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Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^\\infty (\\varOmega )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>\u03a9<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results.<\/jats:p>","DOI":"10.1007\/s00211-021-01222-7","type":"journal-article","created":{"date-parts":[[2021,9,12]],"date-time":"2021-09-12T13:05:37Z","timestamp":1631451937000},"page":"305-340","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations"],"prefix":"10.1007","volume":"149","author":[{"given":"Eduardo","family":"Casas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mariano","family":"Mateos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arnd","family":"R\u00f6sch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,9,12]]},"reference":[{"key":"1222_CR1","volume-title":"Sobolev Spaces, Pure and Applied Mathematics (Amsterdam)","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. 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