# American Institute of Mathematical Sciences

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

## Breaking of resonance for elliptic problems with strong degeneration at infinity

 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.

Citation: Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators, Boll. Unione Mat. Ital., V. Ser., B, 18 (1981), 951-959. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [25] A. Mercaldo, I. Peral and A. Primo, Existence results for noncoercive nonlinear elliptic equations with Hardy potential,, To appear., (). [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., (). [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators, Boll. Unione Mat. Ital., V. Ser., B, 18 (1981), 951-959. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [25] A. Mercaldo, I. Peral and A. Primo, Existence results for noncoercive nonlinear elliptic equations with Hardy potential,, To appear., (). [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., (). [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. [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. [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. [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.
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