Ambrosetti, A.; Calahorrano, M.; Dobarro, F. Remarks on the Grad-Shafranov equation. (English) Zbl 0772.35085 Appl. Math. Lett. 3, No. 3, 9-11 (1990). The paper presents an elegant existence proof for the following free boundary problem related to plasma confinement. Given a smooth bounded domain \(\Omega\subset\mathbb{R}^ N\) and a positive constant \(I\), find \(a\geq 0\), a subset \(\Omega_ p\subset\Omega\) and a function \(v\in C^ 1(r)\cap C^ 2(\Omega\backslash\partial\Omega_ p)\) such that \(-\Delta v=\sigma(v)\), \(v\geq 0\) on \(\Omega_ p\), \(v=0\) on \(\partial\Omega_ p\), \(-\Delta v=0\) on \(\Omega\backslash\Omega_ p\), \(v=-a\) on \(\partial\Omega\), \(-\int_{\partial\Omega}(\partial v/\partial n)=I\), \(n\) representing the outer normal. Typical assumptions on \(\sigma\) (besides monotonicity and regularity) are \(\sigma(s)\leq\alpha s+\beta\), \(\beta>0\), \(\alpha\) less than the first eigenvalue of \(-\Delta\) in \(H^ 1_ 0(\Omega)\); \(\sigma(0+)=b>0\).Under some symmetry assumptions on \(\partial\Omega\) and supposing that \(b\cdot\text{meas}|\Omega|>I\), a solution is shown to exist such that \(\Omega_ p\) is strictly contained in \(\Omega\). The proof is obtained by means of a limiting procedure stemming from bifurcation techniques and topological arguments. Cited in 4 Documents MSC: 35R35 Free boundary problems for PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 82D10 Statistical mechanics of plasmas 35B32 Bifurcations in context of PDEs Keywords:existence proof; limiting procedure PDFBibTeX XMLCite \textit{A. Ambrosetti} et al., Appl. Math. Lett. 3, No. 3, 9--11 (1990; Zbl 0772.35085) Full Text: DOI References: [1] Ambrosetti, A.; Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73, 411-422 (1980) · Zbl 0433.35026 [2] Amick, C. J., A global branch of steady vortex rings, J. Rein. Angew. Math., 384, 1-23 (1988) · Zbl 0628.76032 [3] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 [4] Polya, G.; Szego, G., Isoperimetric inequalities in mathematical physics (1951), Princeton University Press · Zbl 0044.38301 [5] Puel, J. P., Sur un probleme de valeur propre non lineaire et de frontiere libre, C.R.A.S., 284, 861-863 (1977) · Zbl 0362.35024 [6] Temam, R., A nonlinear eigenvalue problem: the shape at equilibrium of a confined plasma, Arch. Rat. Mech. & Analysis, 60, 51-73 (1975) · Zbl 0328.35069 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.