Positive \(L^ p\)-solution of subcritical nonlinear problems (Q2638429)

The authors deal with the existence of positive solutions for the semilinear elliptic equation \[ Lu=f(x,u)\text{ in } \Omega;\quad u=0\text{ on } \partial \Omega,\quad \lim_{| x| \to \infty}u=0, \] where \(\Omega\) is a smooth unbounded domain in \({\mathbb{R}}^ n\), \(n\geq 3\), and \[ Lu=-\sum D_ i(a_{ij}(x)D_ ju)+b(x)u, \] is uniformly elliptic differential operator with \(a_{ij}=a_{ji}\in C^{1+\alpha}_{loc}\cap L^{\infty}({\bar \Omega})\), b(x)\(\geq 0\) and \(b\in C^{\alpha}_{loc}\cap L^{\infty}({\bar \Omega})\), under some rather general conditions for f. Also the corresponding eigenvalue problem is discussed. The main tools are variants of Mountain Pass Theorem and weighted spaces.