
Previous Article
Positive solutions for critical elliptic systems in noncontractible domains
 CPAA Home
 This Issue

Next Article
Hyperbolic balance laws with a dissipative non local source
On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity
1.  Department of Mathematics, Henan Normal University, Xinxiang, 453007, China 
2.  Department of Mathematics, Dong Hua University, Shanghai, 200051, China 
$\Delta u=\lambda [\frac{1}{u^p}\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p1}{q1})^{1/(pq)} ]$ on $\partial B$, $0 < u < \kappa$
in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.
[1] 
Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems  A, 2013, 33 (1) : 321333. doi: 10.3934/dcds.2013.33.321 
[2] 
Philip Korman. Infinitely many solutions and Morse index for nonautonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 3146. doi: 10.3934/cpaa.2020003 
[3] 
Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems  A, 2015, 35 (7) : 32393252. doi: 10.3934/dcds.2015.35.3239 
[4] 
Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems  A, 2020, 40 (4) : 21892212. doi: 10.3934/dcds.2020111 
[5] 
Jungsoo Kang. Survival of infinitely many critical points for the Rabinowitz action functional. Journal of Modern Dynamics, 2010, 4 (4) : 733739. doi: 10.3934/jmd.2010.4.733 
[6] 
Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823842. doi: 10.3934/cpaa.2017039 
[7] 
Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems  A, 2014, 34 (2) : 461475. doi: 10.3934/dcds.2014.34.461 
[8] 
Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete & Continuous Dynamical Systems  S, 2019, 12 (2) : 401411. doi: 10.3934/dcdss.2019026 
[9] 
Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 623634. doi: 10.3934/cpaa.2014.13.623 
[10] 
Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical HardySobolevMaz'ya terms. Discrete & Continuous Dynamical Systems  A, 2016, 36 (3) : 16031628. doi: 10.3934/dcds.2016.36.1603 
[11] 
Shaodong Wang. Infinitely many blowingup solutions for Yamabetype problems on manifolds with boundary. Communications on Pure & Applied Analysis, 2018, 17 (1) : 209230. doi: 10.3934/cpaa.2018013 
[12] 
Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a Sshaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems  B, 2017, 22 (5) : 17571778. doi: 10.3934/dcdsb.2017105 
[13] 
Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 327338. doi: 10.3934/cpaa.2011.10.327 
[14] 
Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624633. doi: 10.3934/proc.2005.2005.624 
[15] 
Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete & Continuous Dynamical Systems  A, 2011, 29 (3) : 693736. doi: 10.3934/dcds.2011.29.693 
[16] 
Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94102. doi: 10.3934/proc.2015.0094 
[17] 
Hua Chen, Huiyang Xu. Global existence and blowup of solutions for infinitely degenerate semilinear pseudoparabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems  A, 2019, 39 (2) : 11851203. doi: 10.3934/dcds.2019051 
[18] 
V. Lakshmikantham, S. Leela. Generalized quasilinearization and semilinear degenerate elliptic problems. Discrete & Continuous Dynamical Systems  A, 2001, 7 (4) : 801808. doi: 10.3934/dcds.2001.7.801 
[19] 
Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289292. doi: 10.3934/amc.2017020 
[20] 
José Carmona, Pedro J. MartínezAparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete & Continuous Dynamical Systems  A, 2017, 37 (1) : 1531. doi: 10.3934/dcds.2017002 
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]