caa a22 4500 606193928 CHVBK 20210128100911.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00030-015-0351-0 doi (NATIONALLICENCE)springer-10.1007/s00030-015-0351-0 A singular elliptic equation with natural growth in the gradient and a variable exponent [Elektronische Daten] [José Carmona, Pedro Martínez-Aparicio, Julio Rossi] In this paper we consider singular quasilinear elliptic equations with quadratic gradient and a singular term with a variable exponent $$\begin{cases} -\Delta u + \frac{|{\nabla}u|^2}{u^{\gamma(x)}} = f & {\rm in} \, \Omega \\ u = 0 & {\rm on} \, \partial \Omega \end{cases}$$ - Δ u + | ∇ u | 2 u γ ( x ) = f in Ω u = 0 on ∂ Ω Here $${\Omega}$$ Ω is an open bounded set of $${\mathbb{R}^N}$$ R N , $${\gamma(x)}$$ γ ( x ) is a positive continuous function and f is positive function that belongs to a certain Lebesgue space. We show, among other results, that there exists a solution in the natural energy space $${H^1_0 (\Omega)}$$ H 0 1 ( Ω ) to this problem when $${\gamma (x)}$$ γ ( x ) is strictly less than 2 in a strip around the boundary; while there is no solution in the energy space when there exists $${\Gamma \subset \partial \Omega}$$ Γ ⊂ ∂ Ω with $${|\Gamma|_{N-1} > 0}$$ | Γ | N - 1 > 0 such that $${\gamma(x) > 2}$$ γ ( x ) > 2 on $${\Gamma}$$ Γ . Moreover, since we work by approximation we can analyze the behavior of the approximated solutions $${u_n}$$ u n in the case in which there is no solution in $${H_0^1(\Omega)}$$ H 0 1 ( Ω ) . Springer Basel, 2015 Nonlinear elliptic equations nationallicence Singular natural growth gradient terms nationallicence Positive solutions nationallicence Variable exponent nationallicence Carmona José Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano, 04120, Almería, Spain aut Martínez-Aparicio Pedro Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universidad Politécnica de Cartagena, 30203, Murcia, Spain aut Rossi Julio Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1, 1428, Buenos Aires, Argentina aut Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/6(2015-12-01), 1935-1948 1021-9722 22:6<1935 2015 22 30 https://doi.org/10.1007/s00030-015-0351-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00030-015-0351-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Carmona José Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano, 04120, Almería, Spain aut NATIONALLICENCE 700 1- Martínez-Aparicio Pedro Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universidad Politécnica de Cartagena, 30203, Murcia, Spain aut NATIONALLICENCE 700 1- Rossi Julio Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1, 1428, Buenos Aires, Argentina aut NATIONALLICENCE 773 0- Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/6(2015-12-01), 1935-1948 1021-9722 22:6<1935 2015 22 30