# American Institute of Mathematical Sciences

• Previous Article
Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space
• CPAA Home
• This Issue
• Next Article
Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems
July  2010, 9(4): 885-904. doi: 10.3934/cpaa.2010.9.885

## Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations

 1 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received  July 2009 Revised  November 2009 Published  April 2010

We study weighted Sobolev embeddings in radially symmetric function spaces and then investigate the existence of nontrivial radial solutions of inhomogeneous quasilinear elliptic equation with singular potentials and super-$(p, q)$-linear nonlinearity. The model equation is of the form

$-\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$

$u(x) \rightarrow 0,$ as $|x|\rightarrow\infty.$

Citation: 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
 [1] Filippo Gazzola. Critical exponents which relate embedding inequalities with quasilinear elliptic problems. Conference Publications, 2003, 2003 (Special) : 327-335. doi: 10.3934/proc.2003.2003.327 [2] Sang-Gyun Youn. On the Sobolev embedding properties for compact matrix quantum groups of Kac type. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3341-3366. doi: 10.3934/cpaa.2020148 [3] Federica Mennuni, Addolorata Salvatore. Existence of minimizers for a quasilinear elliptic system of gradient type. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022013 [4] Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 [5] Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $R^N$. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577 [6] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [7] Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks and Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 [8] Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control and Related Fields, 2021, 11 (3) : 521-554. doi: 10.3934/mcrf.2020052 [9] Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1071-1091. doi: 10.3934/cpaa.2022010 [10] Takahiro Hashimoto. Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients. Conference Publications, 2011, 2011 (Special) : 643-652. doi: 10.3934/proc.2011.2011.643 [11] Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155 [12] 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 [13] Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907 [14] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, 2021, 29 (3) : 2359-2373. doi: 10.3934/era.2020119 [15] Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations and Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023 [16] Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 [17] Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 387-407. doi: 10.3934/dcdss.2021068 [18] Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 [19] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [20] Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357

2020 Impact Factor: 1.916