-
Previous Article
Positive solutions for critical elliptic systems in non-contractible 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 $B$, $u=\kappa \in (0,(\frac{p-1}{q-1})^{-1/(p-q)} ]$ 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 and Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321 |
[2] |
Joseph Iaia. Existence of infinitely many solutions for semilinear problems on exterior domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4269-4284. doi: 10.3934/cpaa.2020193 |
[3] |
Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003 |
[4] |
Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111 |
[5] |
Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239 |
[6] |
Jungsoo Kang. Survival of infinitely many critical points for the Rabinowitz action functional. Journal of Modern Dynamics, 2010, 4 (4) : 733-739. doi: 10.3934/jmd.2010.4.733 |
[7] |
Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure and Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039 |
[8] |
Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461 |
[9] |
Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure and Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623 |
[10] |
Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026 |
[11] |
Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080 |
[12] |
Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603 |
[13] |
Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure and Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013 |
[14] |
Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105 |
[15] |
Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327 |
[16] |
Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624 |
[17] |
Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693 |
[18] |
Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094 |
[19] |
Vilmos Komornik, Anna Chiara Lai, Paola Loreti. Simultaneous observability of infinitely many strings and beams. Networks and Heterogeneous Media, 2020, 15 (4) : 633-652. doi: 10.3934/nhm.2020017 |
[20] |
Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289-292. doi: 10.3934/amc.2017020 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]