Singular solutions for semilinear elliptic equations with general supercritical growth
Yasuhito Miyamoto, Yūki Naito
Abstract
Abstract A positive radial singular solution for $$\Delta u+f(u)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem $$\Delta u+\lambda f(u)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., $$u^p\log u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> , $$\exp (u^p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\exp (\cdots \exp (u)\cdots )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> as well as $$u^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> and $$e^u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>u</mml:mi> </mml:msup> </mml:math> . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.