We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

中文翻译：

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关于涉及拉普拉斯算子的超临界问题

我们讨论了具有零狄利克雷边界条件的单位球中一类椭圆方程的解的存在性、不存在性和多重性，这些条件涉及具有超临界增长的非线性。通过使用 Pohozaev 类型恒等式，我们证明了一类具有可变指数的超临界问题的不存在结果，这使我们能够补充 (Calc. Var. (2016) 55:83) 中开发的分析。此外，我们建立了涉及非线性的半线性椭圆方程的正解的存在性结果，这些非线性仅在域的一部分中在无穷远处是亚临界的，并且在适当的意义上可以是超临界的。