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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

Pierre Baras 1,

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 $.

Keywords: Convex analysis, semilinear elliptic problems, partial differential equations, Hamilton-Jacobi equations.
Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20, 47N10, 35F21.
Citation: Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[7]

P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Mathematics, Pitman, 58, Boston, Mass.-London, 1982.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

S. S. Kutateladze, Convex operators, Russian Uspekhi Mat. Nauk, 34 (1979), 167-196.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. V. Rutski, Linear selections of superlinear set-valued maps with some applications to analysis, arXiv: 1206.3337, (2012). Google Scholar

[19]

M. Théra, Subdifferential calculus for convex operators, J. Math. Anal. Appl., 80 (1981), 78-91.  doi: 10.1016/0022-247X(81)90093-7.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[7]

P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. R. Giles, Convex Analysis with Application in Differentiation of Convex Functions, Research Notes in Mathematics, Pitman, 58, Boston, Mass.-London, 1982.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

S. S. Kutateladze, Convex operators, Russian Uspekhi Mat. Nauk, 34 (1979), 167-196.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. V. Rutski, Linear selections of superlinear set-valued maps with some applications to analysis, arXiv: 1206.3337, (2012). Google Scholar

[19]

M. Théra, Subdifferential calculus for convex operators, J. Math. Anal. Appl., 80 (1981), 78-91.  doi: 10.1016/0022-247X(81)90093-7.  Google Scholar

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