In this article we analyze existence and nonexistence of positive solutions to problem(P±) - Δ u ± | ∇ u |2 = λ frac(u, | x |2) + f in Ω, u = 0 on ∂ Ω . The main results are the following: (i)If the quadratic term in the gradient appears in the equation as a reaction term (- | ∇ u |2) and λ > 0, then there is no solution to problem (P-) (even in a very weak sense).(ii)If the quadratic term in the gradient appears in the equation as an absorption term (+ | ∇ u |2), then there exists a positive solution to (P+) for all λ > 0 and f ∈ L1 (Ω)
Abdellaoui, B., Peral, I., PRIMO RAMOS, A. (2007). Elliptic Problems with a Hardy Potential and Critical Growth in the Gradient: Non-resonance and Blow-up results. JOURNAL OF DIFFERENTIAL EQUATIONS, 239(2), 386-416 [10.1016/j.jde.2007.05.010].
Elliptic Problems with a Hardy Potential and Critical Growth in the Gradient: Non-resonance and Blow-up results
PRIMO RAMOS, ANA
2007
Abstract
In this article we analyze existence and nonexistence of positive solutions to problem(P±) - Δ u ± | ∇ u |2 = λ frac(u, | x |2) + f in Ω, u = 0 on ∂ Ω . The main results are the following: (i)If the quadratic term in the gradient appears in the equation as a reaction term (- | ∇ u |2) and λ > 0, then there is no solution to problem (P-) (even in a very weak sense).(ii)If the quadratic term in the gradient appears in the equation as an absorption term (+ | ∇ u |2), then there exists a positive solution to (P+) for all λ > 0 and f ∈ L1 (Ω)
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