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A generalization of a criterion for the existence of solutions to semilinear elliptic equations
Université de Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France |
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 $.
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). 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. |
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. |
[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). 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. |
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