We investigate here the properties of extremal solutions for
semilinear elliptic equation <mml:math alttext="$-\Delta u=\lambda f(u)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2212;</mml:mo><mml:mi>&#x394;</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x3BB;</mml:mi><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> posed on a bounded smooth domain of <mml:math alttext="${\mathbb R}^n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&#x211D;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> with Dirichlet boundary condition and with <mml:math alttext="$f$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> exploding at a finite positive value <mml:math alttext="$a$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math>.

Abstract and Applied Analysis

On the extremal solutions of semilinear elliptic problems

Abstract

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