Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter

TL;DR

This paper investigates the uniqueness of solutions for certain elliptic equations and systems involving a parameter, providing conditions under which solutions are unique for specific boundary conditions.

Contribution

It establishes new uniqueness results for elliptic systems and fourth order equations with parameter constraints, extending previous understanding of solution behavior.

Findings

Uniqueness results depend on parameter constraints.

Similar results obtained for systems of elliptic equations.

Conditions identified for boundary value problems with Navier or Dirichlet conditions.

Abstract

We examine the equation \[\Delta^2 u = \lambda f(u) \qquad \Omega, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for the sytem {equation*} \{{array}{rrl} -\Delta u &=& \lambda f(v) \qquad \Omega, -\Delta v &=& \gamma g(u) \qquad \Omega, u&=& v = 0 \qquad \partial Omega. {array}. {equation*}