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Luis Yanka Annalisc
Luis Yanka Annalisc
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Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>1p>2 $

Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>1 $$ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.

Since we have already known that the harmonic function have the unique continuation property, I guess such result is still true for the semilinear case here. However I do not know how to go on. Can you give me some hints or references?

Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>1 $

Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>1 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.

Since we have already known that the harmonic function have the unique continuation property, I guess such result is still true for the semilinear case here. However I do not know how to go on. Can you give me some hints or references?

Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $

Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.

Since we have already known that the harmonic function have the unique continuation property, I guess such result is still true for the semilinear case here. However I do not know how to go on. Can you give me some hints or references?

Source Link
Luis Yanka Annalisc
Luis Yanka Annalisc
  • 1.2k
  • 4
  • 13

Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>1 $

Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>1 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.

Since we have already known that the harmonic function have the unique continuation property, I guess such result is still true for the semilinear case here. However I do not know how to go on. Can you give me some hints or references?