# American Institute of Mathematical Sciences

May  2014, 34(5): 1747-1774. doi: 10.3934/dcds.2014.34.1747

## Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

 1 Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000, Algeria 2 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma Sapienza, via Scarpa 16, 00161 Roma, Italy 3 Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid 4 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received  July 2013 Revised  August 2013 Published  October 2013

In this article we consider the following family of nonlinear elliptic problems,
$-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x).$
We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential.
Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality.
1.For $0 < m \le 2$ we prove the existence of a critical exponent $q_+ \le 2$ such that for $q > q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments.
Moreover if $q = q_+ > 1$ we get the following alternative results.
(a) If $m < 2$ and $q=q_+$ there is no solution.
(b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions.
2. If $m > 2$ we obtain some partial results on existence and nonexistence.
We emphasize that if $q(\frac{1}{m}-1)<-1$ and $1 < q \le 2$, there exists positive solutions for any $f \in L^1(Ω)$.
Citation: Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
##### References:
 [1] B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Diff. Eq., 222 (2006), 21-62. doi: 10.1016/j.jde.2005.02.009.  Google Scholar [2] B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371. doi: 10.1016/j.na.2010.10.008.  Google Scholar [3] B. Abdellaoui and I. Peral, Nonexistence results for quasilinear elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Communications in Pure and Applied Analysis, 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539.  Google Scholar [4] B. Abdellaoui and I. Peral, The Equation $-\Delta u - \lambda \frac u{|x|^2} = |D u|^p + c f(x)$, the optimal power, Ann. Scuola Norm. Sup. Pisa, VI, (2007), 159-183.  Google Scholar [5] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Annali di Matematica Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar [6] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc., 8 (2006), 157-170. doi: 10.4171/JEMS/43.  Google Scholar [7] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002.  Google Scholar [8] D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J.diff. equation, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar [9] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier, 34 (1984), 185-206. doi: 10.5802/aif.956.  Google Scholar [10] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM - Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar [11] L.Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.  Google Scholar [12] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513.  Google Scholar [13] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.  Google Scholar [14] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar [15] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C.R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032.  Google Scholar [16] L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [17] G. Dal Maso, F. Murat, L. Orsina, Luigi and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808.  Google Scholar [18] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar [19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [20] F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981), 309-322.  Google Scholar

show all references

##### References:
 [1] B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Diff. Eq., 222 (2006), 21-62. doi: 10.1016/j.jde.2005.02.009.  Google Scholar [2] B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371. doi: 10.1016/j.na.2010.10.008.  Google Scholar [3] B. Abdellaoui and I. Peral, Nonexistence results for quasilinear elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Communications in Pure and Applied Analysis, 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539.  Google Scholar [4] B. Abdellaoui and I. Peral, The Equation $-\Delta u - \lambda \frac u{|x|^2} = |D u|^p + c f(x)$, the optimal power, Ann. Scuola Norm. Sup. Pisa, VI, (2007), 159-183.  Google Scholar [5] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Annali di Matematica Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar [6] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc., 8 (2006), 157-170. doi: 10.4171/JEMS/43.  Google Scholar [7] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002.  Google Scholar [8] D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J.diff. equation, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar [9] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier, 34 (1984), 185-206. doi: 10.5802/aif.956.  Google Scholar [10] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM - Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar [11] L.Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.  Google Scholar [12] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513.  Google Scholar [13] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.  Google Scholar [14] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar [15] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C.R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032.  Google Scholar [16] L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [17] G. Dal Maso, F. Murat, L. Orsina, Luigi and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808.  Google Scholar [18] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar [19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [20] F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981), 309-322.  Google Scholar
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