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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 |
        $-\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 $f \in L^1(Ω)$.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262. |
[14] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N2$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[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. |
[16] |
L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275. |
[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. |
[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. |
[19] |
T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.
doi: 10.1007/BF02760233. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262. |
[14] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N2$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[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. |
[16] |
L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275. |
[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. |
[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. |
[19] |
T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.
doi: 10.1007/BF02760233. |
[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. |
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