Gelfand type quasilinear elliptic problems with quadratic gradient terms
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URI: http://hdl.handle.net/10835/15697
DOI: 10.1016/J.ANIHPC.2013.03.002
DOI: 10.1016/J.ANIHPC.2013.03.002
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In this paper, for 0 < m1 m(x) m2 and positive parameters λ and p, we study the existence of positive solution for the
quasilinear model problem
⎧
⎨
⎩
−u + m(x)|∇u|
2
1 + u = λ(1 + u)p in Ω,
u = 0 on ∂Ω.
We prove that the maximal set of λ for which the problem has at least one positive solution is an interval (0,λ∗], with λ∗ > 0,
and there exists a minimal regular positive solution for every λ ∈ (0,λ∗). We also prove, under suitable conditions depending on
the dimension N and the parameters p, m1, m2, that for λ = λ∗ there exists a minimal regular positive solution. Moreover we
characterize minimal solutions as those solutions satisfying a stability condition in the case m1 = m2
Palabra/s clave
Gelfand problem
Quasilinear elliptic equations
Quadratic gradient
Stability condition
Extremal solutions