Symmetries, Hopf fibrations and supercritical elliptic problems

We consider the semilinear elliptic boundary value problem <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u equals StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u in normal upper Omega comma reverse-solidus quad u equals 0 on partial-differential normal upper Omega comma"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mtext> in </mml:mtext> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> <mml:mtext>\quad </mml:mtext> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mtext> on </mml:mtext> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-\Delta u=\left \vert u\right \vert ^{p-2}u\text { in }\Omega ,\text {\quad }u=0\text { on }\partial \Omega ,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> in a bounded smooth domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for supercritical exponents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than StartFraction 2 upper N Over upper N minus 2 EndFraction period"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;\frac {2N}{N-2}.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.