On the existence of positive solutions of second-order semilinear elliptic equations in cylindrical domains (Q706268)

Consider the equation \[ \sum_{i,j=1}^n\frac{\partial}{\partial x_i} \Bigl(a_{i,j}(\widehat x)\frac{\partial u}{\partial x_j}\Bigr)+ \sum_{i=1}^{n-1} a_{i}(\widehat x)\frac{\partial u}{\partial x_i} + a_0(x)| u| ^{q-1}u=0, \] where \([a_{ij}]\) is a symmetric positive definite matrix with \(a_{nn}=1\), \(a_{in}=0\) for \(i<n\), \(a_i(\widehat x)\) and \(a_0(x)\) are bounded measurable functions with \(a_0(x)\geq\) constant \(>0\); \(x=(\widehat x,x_n)\), \(\hat x=(x_1,\dots,x_{n-1})\), \(q=\) constant \(>1\). This paper concerns existence (or non existence) of positive solutions of this equation in the cylindrical domain \(\Pi_{a,b}=\Omega\times(a,b)\), where \(\Omega\) is a bounded Lipschitz domain in \(\mathbb R^{n-1}\) It is proved that a positive (generalized) solution exists under homogeneous Dirichlet boundary conditions, and that no positive solutions exist under homogeneous Neumann boundary conditions.