Existence results for semilinear elliptic equations with small measure data

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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“…Thus we obtain (6.10). Inequality (6.10) also holds for p ≥ n; see for example [Gre,Lemma 2.1]. This completes the proof of the lemma.…”

Section: Renormalized Solutions Of Quasilinear Dirichlet Problemssupporting

confidence: 55%

Quasilinear and Hessian equations of Lane–Emden type

2008

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“…where µ ∈ M b (Ω), and |h(x, U )| ≦ f (x)(1 + |U | Q ), precising and improving the results announced in [39], see Theorem 6.2.…”

Section: Theorem 13mentioning

confidence: 60%

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Hamid

Bidaut‐Véron

2010

Commun. Contemp. Math.

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We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of R N . The first one, of the forminvolves a source gradient term with natural growth, where β is nonnegative, λ > 0, f (x) ≧ 0, and α is a nonnegative measure. The second one, of the formpresents a source term of order 0, where g is nondecreasing, and µ is a nonnegative measure. Here β and g can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.

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“…The constant on the right-hand side of (1.3) is obtained in the proof of Theorem 1.1 (see step 1 below). It could also been deduced from an argument by Grenon in [18,19], checking the dependence on p > of every involved constant and letting p go to . (It should be mentioned that Grenon assumes p > N N+ , but this hypothesis can be removed.)…”

Section: Multiplicity Of Solutionsmentioning

confidence: 99%

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

Abdellaoui

Dall’Aglio

León

2017

Advanced Nonlinear Studies

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In the present paper we study the Dirichlet problem for an equation involving the -Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the -Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

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Existence results for semilinear elliptic equations with small measure data

Cited by 14 publications

References 9 publications

Quasilinear and Hessian equations of Lane–Emden type

Quasilinear and Hessian equations of Lane–Emden type

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

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