A mass supercritical problem revisited

In any dimension N >= 1 and for given mass m>0, we revisit the nonlinear scalar field equation with an L2 constraint: <disp-formula id="Equ51"><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mfenced open="{"><mml:mtable><mml:mtr><mml:mtd columnalign="right">-Delta u</mml:mtd><mml:mtd columnalign="left">=f(u)-mu u<mml:mspace width="1em"></mml:mspace>in<mml:mspace width="3.33333pt"></mml:mspace>RN,</mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right">uL2(RN)2</mml:msubsup></mml:mtd><mml:mtd columnalign="left">=m,</mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right">u</mml:mtd><mml:mtd columnalign="left">is an element of H1(RN),</mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace><mml:mspace width="2em"></mml:mspace>(Pm)</mml:mtd></mml:mtr></mml:mtable><graphic position="anchor" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="526_2020_1828_Article_Equ51.gif"></graphic></disp-formula>where mu is an element of R will arise as a Lagrange multiplier. Assuming only that the nonlinearity f is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states to (Pm) and reveal the basic behavior of the ground state energy <mml:msub>Em as m>0 varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other L2 constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any N >= 2 and establish the existence and multiplicity of nonradial sign-changing solutions when N >= 4. Finally we propose two open problems.