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 & 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. 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. 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. 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 Poincar\'e, 25 (2008), 969. 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. 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. 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), 81. Google Scholar [8] P. Baras and J. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 294 (1984), 121. 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, 22 (1995), 241. Google Scholar [10] L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators,, Boll. Unione Mat. Ital., 18 (1981), 951. Google Scholar [11] L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form,, in, (1994), 43. Google Scholar [12] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411. 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. 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. 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, 46 (1998), 51. 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. 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, 11 (1984), 213. 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. 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. 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. 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. Google Scholar [22] F. Della Pietra and G. di Blasio, Existence and comparison results for non-uniformly parabolic problems,, Mediterr. J. Math. 7 (2010), (2010), 323. Google Scholar [23] J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 441. 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. 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. 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. 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. 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. 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. 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. 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. 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 Poincar\'e, 25 (2008), 969. 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. 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. 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), 81. Google Scholar [8] P. Baras and J. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 294 (1984), 121. 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, 22 (1995), 241. Google Scholar [10] L. Boccardo, Positive eigenfunctions for a class of quasi-linear operators,, Boll. Unione Mat. Ital., 18 (1981), 951. Google Scholar [11] L. Boccardo, Some nonlinear Dirichlet problems in $L^1$ involving lower order terms in divergence form,, in, (1994), 43. Google Scholar [12] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411. 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. 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. 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, 46 (1998), 51. 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. 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, 11 (1984), 213. 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. 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. 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. 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. Google Scholar [22] F. Della Pietra and G. di Blasio, Existence and comparison results for non-uniformly parabolic problems,, Mediterr. J. Math. 7 (2010), (2010), 323. Google Scholar [23] J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 441. 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. 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. 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. 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. 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. Google Scholar
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