Díaz Díaz, Jesús Ildefonso and Rakotoson, Jean Michel Theresien (2010) On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete and Continuous Dynamical Systems. Series A., 27 (3). pp. 1037-1058. ISSN 1553-5231
Restringido a Repository staff only
We prove the existence of an appropriate function (very weak solution) u in the Lorentz space L(N') (,infinity)(Omega), N' = (N)(N - 1) satisfying Lu - Vu + g (x, u, del u) = mu in Omega an open bounded set of R(N), and u = 0 on partial derivative Omega in the sense that (u, L phi)(0) - (Vu, phi)(0) + (g(., u, del u),phi)(0) = mu(phi), for all phi is an element of C(c)(2)(Omega). The potential V <= lambda < lambda(1) is assumed to be in the weighted Lorentz space L(N,1)(Omega, delta), where delta(x) = dist (x, partial derivative Omega), mu is an element of M(1)(Omega, delta), the set of weighted Radon measures containing L(1)(Omega, delta), L is an elliptic linear self adjoint second order operator, and lambda(1) is the first eigenvalue of L with zero Dirichlet boundary conditions. If mu is an element of L(1)(Omega, delta) we only assume that for the potential V is in L(loc)(1) (Omega), V <= lambda < lambda(1). If mu is an element of M(1)(Omega, delta(alpha)), alpha is an element of [0, 1[, then we prove that the very weak solution vertical bar del u vertical bar is in the Lorentz space L(N/N-1+alpha,infinity)(Omega). We apply those results to the existence of the so called large solutions with a right hand side data in L(1)(Omega, delta). Finally, we prove some rearrangement comparison results.
|Uncontrolled Keywords:||Very weak solutions; semilinear elliptic equations distance to the boundary; weighted spaces measure unbounded potentials|
|Subjects:||Sciences > Mathematics > Differential geometry|
Sciences > Mathematics > Topology
B. Alziary, J. Fleckinger-Pellé and P. Takac, Ground-state positivity, negativity, and compactness for a Schrödinger operator in IRN, J. Funct. Anal., 245 (2007), 213–248.
C. Bandle, R. P. Sperb and I. Stakgold, Diffusion and reaction with monotone kinetics, Nonlinear Anal., 8 (1984), 321–333.
Ph. Benilan and H. Brezis Nonlinear problems related to the Thomas-Fermi equation, J. Evol. Equations, 3 (2003), 673–770.
C. Bennett and R. Sharpley, “Interpolation of Operators,” Academic Press London, 1983.
H. Berestycki, S. Kamin and G. Sivaskinsky, Metastability in a flame front evolution equation,Interfaces Free Bound, 3 (2001), 361–392.
M. Betta and A. Mercaldo, Geometric inequalities related to Steiner symmetriszation, Differential Integral Equations, 10 (1997), 473–486.
L. Boccardo and T. Galloüet, Non linear elliptic and parabolic equations involving measure as data, J. Funct. Anal., 87 (1989), 149–169.
H. Brezis, Semilinear equations in IRN without conditions at infinity, Appl. Math. Optim.,12 (1984), 271–282.
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for ut−_u = g(u) revisited, Advance in Diff. Eq., 1 (1996), 73–90.
X. Cabr´e and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal., 156 (1998), 30–56.
Capuzzo Dolcetta, F. Leoni and A. Porreta, Hölder estimates for degenerate elliptic equations with coercive Hamiltonians, to appear in Transactions Amer. Math. Soc.
J. I. Díaz and O. A. Oleinik, Nonlinear elliptic boundary-value problem in unbounded domains and the asymptotic behaviour of its solutions, C.R.A.S. 315, S´erie I, (1992), 787–792.
J. I. Díaz, Nonlinear partial differential equations and free boundaries, Research Notes in Math., 106 Pitman, London, 1985.
G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal., 20 (1993), 97–125.
J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Functional Analysis, 257(2009), 807–831.
L. C. Evans, “Weak Convergence Methods for Nonlinear Partial Diff. Equations,” AMS,Providence, 1990.
V. Ferone and M. R. Posteraro, Symmetrization results for elliptic equations with lower-order terms, Atti. Sem. Mat. Fis. Univ Modena, 40 (1992), 47–61.
T. Galloüet and J.-M. Morel, On some semilinear problem in L1, Boll. Un. Mat. Ital. A (6), 4 (1985), 123–131.
Y. Haitao, Positive versus compact support solutions to a singular elliptic problem, J. Math.Anal. Appl., 319 (2006), 830–840.
J. Hernández and F. J. Mancebo, Singular elliptic and parabolic equations, in “Handbook of Differential Equations” (eds. M. Chipot and P. Quittner), vol. 3, Elsevier Amsterdam, (2006),317–400.
J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Annalen, 283 (1989), 583–630.
J.-M. Morel and S. Solimini, Optimal conditions for solving a semilinear elliptic equation in L1 with “absorbing” nonlinearity, Houston J. Math., 12 (1986), 405–409.
M. R. Posteraro, On the solutions of the equation _u = eu blowing up on the boundary, C.R.A.S. Paris S´erie Math., 332 (1996), 445–450.
J. E. Rakotoson and J. M. Rakotoson,“Analyse Fonctionnelle Appliqu´ee aux ´Equations aux Dérivées Partielles,” P.U.F. Paris, 1999.
J. M. Rakotoson,“R´earrangement Relatif: Un Instrument D’estimation dans les Problèmes aux Limites,” 2008, Springer Verlag Berlin.
J. M. Rakotoson, Quasilinear elliptic problems with measure as data, Differential and Integral Equations, 4 (1991), 449–457.
] L. Veron, “Singularities of Solutions of Second Order Quasilinear Equations,” Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996.
] L. Veron, Elliptic equations involving measures, Stationary Partial Differential Equations,Vol. I, 593–712, Handb. Differ. Equ., North-Holland, Amsterdam, (2004).
|Deposited On:||27 Apr 2012 08:40|
|Last Modified:||20 Apr 2015 12:00|
Repository Staff Only: item control page