Some Weighted Inequalities on Polyharmonic Operators and Applications
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摘要: 研究了关于调和算子的加权不等式,推广了Bernis等人关于类似问题的结果,并且得到一些新结果．利用这些新结果,讨论了关于半线性部分是多调和算子的临界指数问题. 改进了问题存在非平凡径向解的必要条件,使该问题存在非平凡径向解的范围比已知的范围增大一倍.Abstract: This work is devoted to weighted inequalities on polyharmonic operators and their applications to the critical exponent problems of semilinear polyharmonic operator with Dirichlet boundary value conditions. Some newer weighted inequalities on polyharmonic operator are obtained. By applying these new weighted inequalities to the critical exponent problems, it obtains a preferable necessary condition for nontrivial radial solutions of the critical exponent problems of semilinear polyharmonic operator with Dirichlet boundary conditions existing. This preferable necessary condition extends the scope in which the critical exponent problems has nontrivial radial solutions to two times of that in Bernis' paper.
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Key words:
- harmonic analysis /
- operators /
- critical exponent /
- boundary value problems
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1. Bernis F, Grunau H C. Critical exponents and multiple critical dimensions for polyharmonic operators. J Diff Equations, 1995, 117(2):469~486 2. Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36(3):437~477 3. Edmunds D, Fortunato D, Jannelli E. Critical exponents, critical dimensions and the biharmonic operator. Arch Rational Mech Anal, 1990, 112(2):269~289 4. Noussair E S, Swanson C A, Yang J F. Critical semilinear biharmonic equations in RN. Proc Royal Soc Edinburgh Sect, 1992(A),121(1):139~148 5. Pucci P, Serrin J. Critical exponents and critical dimensions for polyharmonic operators. J Math Pure Appl, 1990, 69:58~83 -
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