Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term

By Lucio Boccardo, Sergio Segura de León, Cristina Trombetti. 2001

71 citations recorded by the source corpus.

Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/edebd0c98654e72a3f8edbdcabc0dced26cfdc65

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Citations in the corpus

Nonlinear elliptic equations having a gradient term with natural growth (2006, methodology): “In this work we prove that this is not the case and that the results in [8] can be extended relaxing assumption (2) in dependence on the regularity of the data.”
On the connection between two quasilinear elliptic problems with source terms of order 0 or 1 (2008, background): “Among them, let us mention the results of [13], [14] for the case p = 2, [29], [30] for general quasilinear operators, when β is defined on R, not necessarily positive, but bounded.”
Nonlinear equations with natural growth terms and measure data (2002, unknown): “For fixed n ∈ N, since μn ∈ L∞(Ω), under the previous assumptions it is proved in [19] that there exists a weak solution un ∈ W 1,p 0 (Ω) ∩ L∞(Ω) of the”
UNIQUENESS OF SOLUTIONS FOR SOME ELLIPTIC EQUATIONS WITH A QUADRATIC GRADIENT TERM (2010, background): “(Similar cancellation results can be found in [20] and [31].”