带有变号对数非线性项的p-Laplacian方程解的多重性
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摘要
利用变分方法、Nehari流形和对数Sobolev不等式,研究一类带有变号对数非线性项的p-Laplacian方程解的多重性问题,将Nehari流形N分为N~+、N~-和N~0 3个部分,证明N~+有界,并且相应的能量泛函在N~+上有一个极小元,证明泛函在N~-上的极小化序列有界并有一个极小元。结果表明,该p-Laplacian方程至少有2个非平凡解。
The multiplicity of solutions for a class of p-Laplacian equation with sign-changing logarithmic nonlinearity was studied by using variational methods,Nehari manifold, and logarithmic Sobolev inequality. The Nehari manifold N was divided into 3 parts of N~+, N~-, and N~0 to prove that N~+ was bounded, the energy functional had a minimizer on N~+, and a minimizing sequence of the energy functional was bounded having a minimizer on N~-. The results show that the p-Laplacian equation has at least 2 nontrivial solutions.
The multiplicity of solutions for a class of p-Laplacian equation with sign-changing logarithmic nonlinearity was studied by using variational methods,Nehari manifold, and logarithmic Sobolev inequality. The Nehari manifold N was divided into 3 parts of N~+, N~-, and N~0 to prove that N~+ was bounded, the energy functional had a minimizer on N~+, and a minimizing sequence of the energy functional was bounded having a minimizer on N~-. The results show that the p-Laplacian equation has at least 2 nontrivial solutions.
引文
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[14] TIAN S. Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity[J]. Journal of Mathematical Analysis and Applications, 2017, 454(2): 816-828.
[15] CONG N L,XUAN T L. Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity[J]. Computers and Mathematics with Applications, 2017, 73(9): 1-21.
[2] SHI L, CHANG X. Multiple solutions to p-Laplacian problems with concave nonlinearities[J]. Journal of Mathematical Analysis and Applications, 2010, 363(1): 155-160.
[3] CHENG W S, REN J L. On the existence of periodic solutions for p-Laplacian generalized Liénard equation[J]. Nonlinear Analysis Theory Methods and Applications,2005,60(1): 65-75.
[4] 陈顺清. 二阶p-Laplacian算子方程的正周期解[J]. 四川师范大学学报(自然科学版),2008,31(2): 155-158.
[5] 袁红秋,张绍康,康道坤. 一维p-Laplacian方程多解的存在性[J]. 四川师范大学学报(自然科学版), 2010, 33(5): 635-637.
[6] 蓝永艺, 沈贤端. 一类p-Laplacian方程非平凡解的存在性[J]. 集美大学学报(自然科学版), 2017, 22(6): 66-69.
[7] AMBROSETTI A,AZORERO J G,PERAL I. Multiplicity results for some nonlinear elliptic equations[J]. Journal of Functional Analysis, 1996,137(1): 219-242.
[8] CWISZEWSKI A,MACIEJEWSKI M. Positive stationary solutions for p-Laplacian problems with nonpositive perturbation[J]. Journal of Differential Equations, 2013, 254(3): 1120-1136.
[9] AN Y,KIM C G,SHI J. Exact multiplicity of positive solutions for a p-Laplacian equation with positive convex nonlinearity[J]. Journal of Differential Equations,2016,260(3): 2091-2118.
[10] GARCíA-HUIDOBRO M,WARD J R J R, MANáSEVICH R. Positive solutions for equations and systems with p-Laplace like operators[J]. Advances in Differential Equations,2009,14(5/6): 401-432.
[11] KIM C G,LEE E K,LEE Y H. Existence of the second positive radial solution for a p-Laplacian problem[J]. Journal of Computational and Applied Mathematics, 2011,235(13): 3743-3750.
[12] JI C,SZULKIN A. A logarithmic Schr?dinger equation with asymptotic conditions on the potential[J]. Journal of Mathemati-cal Analysis and Applications, 2016,437(1): 241-254.
[13] SQUASSINA M,SZULKIN A. Multiple solutions to logarithmic Schr?dinger equations with periodic potential[J]. Calculus of Variations and Partial Differential Equations,2015,54(1): 585-597.
[14] TIAN S. Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity[J]. Journal of Mathematical Analysis and Applications, 2017, 454(2): 816-828.
[15] CONG N L,XUAN T L. Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity[J]. Computers and Mathematics with Applications, 2017, 73(9): 1-21.