Nonexistence results for a class of nonlinear elliptic equations involving critical Sobolev exponents

In this paper we prove nonexistence results for some classes of nonlinear elliptic equations with critical growth of the form {−Δu=λg(x,u)+|u|2∗−2u,x∈Ωu>0x∈Ωu=0x∈∂Ω where 2∗=2N/(N−2)2∗=2N/(N−2), g(x,u)g(x,u) is a lower-order perturbation of u2∗−1u2∗−1 and ΩΩ is a bounded, strictly star-shaped domain in RNRN, N≥3N≥3. Combining Pohozaev’s identity with classical and weak interpolation inequality, we are able to exhibit examples of nonlinear problems without any (nontrivial) solution bifurcating from infinity in λ=0λ=0, for N=3N=3 and N=4N=4.