Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions

Abstract We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $$\lambda =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in N dimensions with $$3\le N \le 9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:math> and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $$2\le N\le 9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:math> .