{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms |
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Authors: | David Arcoya Lucio Boccardo Tommaso Leonori |
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Institution: | 1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva S/N, 18071, Granada, Spain 2. Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazza A. Moro 2, 00185, Rome, Italy 3. Departamento di Matematicas, Universidad Carlos III, Avda. de la Universidad 30, 28911, Leganés Madrid, Spain
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Abstract: | In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0. |
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