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We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.<\/jats:p>","DOI":"10.1007\/s10589-020-00223-w","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T17:02:51Z","timestamp":1600448571000},"page":"831-869","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems"],"prefix":"10.1007","volume":"77","author":[{"given":"Veronika","family":"Karl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ira","family":"Neitzel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Wachsmuth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,9,18]]},"reference":[{"key":"223_CR1","volume-title":"Sobolev Spaces, Volume 140 of Pure and Applied Mathematics (Amsterdam)","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. 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