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ON THE MULTIPLE SOLUTIONS OF THE ELLIPTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS NONLINEAR TERMS
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1978
Year
Spectral TheoryEquilibrium EquationsElliptic EquationEngineeringFree Boundary ProblemHilbert SpaceConstrained FunctionalThe Multiple SolutionsElliptic FunctionNonlinear EquationNonlinear Hyperbolic ProblemFunctional AnalysisCalculus Of VariationVariational InequalitiesNonlinear Functional Analysis
We study the EVP of the PDE: Lu=λφ(x, u(x)), where L is a second order elliptic differential operator, and φ(x, t) is a function in x and t, which may be discontinuous in t. Under certain conditions of φ, the structure of the positive spectrum and the existence of at least three distinct solutions of this equation are treated. These results are applied to a class of free boundary problems of the equilibrium equations in nuclear fusion. We obtain at least two distinct non-trivial free boundaries. In addition, the Lagrangian multiplier theorem in a Hilbert space covers the case where the constrained functional is continuously convex and nondifferentiable.