Positive solutions to a class of Kirchhoff-type problems with Hardy-Sobolev critical exponent in \(\mathbb{R}^N\). (Chinese. English summary) Zbl 07802330

Duan, Yu; Sun, Xin; Liao, Jiafeng

Chin. Ann. Math., Ser. A 44, No. 3, 323-334 (2023).

Summary: In this article, the following class of Kirchhoff-type problems with Hardy-Sobolev critical exponent is considered \[ \begin{cases} - \left[ a+b \left(\int_{\mathbb{R}^N} |\nabla u|^2\mathrm{d}x \right)^m \right] \Delta u= \frac{|u|^{(s)-2} u}{|x|^s} + \lambda h(x), \quad x\in \mathbb{R}^N,\\ u\in D^{1,2} (\mathbb{R}^N), \end{cases} \] where \(N \geqslant 3\), \(a\), \(\lambda \geqslant 0\), \(b\), \(m>0\), \(0 \leqslant s < 2\), \(h\in L \frac{2^*}{2^*-1} (\mathbb{R}^N)\) is nonzero and nonnegative. When \(\lambda = 0\), some existence results are obtained by some analysis techniques. When \(\lambda > 0\), by using the variational method, the existence of positive solutions is obtained. Particularly, when \(0<m < \frac{2-s}{N-2}\), at least two positive solutions are obtained.

MSC:

35B09	Positive solutions to PDEs
35J15	Second-order elliptic equations
35J20	Variational methods for second-order elliptic equations

Keywords:

Hardy-Sobolev critical exponent; Kirchhoff-type problem; positive solution; variational method

References:

[1]

Refernce(1):

[2]

Liu J, Liao J F, Tang C L. Positive solutions for Kirchhoff-type equations with critical exponent in RN [J]. J Math Anal Appl, 2015, 429:1153-1172. [2] 丁凌, 汪继秀, 肖氏武. 全空间上具有临界指数的Kirchhoff类方程无穷多个正解的存在性[J]. 南昌大学学报(自然科学版), 2017, 41(5):414-417. [3] 丁凌, 汪继秀, 张丹丹. 全空间上具有临界指数的Kirchhoff类方程两个正解的存在性[J]. 四川大学学报(自然科学版), 2018, 55(3):457-461. [4] 刘选状, 吴行平. 两类带有临界指数的Kirchhoff型方程的解的存在性和多重性 [D]. 重庆: 西南大学, 2015. [5] 朱同亮, 吴行平. 两类带有临界指数的Kirchhoff型方程的解的存在性和多重性 [D]. 重庆: 西南大学, 2016. [6] Lei C Y, Suo H M, Chu C M, et al. On ground state solutions for a Kirchhoff type equation with critical growth [J]. Comput Math Appl, 2016, 72(3):729-740. [7] Ke X F, Liu J, Liao J F. Positive solutions for a critical p-Laplacian problem with a Kirchhoff term [J]. Comput Math Appl, 2019, 77(9):729-740. [8] Guo Z J, Zhang X G. Schr¨odinger-Kirchhoff equation involving double critical nonlinearities [J]. J Math Anal Appl, 2019, 471(1-2):358-377. [9] Zeng L, Tang C L. Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent [J]. Ann Polon Math, 2016, 117(1):163-180. [10] Ghoussoub N, Yuan C. Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents [J]. Trans Amer Math Soc, 2000, 352(12):5703-5743. [11] Kang D S, Peng S J. Existence of solutions for elliptic problems with critical SobolevHardy exponents [J]. Israel J Mathematics, 2004, 143:281-297. [12] Br′ezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals [J]. Proc Amer Math Soc, 1983, 88:486-490.