(untitled) (Journal of Partial Differential Equations, 1991)
In this paper, we study the following Eigen-problem -&#8706&#8706x_i(a_ij(x, u)&#8706u&#8706x_j) + 12a_iju(x,u)&#8706u&#8706x_i&#8706u&#8706x_j + h(x)u = &#956&#956n+2n-2 in &#937 (0.1) u = 0 on &#8706&#937 u > 0 in &#937 &#8834 R^n under some assumptions. First. we minimize I(u) = 12&#8747_&#937a_ij(x, u)&#8706u&#8706x_i&#8706u&#8706x_j + h(x)u&sup2 over E_&#945 = u &#8712 H&sup1_0(&#937); &#8747_&#937u^&#945 = 1 ( 2 < &#945 < N = 2nn-2) to give a H&sup1_0-solution U_&#945 of the perturbation problems of (0.1). Since I is not differentiable in H&sup1_0(&#937), the key point is the estimate of U_&#945. Then, we derive local uniform bounds of (U_&#945) 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.