Isolated Singularities of Positive Solutions of a Superlinear Biharmonic Equation
Tóm tắt
This paper is mainly concerned with the local behavior of singular solutions of the biharmonic equation
$$\Delta ^2 u = |x|^\sigma u^p $$
with u ≤ 0 ,
$$- \Delta u $$
≤ 0 in
$$\Omega \backslash \{ 0\} \subset \mathbb{R}^N ,N$$
≥ 4, and Ω = B(0, R) is a ball centered at the origin of radius R > 0. The complete description of the singularity together with an existence result will be given when
$$$$
≤ 0, for N > 4, or 1 < p < +∞, for N = 4. Moreover, an a priori estimate of the radially symmetric solutions will be established when p ≥
$$\frac{{N+ \sigma}}{{N-4}}, -4 < \sigma$$
≤ 0, N > 4. This paper generalizes the results in Brezis and Lions (1981) and Lions (1980) for the corresponding Laplace equation.
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