Critère d’existence de solutions positives pour des équations semi-linéaires non monotones

Baras, Pierre; Pierre, Michel

doi:10.1016/s0294-1449(16)30402-4

Cited by 152 publications

(135 citation statements)

References 8 publications

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“…Let µ[u] be the Radon measure on Ω associated with u, and let ϕ be an arbitrary nonnegative function in C ∞ 0 (Ω). As in [BP,Lemme 2.2], we can find a sequence {ϕ n } of nonnegative functions in…”

Section: Renormalized Solutions Of Quasilinear Dirichlet Problemsmentioning

confidence: 99%

Quasilinear and Hessian equations of Lane–Emden type

2008

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The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2, . . . , n); µ is a nonnegative measurable function (or measure) on Ω.The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ L s (Ω), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given.Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Malý, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampère type.

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Section: Renormalized Solutions Of Quasilinear Dirichlet Problemsmentioning

confidence: 99%

Quasilinear and Hessian equations of Lane–Emden type

2008

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“…This problem has been considered in several places in the literature (cf. [3], [6], [29], [32], [52] ). In the examples we wish to consider, the kernel K has an additional property which we term the quasi-metric assumption i.e.…”

Section: K(x Y) Dω(y)mentioning

confidence: 99%

“…We will also give an alternative approach to some results of Baras and Pierre [6] on the solvability of the "superlinear" problem u = T u q + f discussed in the introduction. Let X be a metric space and suppose σ is a σ-finite measure on X.…”

Section: Superlinear Problems and Related Function Spacesmentioning

confidence: 99%

Nonlinear equations and weighted norm inequalities

Kalton

Verbitsky

1999

Trans. Amer. Math. Soc.

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Abstract. We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem −∆u = v u q + w, u ≥ 0 on Ω,on a regular domain Ω in R n in the "superlinear case" q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space.The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary "integration by parts" argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

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“…In the case when the principal part of equation (1.2) is replaced by the standard heat operator, we quote the papers [6], [13], [14], [16]. We refer to [10], [14], [17] and to the literature quoted therein, for a discussion of the Morrey norms [µ] defined above, whose connection with initial value problems (mainly for semilinear parabolic equations) has been recently the subject of considerable interest.…”

Section: Notation Let µ Be Any Radon Measure In Rmentioning

confidence: 99%

Degenerate parabolic equations with initial data measures

Andreucci

1997

Trans. Amer. Math. Soc.

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Abstract. We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.

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Critère d’existence de solutions positives pour des équations semi-linéaires non monotones

Cited by 152 publications

References 8 publications

Quasilinear and Hessian equations of Lane–Emden type

Quasilinear and Hessian equations of Lane–Emden type

Nonlinear equations and weighted norm inequalities

Degenerate parabolic equations with initial data measures

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