The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

作者：Deng, Yinbin; He, Qihan^*; Pan, Yiqing; Zhong, Xuexiu

来源：ADVANCED NONLINEAR STUDIES, 2023, 23(1): 20220049.

DOI：10.1515/ans-2022-0049

• Brezis-Nirenberg problem
• critical exponents
• positive solution
• logarithmic perturbation

We consider the existence and nonexistence of the positive solution for the following Br & eacute;zis- Nirenberg problem with logarithmic perturbation: ?-delta u= |u| (2*-2)u+ lambda u+ mu u u log(2) xE omega, u=0 xE 8 omega, where omega c RN is a bounded open domain, lambda,mu ER,N >_3 and 2 & lowast; := 2 - N is the critical Sobolev exponent for N 2 the embedding H0 omega L omega 1( ) y & lowast;( ). The uncertainty of the sign of s logs2 in (0, +oo) has some interest in itself. 2 We will show the existence of positive ground state solution, which is of mountain pass type provided lambda E R, mu > 0 and N >_ 4. While the case of mu < 0 is thornier. However, for N = 3, 4, lambda E (-oo, lambda 1(omega)), we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for mu < 0 and - (N-2)mu/2 (N-2)mu /2(-(N-2)mu/2) log lambda lambda(1) omega 0 ( ) >_ if N >_ 3. Comparing with the results in the study by Br & eacute;zis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477), some new interesting phenomenon occurs when the parameter mu on logarithmic perturbation is not zero.

• 单位
  广西大学