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This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.
elliptic problem, nonlinear boundary conditions, maximal and minimal weak solution, pseudomonotone operator, Kato's inequality
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S. Bandyopadhyay, M. Chhetri, B. B. Delgado, Nsoki Mavinga, and R. Pardo.
"Maximal And Minimal Weak Solutions For Elliptic Problems With Nonlinearity On The Boundary".
Electronic Research Archive.