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March  2011, 10(2): 593-612. doi: 10.3934/cpaa.2011.10.593

Breaking of resonance for elliptic problems with strong degeneration at infinity

Francesco Della Pietra 1, and  Ireneo Peral 2,

1. 

Università degli Studi del Molise, Dipartimento S.A.V.A., Facoltà di Ingegneria, Via Duca degli Abruzzi, 86039 Termoli (CB), Italy
2. 

Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid

Received  June 2010 Revised  October 2010 Published  December 2010

In this paper we study the problem

-div$(\frac{Du}{(1+u)^\theta})+|Du|^q =\lambda g(x)u +f$ in $\Omega,$

$u=0$ on $\partial \Omega, $

$u\geq 0$ in $\Omega,$

where $\Omega$ is a bounded open set of $R^n$, $1 < q \leq 2$, $\theta\geq 0$, $f\in L^1(\Omega)$, and $f>0$. The main feature is to show that even for large values of $\theta$ there is solution for all $\lambda>0$.
The problem could be seen as a reaction-diffusion model which produces a saturation effect, that is, the diffusion goes to zero when $u$ go to infinity.

Keywords: Non-coercive non-linear elliptic equations, degeneration at infinity, existence and nonexistence..
Mathematics Subject Classification: Primary: 35D05, 35J20; Secondary: 35J25, 35J7.
Citation: Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593
References:
[1]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-laplacian with a critical potential, Ann. Mat. Pura Appl., 182 (2003), 247-270. doi: doi:10.1007/s10231-002-0064-y.  Google Scholar

[2]

B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Differential Equations, 222 (2006), 21-62. doi: doi:10.1016/j.jde.2005.02.009.  Google Scholar

[3]

B. Abdellaoui, I. Peral and A. Primo, Some elliptic problems with Hardy potential and critical growth in the gradient: non-resonance and blow-up results, J. Differential Equations, 239 (2007), 386-416. doi: doi:10.1016/j.jde.2007.05.010.  Google Scholar

[4]

B. Abdellaoui, I. Peral and A. Primo, Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations, Ann. Inst. Henri Poincaré, 25 (2008), 969-985.  Google Scholar

[5]

N. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations wit data measures, SIAM J. Math. Anal., 24 (1993) 23-35. doi: doi:10.1137/0524002.  Google Scholar

[6]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl., 182 (2003), 53-79. doi: doi:10.1007/s10231-002-0056-y.  Google Scholar

[7]

A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of nonuniformly elliptic equations, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), supl. 81-391.  Google Scholar

[8]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 294 (1984), 121-139. doi: doi:10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[9]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.-L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 22 (1995), 241-273.  Google Scholar

[10]

L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators, Boll. Unione Mat. Ital., V. Ser., B, 18 (1981), 951-959.  Google Scholar

[11]

L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form, in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), volume 350 of Pitman Res. Notes Math. Ser., pages 43-57. Longman, Harlow, 1996.  Google Scholar

[12]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: doi:10.1051/cocv:2008031.  Google Scholar

[13]

L. Boccardo, Quasilinear elliptic equations with natural growth terms: the regularizing effect of the lower order terms, J. Nonlinear Convex Anal., 7 (2006), 355-365.  Google Scholar

[14]

L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 521-530.  Google Scholar

[15]

L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), suppl., 51-81.  Google Scholar

[16]

L. Boccardo and T. Gallouet, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal TMA, 19 (1992), 573-579. doi: doi:10.1016/0362-546X(92)90022-7.  Google Scholar

[17]

L. Boccardo, F. Murat and J. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 11 (1984), 213-235.  Google Scholar

[18]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving hardy potential, Discrete and Continuous Dynamical Systems, 16 (2006), 513-523. doi: doi:10.3934/dcds.2006.16.513.  Google Scholar

[19]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (1998), 223-262.  Google Scholar

[20]

G. Croce, The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity, Rend. Mat. Appl., 27 (2007), 299-314.  Google Scholar

[21]

F. Della Pietra and G. di Blasio, Comparison, existence and regularity results for a class of non-uniformly elliptic equations, Differ. Equ. Appl., 2 (2010), 79-103.  Google Scholar

[22]

F. Della Pietra and G. di Blasio, Existence and comparison results for non-uniformly parabolic problems, Mediterr. J. Math. 7 (2010), no. 3, 323-340.  Google Scholar

[23]

J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: doi:10.1006/jdeq.1997.3375.  Google Scholar

[24]

A. Mercaldo and I. Peral, Existence results for semilinear elliptic equations with some lack of coercivity, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 569-595. doi: doi:10.1017/S0308210506000126.  Google Scholar

[25]

A. Mercaldo, I. Peral and A. Primo, Existence results for noncoercive nonlinear elliptic equations with Hardy potential,, To appear., ().   Google Scholar

[26]

A. Perrotta and A. Primo, Breaking of resonance and regularizing effect of the gradient with the $p$-Laplacian Operator,, To appear in Advanced Nonlinear Studies., ().   Google Scholar

[27]

M. M. Porzio and M. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators, J. Evol. Equ., 8 (2008), 31-70. doi: doi:10.1007/s00028-007-0317-8.  Google Scholar

[28]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66. doi: doi:10.1016/j.jde.2003.05.001.  Google Scholar

[29]

P. Pucci and J. Serrin, Erratum to The strong maximum principle revisited, J. Differential Equations, 196 (2004) 1-66, 2004, J. Differential Equations, 207 (2004), 226-227. doi: doi:10.1016/j.jde.2004.09.002.  Google Scholar

[30]

G. Stampacchia, Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  Google Scholar

show all references

References:
[1]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the $p$-laplacian with a critical potential, Ann. Mat. Pura Appl., 182 (2003), 247-270. doi: doi:10.1007/s10231-002-0064-y.  Google Scholar

[2]

B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Differential Equations, 222 (2006), 21-62. doi: doi:10.1016/j.jde.2005.02.009.  Google Scholar

[3]

B. Abdellaoui, I. Peral and A. Primo, Some elliptic problems with Hardy potential and critical growth in the gradient: non-resonance and blow-up results, J. Differential Equations, 239 (2007), 386-416. doi: doi:10.1016/j.jde.2007.05.010.  Google Scholar

[4]

B. Abdellaoui, I. Peral and A. Primo, Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations, Ann. Inst. Henri Poincaré, 25 (2008), 969-985.  Google Scholar

[5]

N. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations wit data measures, SIAM J. Math. Anal., 24 (1993) 23-35. doi: doi:10.1137/0524002.  Google Scholar

[6]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl., 182 (2003), 53-79. doi: doi:10.1007/s10231-002-0056-y.  Google Scholar

[7]

A. Alvino, V. Ferone and G. Trombetti, A priori estimates for a class of nonuniformly elliptic equations, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), supl. 81-391.  Google Scholar

[8]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 294 (1984), 121-139. doi: doi:10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[9]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.-L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 22 (1995), 241-273.  Google Scholar

[10]

L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators, Boll. Unione Mat. Ital., V. Ser., B, 18 (1981), 951-959.  Google Scholar

[11]

L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form, in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), volume 350 of Pitman Res. Notes Math. Ser., pages 43-57. Longman, Harlow, 1996.  Google Scholar

[12]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426. doi: doi:10.1051/cocv:2008031.  Google Scholar

[13]

L. Boccardo, Quasilinear elliptic equations with natural growth terms: the regularizing effect of the lower order terms, J. Nonlinear Convex Anal., 7 (2006), 355-365.  Google Scholar

[14]

L. Boccardo and H. Brezis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 521-530.  Google Scholar

[15]

L. Boccardo, A. Dall'Aglio and L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Dedicated to Prof. C. Vinti (Italian) (Perugia, 1996). Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), suppl., 51-81.  Google Scholar

[16]

L. Boccardo and T. Gallouet, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data, Nonlinear Anal TMA, 19 (1992), 573-579. doi: doi:10.1016/0362-546X(92)90022-7.  Google Scholar

[17]

L. Boccardo, F. Murat and J. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 11 (1984), 213-235.  Google Scholar

[18]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving hardy potential, Discrete and Continuous Dynamical Systems, 16 (2006), 513-523. doi: doi:10.3934/dcds.2006.16.513.  Google Scholar

[19]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (1998), 223-262.  Google Scholar

[20]

G. Croce, The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity, Rend. Mat. Appl., 27 (2007), 299-314.  Google Scholar

[21]

F. Della Pietra and G. di Blasio, Comparison, existence and regularity results for a class of non-uniformly elliptic equations, Differ. Equ. Appl., 2 (2010), 79-103.  Google Scholar

[22]

F. Della Pietra and G. di Blasio, Existence and comparison results for non-uniformly parabolic problems, Mediterr. J. Math. 7 (2010), no. 3, 323-340.  Google Scholar

[23]

J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: doi:10.1006/jdeq.1997.3375.  Google Scholar

[24]

A. Mercaldo and I. Peral, Existence results for semilinear elliptic equations with some lack of coercivity, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 569-595. doi: doi:10.1017/S0308210506000126.  Google Scholar

[25]

A. Mercaldo, I. Peral and A. Primo, Existence results for noncoercive nonlinear elliptic equations with Hardy potential,, To appear., ().   Google Scholar

[26]

A. Perrotta and A. Primo, Breaking of resonance and regularizing effect of the gradient with the $p$-Laplacian Operator,, To appear in Advanced Nonlinear Studies., ().   Google Scholar

[27]

M. M. Porzio and M. Pozio, Parabolic equations with non-linear, degenerate and space-time dependent operators, J. Evol. Equ., 8 (2008), 31-70. doi: doi:10.1007/s00028-007-0317-8.  Google Scholar

[28]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66. doi: doi:10.1016/j.jde.2003.05.001.  Google Scholar

[29]

P. Pucci and J. Serrin, Erratum to The strong maximum principle revisited, J. Differential Equations, 196 (2004) 1-66, 2004, J. Differential Equations, 207 (2004), 226-227. doi: doi:10.1016/j.jde.2004.09.002.  Google Scholar

[30]

G. Stampacchia, Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  Google Scholar

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