×

A uniqueness result for Kirchhoff type problems with singularity. (English) Zbl 1344.35039

The authors consider the following Kirchhoff type problem with singularity \[ \begin{cases} -\left(a+b\int_\Omega |\nabla u|^2\,dx \right)\Delta u = f(x) u^{-\gamma}-\lambda u^p & \text{ in } \Omega,\cr u>0 & \text{ in } \Omega,\cr u=0 & \text{ on } \partial \Omega, \end{cases} \] where \(\Omega\subset \mathbb R^N, N\geq 3\) is a bounded domain, \(0<\gamma<1,\) \(\lambda\geq 0,\) \(0<p\leq 2^*-1,\) \(a,b\geq 0\) and \(f\in L^{\frac{2^*}{2^*+\gamma-1}}(\Omega).\) By using the minimax method and some analysis techniques, they obtain uniqueness of positive solutions.

MSC:

35J60 Nonlinear elliptic equations
35J99 Elliptic equations and elliptic systems
35B09 Positive solutions to PDEs
34K10 Boundary value problems for functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anello, G., A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problems, J. Math. Anal. Appl., 373, 248-251 (2011) · Zbl 1203.35287
[2] Autuori, G.; Fiscella, A.; Pucci, P., Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125, 699-714 (2015) · Zbl 1323.35015
[3] Chen, C. Y.; Kuo, Y. C.; Wu, T. F., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250, 1876-1908 (2011) · Zbl 1214.35077
[4] Ma, T. F.; Muñoz Rivera, J. E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16, 243-248 (2003) · Zbl 1135.35330
[5] Naimen, D., The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257, 1168-1193 (2014) · Zbl 1301.35022
[6] He, X. M.; Zou, W. M., Infnitely many solutions for Kirchhoff-type problems, Nonlinear Anal., 70, 1407-1414 (2009) · Zbl 1157.35382
[7] Li, Y. H.; Li, F. Y.; Shi, J. P., Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253, 2285-2294 (2012) · Zbl 1259.35078
[8] Liu, X.; Sun, Y. J., Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12, 721-733 (2013) · Zbl 1270.35242
[9] Lei, C. Y.; Liao, J. F.; Tang, C. L., Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421, 521-538 (2015) · Zbl 1323.35016
[10] Liao, J. F.; Zhang, P.; Liu, J.; Tang, C. L., Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430, 1124-1148 (2015) · Zbl 1322.35010
[11] Mao, A. M.; Luan, S. X., Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383, 239-243 (2011) · Zbl 1222.35092
[12] Sun, J. J.; Ma, S. W., Nontrivial solutions for Kirchhoff type equations via Morse theory, Commun. Pure Appl. Anal., 13, 483-494 (2014) · Zbl 1279.35047
[13] Sun, J. J.; Tang, C. L., Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33, 2139-2154 (2013) · Zbl 1271.35031
[14] Wei, S., Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259, 1256-1274 (2015) · Zbl 1319.35023
[15] Wu, X., Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in \(R^3\), Nonlinear Anal. RWA, 12, 1278-1287 (2011) · Zbl 1208.35034
[16] Zhang, Z. T.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317, 456-463 (2006) · Zbl 1100.35008
[17] del Pino, M., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburg Sect. A, 122, 341-351 (1992) · Zbl 0791.35046
[18] Rudin, W., Real and complex analysis (1966), McGraw-Hill: McGraw-Hill New York, London etc · Zbl 0142.01701
[19] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 486-490 (1983) · Zbl 0526.46037
[20] Sun, Y. J.; Wu, S. P.; Long, Y. M., Combined effects of singular and superlinear non- linearities in some singular boundary value problems, J. Differential Equations, 176, 511-531 (2001) · Zbl 1109.35344
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.