Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents (Journal of Partial Differential Equations, 1991)
Let &#937 be a bounded domain in R^4(n &#8805 4) with smooth boundary &#8706&#937. We discuss the existence of nontrivial solutions of the Dirichlet problem {- &#916u = a(x) |u|^{4/(a-2)}u + &#955u + g(x, u), \quad x &#8712 &#937 u = 0, \quad x &#8712 &#8706&#937 where a(x) is a smooth function which is nonnegative on \overline{&#937} and positive somewhere, &#955> 0 and &#955 &#8713 &#963(-&#916). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.
Keywords: Semilinear elliptic equation; Sobolev exponent; Critical value; Critical point; (P &#8226 S) condition