On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators

In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.