[PDF][PDF] Some simple nonlinear PDE's without solutions
H Brezis, X Cabré - Bollettino della Unione Matematica Italiana-B, 1998 - researchgate.net
H Brezis, X Cabré
Bollettino della Unione Matematica Italiana-B, 1998•researchgate.net(0.1){−∆ u= a (x) u2+ f (x) in Ω u= 0 on∂ Ω where Ω is a smooth bounded domain in RN, N≥
3. If a (x)∈ Lp (Ω) and p> N/2, then for any f∈ Lp (Ω) with fp small, problem (0.1) has a
unique small solution u in W2, p (Ω). This is an easy consequence of the Inverse Function
Theorem applied to F (u)=−∆ u− a (x) u2 which maps X= W2, p (Ω)∩ W 1, p
3. If a (x)∈ Lp (Ω) and p> N/2, then for any f∈ Lp (Ω) with fp small, problem (0.1) has a
unique small solution u in W2, p (Ω). This is an easy consequence of the Inverse Function
Theorem applied to F (u)=−∆ u− a (x) u2 which maps X= W2, p (Ω)∩ W 1, p
(0.1){−∆ u= a (x) u2+ f (x) in Ω u= 0 on∂ Ω where Ω is a smooth bounded domain in RN, N≥ 3. If a (x)∈ Lp (Ω) and p> N/2, then for any f∈ Lp (Ω) with fp small, problem (0.1) has a unique small solution u in W2, p (Ω). This is an easy consequence of the Inverse Function Theorem applied to F (u)=−∆ u− a (x) u2 which maps X= W2, p (Ω)∩ W 1, p
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