On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators (Journal of Partial Differential Equations, 1991)
In this paper, we study the following Eigen-problem {-\frac{&#8706}{&#8706x_i}(a_{ij}(x, u)\frac{&#8706u}{&#8706x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{&#8706u}{&#8706x_i}\frac{&#8706u}{&#8706x_j} + h(x)u = &#956&#956\frac{n+2}{n-2} \quad in &#937 \qquad (0.1) u = 0 \quad on &#8706&#937 u > 0 \quad in &#937 &#8834 R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}&#8747_&#937a_{ij}(x, u)\frac{&#8706u}{&#8706x_i}\frac{&#8706u}{&#8706x_j} + h(x)u&sup2 over E_&#945 = {u &#8712 H&sup1_0(&#937); &#8747_&#937u^&#945 = 1} ( 2 < &#945 < N = \frac{2n}{n-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.
Keywords: Non-differentiable; critical; regularity