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Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

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Abstract

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem

$$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$

whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have\(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u p possesses a positive solution inH 10 (Ω).

We also prove the existence of nonradial solutions to the problem −Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ∂Ω, Ω=B(0,1). for a class off(r, u).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, P.R. China

    Wang Xu-jia

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  1. Wang Xu-jia
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Project supported by the National Natural Science Foundation of China.

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Xu-jia, W. Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents. Acta Mathematica Sinica 8, 273–291 (1992). https://doi.org/10.1007/BF02582916

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  • DOI: https://doi.org/10.1007/BF02582916

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