Continuous dependence of solutions for indefinite semilinear elliptic problems

Abstract

We consider the superlinear elliptic problem $$ -\Delta u + m(x)u = a(x)u^p $$ in a bounded smooth domain under Neumann boundary conditions, where $m \in L^{\sigma}(\Omega)$, $\sigma\geq N/2$ and $a\in C(\overline{\Omega})$ is a sign changing function. Assuming that the associated first eigenvalue of the operator $-\Delta + m $ is zero, we use constrained minimization methods to study the existence of a positive solution when $\widehat{m}$ is a suitable perturbation of m.

Keywords

• Positive solution
• constrained minimization
• eigenvalue problem
• Neumann boundary condition
• unique continuation