Existence and multiplicity of solutions to semilinear elliptic equations of the form
\[ -\Delta u(x)+f(x,u(x))=0, \quad x\in\Omega, \]
where \(\Omega \subset {\mathbb R}^n\) is a domain, and \(f: \Omega \times {\mathbb R} \to {\mathbb R}\) is a given nonlinearity function, with Dirichlet type boundary conditions \(u(x)=0\) on \(\partial\Omega\), are studied. A novel computer assisted approach, exploiting the knowledge of approximate solutions to obtain a rigorous proof of existence of an exact solution, is presented.
The method starts with an approximate solution \(\omega\), which can be obtained by using any numerical method giving approximations in the function space required. Then the boundary value problem for the error function \(v=u-\omega\) is considered. This problem can be rewritten as a fixed point equation \(v\in X\), \(v=Tv\), in a Banach space \(X\), and then treated by using well-known fixed point theorems.
In addition, several interesting examples about using this method to prove existence and multiplicity results for various problems of the above type, including cases where purely analytical methods have not been successful, are presented.