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Existence and nonexistence of solutions for singular quadratic quasilinear equations. (English) Zbl 1173.35051

The paper is concerned with quasilinear elliptic problems of the form
\[ \begin{cases}-\text{div}(M(x,u)\nabla u)+g(x,u)|\nabla u|^2=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{1} \]
Here \(\Omega\subseteq\mathbb{R}^N\) is open and bounded, and \(N\geq3\). The coefficients are Caratheodory functions, and such that the principal part is uniformly elliptic with bounded coefficients. For the nonhomogeneous part \(f\) it is assumed that it lies in a suitable Lebesgue space, and that it is uniformly bounded from below by positive constants on compact subsets of \(\Omega\).
The main interest lies in nonnegative functions \(g\) with a singularity in \(u=0\) that is uniform in \(x\). Suppose that \(h:(0,\infty)\to[0,\infty)\) is continuous, nonincreasing in a neighborhood of zero, and satisfies
\[ \lim_{s\to0+}\int_s^1\sqrt{h(t)}\,dt<\infty. \]
If
\[ g(x,s)\leq h(s)\qquad\text{for a.e.\;}x\in\Omega,\;\forall s>0, \]
then it is proved that (1) has a weak positive solution in \(H^1_0(\Omega)\).
Conversely, a nonexistence result is given in the case that \(g\) grows faster in \(s\) than a function \(h\) whose square root is not integrable near \(0\). For the model problem
\[ \begin{cases}-\Delta u+\frac{|\nabla u|^2}{u^\gamma}=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{2} \]
this amounts to the following assertion: Suppose that \(\gamma>0\). Then Eq. (2) has a positive solution if and only if \(\gamma<2\).
Existence of a solution to (1) is proved by applying classical results for quasilinear equations to a family of problems with truncated coefficients and then passing to the limit.
The regularity of solutions to (1) is also considered. Moreover, the authors treat a general semilinear variant of (1).

MSC:

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B45 A priori estimates in context of PDEs
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References:

[1] Arcoya, D.; Barile, S.; Martínez-Aparicio, P.J., Singular quasilinear equations with quadratic growth in the gradient without sign condition, J. math. anal. appl., 350, 401-408, (2009) · Zbl 1161.35013
[2] Arcoya, D.; Carmona, J.; Martínez-Aparicio, P.J., Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms, Adv. nonlinear stud., 7, 299-317, (2007) · Zbl 1189.35136
[3] Arcoya, D.; Martínez-Aparicio, P.J., Quasilinear equations with natural growth, Rev. mat. iberoamericana, 24, 597-616, (2008) · Zbl 1151.35343
[4] D. Arcoya, S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var. (2008), doi:10.1051/cocv:2008072
[5] Bandle, C.; Marcus, M., Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. anal. math., 58, 9-24, (1992) · Zbl 0802.35038
[6] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vázquez, J.L., An \(L^1\)-theory of existence and uniqueness of nonlinear elliptic equations, Ann. sc. norm. super. Pisa cl. sci., 22, 241-273, (1995) · Zbl 0866.35037
[7] Bensoussan, A.; Boccardo, L.; Murat, F., On a nonlinear PDE having natural growth terms and unbounded solutions, Ann. inst. H. Poincaré anal. non linéaire, 5, 347-364, (1988) · Zbl 0696.35042
[8] Boccardo, L., Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM control optim. calc. var., 14, 411-426, (2008) · Zbl 1147.35034
[9] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. funct. anal., 87, 149-169, (1989) · Zbl 0707.35060
[10] Boccardo, L.; Gallouët, T., Nonlinear elliptic equations with right-hand side measures, Comm. partial differential equations, 17, 641-655, (1992) · Zbl 0812.35043
[11] Boccardo, L.; Gallouët, T., Strongly nonlinear elliptic equations having natural growth terms and \(L^1\) data, Nonlinear anal., 19, 573-579, (1992) · Zbl 0795.35031
[12] Boccardo, L.; Gallouët, T.; Murat, F., A unified presentation of two existence results for problems with natural growth, (), 127-137 · Zbl 0806.35033
[13] Boccardo, L.; Gallouët, T.; Vázquez, J.L., Nonlinear elliptic equations in \(\mathbb{R}^N\) without growth conditions on the data, J. differential equations, 105, 334-363, (1993) · Zbl 0810.35021
[14] Boccardo, L.; Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear anal., 19, 581-597, (1992) · Zbl 0783.35020
[15] Boccardo, L.; Murat, F.; Puel, J.-P., Existence de solutions non bornees pour certaines équations quasi-linéaires, Port. math., 41, 507-534, (1982) · Zbl 0524.35041
[16] Boccardo, L.; Murat, F.; Puel, J.-P., \(L^\infty\) estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. math. anal., 23, 326-333, (1992) · Zbl 0785.35033
[17] Brezis, H., Semilinear equations in \(\mathbb{R}^N\) without condition at infinity, Appl. math. optim., 12, 271-282, (1984) · Zbl 0562.35035
[18] De Figueiredo, Djairo G., Positive solutions of semilinear elliptic problems, () · Zbl 0607.35037
[19] Gallouët, T.; Morel, J.-M., The equation \(- \operatorname{\Delta} u + | u |^{\alpha - 1} u = f\), for \(0 \leqslant \alpha \leqslant 1\), Nonlinear anal., 11, 893-912, (1987) · Zbl 0659.35036
[20] D. Giachetti, F. Murat, An elliptic problem with a lower order term having singular behaviour, Boll. Unione Mat. Ital. Sez. B, in press · Zbl 1173.35469
[21] Giarrusso, E.; Porru, G., Problems for elliptic singular equations with a gradient term, Nonlinear anal., 65, 107-128, (2006) · Zbl 1103.35031
[22] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 3, 209-243, (1979) · Zbl 0425.35020
[23] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag · Zbl 0691.35001
[24] Keller, J.B., On solutions of \(\operatorname{\Delta} u = f(u)\), Comm. pure appl. math., 10, 503-510, (1957) · Zbl 0090.31801
[25] Ladyzenskaya, O.; Ural’tseva, N., Linear and quasilinear elliptic equations, (1968), Academic Press New York, translated by Scripta Technica
[26] Leone, C.; Porretta, A., Entropy solutions for nonlinear elliptic equations in \(L^1\), Nonlinear anal., 32, 325-334, (1998) · Zbl 1155.35352
[27] Leoni, F., Nonlinear elliptic equations in \(\mathbb{R}^N\) with “absorbing” zero order terms, Adv. differential equations, 5, 681-722, (2000) · Zbl 1018.35031
[28] Leoni, F.; Pellacci, B., Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data, J. evol. equ., 6, 113-144, (2006) · Zbl 1109.35052
[29] Leonori, T., Large solutions for a class of nonlinear elliptic equations with gradient terms, Adv. nonlinear stud., 7, 237-269, (2007) · Zbl 1156.35030
[30] Leray, J.; Lions, J.L., Quelques résultats de višik sur LES problèmes elliptiques non linéaires par LES méthodes de minty – browder, Bull. soc. math. France, 93, 97-107, (1965) · Zbl 0132.10502
[31] Marcus, M.; Veron, L., Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. inst. H. Poincaré, 14, 237-274, (1997) · Zbl 0877.35042
[32] Marcus, M.; Veron, L., Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. evol. equ., 3, 637-652, (2003), dedicated to Philippe Bénilan · Zbl 1160.35408
[33] Osserman, R., On the inequality \(\operatorname{\Delta} u \geqslant f(u)\), Pacific J. math., 7, 1641-1647, (1957) · Zbl 0083.09402
[34] Porru, G.; Vitolo, A., Problems for elliptic singular equations with a quadratic gradient term, J. math. anal. appl., 334, 467-486, (2007) · Zbl 1156.35032
[35] Porretta, A., Existence for elliptic equations in \(L^1\) having lower order terms with natural growth, Port. math., 57, 179-190, (2000) · Zbl 0963.35068
[36] Stampacchia, G., Équations elliptiques du second ordre à coefficients discontinus, (1966), Les Presses de l’Université de Montréal Montréal · Zbl 0151.15501
[37] Vázquez, J.L., An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear anal., 5, 95-103, (1981) · Zbl 0446.35018
[38] Veron, L., Semilinear elliptic equations with uniform blow-up on the boundary, J. anal. math., 59, 231-250, (1992) · Zbl 0802.35042
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