A Global Solution Curve for a Class of Free Boundary Value Problems Arising in Plasma Physics

Abstract

We study the existence and multiplicity of solutions and the global solution curve of the following free boundary value problem, arising in plasma physics, see Temam (Arch Ration Mech Anal 60(1):51---73, 1975---1976), and Berestycki and Brezis (Nonlinear Anal. 4(3):415---436, 1980): find a function $$u(x)$$ u ( x ) and a constant $$b$$ b , satisfying $$\begin{aligned}&\Delta u+g(x,u)=p(x) \; \;\text{ in }\, D \\u \;\; \;\int \limits _{\partial D} \frac{\partial u}{\partial n} \, ds=0. \end{aligned}$$ Δ u + g ( x , u ) = p ( x ) in D u | ? D = b , ? ? D ? u ? n d s = 0 . Here $$D \subset R^n$$ D ? R n , is a bounded domain, with a smooth boundary. This problem can be seen as a PDE generalization of the periodic problem for one-dimensional pendulum-like equations. We use continuation techniques. Our approach is suitable for numerical computations.