.2017.02.011. [4] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence, Nonlinear Differ. Equ. Appl., 23 (2017), 34 pp. doi: 10.1007/s00030-016-0411-0. [5] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: Compactness, Calc. Var. Partial Differ. Equ., 54 (2015), 1061-1090. doi: 10.1007/s00526-015-0817-2. [6] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials, Anal. Appl., 19 (2021), 751-777. doi: 10.1142/S0219530521500020. [7] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA, Nonlinear Differ. Equ. Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y. [8] M. Badiale and F. Zaccagni, Radial nonlinear elliptic problems with singular or vanishing potentials, Adv.Nonlinear Stud., 18 (2018), 409-428. doi: 10.1515/ans-2018-0007. [9] H. Cai, J. Su and Y. Sun, Sobolev type embeddings and an inhomogeneous quasilinear elliptic equation on$\mathbb{R}^N$with singular weights, Nonlinear Anal., 96 (2014), 59-67. doi: 10.1016/j.na.2013.11.002. [10] M. Guida and S. Rolando, Nonlinear Schrödinger equations without compatibility conditions on the potentials, J. Math. Anal. Appl., 439 (2016), 347-363. doi: 10.1016/j.jmaa.2016.02.061. [11] J. Su, Quasilinear elliptic equations on$\mathbb{R}^{N}$with singular potentials and bounded nonlinearity, Z. Angew. Math. Phys., 63 (2012), 51-62. doi: 10.1007/s00033-011-0138-z. [12] J. Su and R. Tian, Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on$\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 140 (2012), 891-903. doi: 10.1090/S0002-9939-2011-11289-9. [13] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. [14] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. doi: 10.1016/j.jde.2007.03.018. [15] J. Su and Z.-Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differ. Equ., 250 (2011), 223-242. doi: 10.1016/j.jde.2010.08.025. [16] Y. Yang and J. Zhang, A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, Bound. Value Probl., 2012 (2012), 7 pp. doi: 10.1186/1687-2770-2012-136. [17] G. Zhang, Weighted Sobolev spaces and ground state solutions for quasilinear elliptic problems with unbounded and decaying potentials, Bound. Value Probl., 2013 (2013), 15 pp. doi: 10.1186/1687-2770-2013-189.  [1] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [2] Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 [3] Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in$L^p$--spaces. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040 [4] Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$coefficients in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011 [5] Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565 [6] T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105 [7] Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2587-2613. doi: 10.3934/cpaa.2022062 [8] Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems and Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115 [9] Lijing Xi, Yuying Zhou, Yisheng Huang. A class of quasilinear elliptic hemivariational inequality problems on unbounded domains. Journal of Industrial and Management Optimization, 2014, 10 (3) : 827-837. doi: 10.3934/jimo.2014.10.827 [10] Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096 [11] Shigeaki Koike, Andrzej Świech. Local maximum principle for$L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 [12] Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted$L^2$spaces. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751 [13] Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581 [14] Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286 [15] Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 [16] Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in$\mathbb{R}^3\$: An approach in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901 [17] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [18] Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2537-2562. doi: 10.3934/dcdsb.2021147 [19] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [20] Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201

2021 Impact Factor: 1.273