On the critical dimension of a fourth order elliptic problem with negative exponent
نویسنده
چکیده
We study the regularity of the extremal solution of the semilinear biharmonic equation β∆u−τ∆u = λ (1−u)2 on a ball B ⊂ R , under Navier boundary conditions u = ∆u = 0 on ∂B, where λ > 0 is a parameter, while τ > 0, β > 0 are fixed constants. It is known that there exists a λ∗ such that for λ > λ∗ there is no solution while for λ < λ∗ there is a branch of minimal solutions. Our main result asserts that the extremal solution u∗ is regular (supB u ∗ < 1) for N ≤ 8 and β, τ > 0 and it is singular (supB u ∗ = 1) for N ≥ 9, β > 0, and τ > 0 with τ β small. Our proof for the singularity of extremal solutions in dimensions N ≥ 9 is based on certain improved Hardy-Rellich inequalities.
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