# American Institute of Mathematical Sciences

February  2021, 14(2): 465-504. doi: 10.3934/dcdss.2020439

## A generalization of a criterion for the existence of solutions to semilinear elliptic equations

 Université de Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

1 Pierre Baras died on April 22nd, 2020. He was still working on a final version of this article but unfortunately could not submit it. This version is essentially the first one he submitted, except for a few technical corrections made with the help of the referee. We would like to express our sincere thanks to the referee for his fruitful help, as well as to "Maha Daoud" who kindly rebuilt the latex source from the pdf submission.
A brief tribute describing some of Pierre Baras' actions may be found just after the preface. The Guest Editors.

Published  February 2021 Early access  October 2020

We prove an abstract result of existence of "good" generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem $-\Delta u = a |\nabla u|^p+ b|u|^q+f$ with Dirichlet boundary conditions where $a,b>0$, $p,q>1$. No other condition is made on $p$ and $q$.

Citation: Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
##### References:
 [1] D. R. Adams and M. Pierre, Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier, 41 (1991), 117-135.  doi: 10.5802/aif.1251. [2] N. E. Alaa and M. Pierre, Weak solutions for some quasi-linear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35.  doi: 10.1137/0524002. [3] B. Abdellaoui, A. Attar and E.-H. Laamri, On the existence of positive solutions to semilinear elliptic systems involving gradient term, Appl. Anal., 98 (2019), 1289-1306.  doi: 10.1080/00036811.2017.1419204. [4] T. Andô, On fundamental properties of a Banach space with cone, Pacific J. Math., 12 (1962), 1163-1169.  doi: 10.2140/pjm.1962.12.1163. [5] A. Attar, R. Bentifour and E.-H. Laamri, Nonlinear elliptic systems with coupled gradient terms, Acta Appl. Math., (2020), https://doi.org/10.1007/s10440-020-00329-7. doi: 10.1007/s10440-020-00329-7. [6] P. Baras, Semilinear problem with convex nonlinearity, Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), 202–215, Pitman Res. Notes Math. Ser., 208, Longman Sci. Tech., Harlow, (1989). [7] P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514. [8] P. Baras and M. Pierre, Critère d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 185-212.  doi: 10.1016/S0294-1449(16)30402-4. [9] A. Brønsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16 (1965), 605-611.  doi: 10.1090/S0002-9939-1965-0178103-8. [10] N. Dunford and J. T. Schwartz, Linear Operators, Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London 1958. [11] J. R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Mathematics, Pitman, 58, Boston, Mass.-London, 1982. [12] N. Grenon, F. Murat and A. Porretta, A priori estimates and existence for elliptic equations with gradient dependent terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 137-205. [13] K. Hansson, V. G. Maz'ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat., 37 (1999), 87-120.  doi: 10.1007/BF02384829. [14] S. S. Kutateladze, Convex operators, Russian Uspekhi Mat. Nauk, 34 (1979), 167-196. [15] T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework, J. Differential Equations, 260 (2016), 5421-5449.  doi: 10.1016/j.jde.2015.12.007. [16] R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc., 123 (1966), 46-63.  doi: 10.1090/S0002-9947-1966-0192318-X. [17] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.  doi: 10.2140/pjm.1970.33.209. [18] D. V. Rutski, Linear selections of superlinear set-valued maps with some applications to analysis, arXiv: 1206.3337, (2012). [19] M. Théra, Subdifferential calculus for convex operators, J. Math. Anal. Appl., 80 (1981), 78-91.  doi: 10.1016/0022-247X(81)90093-7.

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##### References:
 [1] D. R. Adams and M. Pierre, Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier, 41 (1991), 117-135.  doi: 10.5802/aif.1251. [2] N. E. Alaa and M. Pierre, Weak solutions for some quasi-linear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35.  doi: 10.1137/0524002. [3] B. Abdellaoui, A. Attar and E.-H. Laamri, On the existence of positive solutions to semilinear elliptic systems involving gradient term, Appl. Anal., 98 (2019), 1289-1306.  doi: 10.1080/00036811.2017.1419204. [4] T. Andô, On fundamental properties of a Banach space with cone, Pacific J. Math., 12 (1962), 1163-1169.  doi: 10.2140/pjm.1962.12.1163. [5] A. Attar, R. Bentifour and E.-H. Laamri, Nonlinear elliptic systems with coupled gradient terms, Acta Appl. Math., (2020), https://doi.org/10.1007/s10440-020-00329-7. doi: 10.1007/s10440-020-00329-7. [6] P. Baras, Semilinear problem with convex nonlinearity, Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), 202–215, Pitman Res. Notes Math. Ser., 208, Longman Sci. Tech., Harlow, (1989). [7] P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514. [8] P. Baras and M. Pierre, Critère d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 185-212.  doi: 10.1016/S0294-1449(16)30402-4. [9] A. Brønsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16 (1965), 605-611.  doi: 10.1090/S0002-9939-1965-0178103-8. [10] N. Dunford and J. T. Schwartz, Linear Operators, Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London 1958. [11] J. R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Mathematics, Pitman, 58, Boston, Mass.-London, 1982. [12] N. Grenon, F. Murat and A. Porretta, A priori estimates and existence for elliptic equations with gradient dependent terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 137-205. [13] K. Hansson, V. G. Maz'ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat., 37 (1999), 87-120.  doi: 10.1007/BF02384829. [14] S. S. Kutateladze, Convex operators, Russian Uspekhi Mat. Nauk, 34 (1979), 167-196. [15] T. Mengesha and N. C. Phuc, Quasilinear Riccati type equations with distributional data in Morrey space framework, J. Differential Equations, 260 (2016), 5421-5449.  doi: 10.1016/j.jde.2015.12.007. [16] R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc., 123 (1966), 46-63.  doi: 10.1090/S0002-9947-1966-0192318-X. [17] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.  doi: 10.2140/pjm.1970.33.209. [18] D. V. Rutski, Linear selections of superlinear set-valued maps with some applications to analysis, arXiv: 1206.3337, (2012). [19] M. Théra, Subdifferential calculus for convex operators, J. Math. Anal. Appl., 80 (1981), 78-91.  doi: 10.1016/0022-247X(81)90093-7.
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