Constructing solutions of an ill-posed nonlinear singularly perturbed problem for an equation of elliptic type (Q1810311)

Consider the following singularly perturbed problem \[ \varepsilon^2\Delta u= f(u,x),\quad x= (x_1,x_2)\in \Omega\subset \mathbb{R}^2,\tag{1} \] where \(\varepsilon> 0\), with the boundary conditions \[ u|_{\partial\Omega}= 0\quad\text{or}\quad {\partial u\over\partial n}\Biggl|_{\partial\Omega}= 0.\tag{2} \] Regarding \(f(u,x)\) as the right-hand side of the Poisson equation, one can write out the equation of the second kind for \(u(x,\varepsilon)\) in terms of the source function \(G(x,y)\) with known properties \[ -\int_\Omega G(x,y) f(u(y), y)\,d\tau_y= \varepsilon^2 u(x,\varepsilon).\tag{3} \] The limiting case, that is \[ -\int_\Omega G(x,y) f(u(y), y)\,dy= 0\tag{4} \] is ill-posed. Depending on the roots of the equation \(f(u,x)= 0\), the author studies asymptotics of bounded solutions (3) as \(\varepsilon\to 0\).