Existence of solutions for some elliptic problems with critical Sobolev exponents (Q1173659)

The author deals with the existence of solutions \(u\in H_ 0^ 1(\Omega)\) of the boundary value problems (1) \(\Delta u+| u| ^{{n+2\over n-2}}+f(x)=0\) in \(\Omega\), (2) \(\Delta u+u+| u| ^{{n+2\over n-2}}+f| x| =0\) in \(\Omega\) with the boundary condition \(u(x)=0\) on \(\partial \Omega\), where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^ n\), \(n\geq 3\), and \(f\in L^ \infty(\Omega)\). It is shown that if \(f\neq 0\), then (1), (2) admit nontrivial solutions and if \(f>0\), there are positive solutions, respectively. Moreover, if \(f=0\), (2) admits a positive solution, provided, \(1\in ]0,\lambda_ 1[\), where \(\lambda_ 1\) denotes the first eigenvalue of \(-\Delta\) with zero Dirichlet boundary condition.