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On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. (English) Zbl 1156.35046

The author considers the problem

-Δu=λf(x)|u| q-2 u+|u| 2 * -2 uinΩ,uH 0 1 (Ω),

where Ω is a bounded domain in N (N3), 1<q<2<2 * =2N/(N-2), λ>0 and f:Ω ¯ is a continuous function with f + (x)=max{f(x),0}¬0 in Ω ¯. By using variational methods he proves that, for λ>0 small, the problem possesses at least two positive solutions. He also studies the asymptotic behavior of the obtained solutions as λ0.

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
35B40Asymptotic behavior of solutions of PDE