Local minimizers of the mass constrained problem for the mass-supercritical bi-harmonic Schrödinger equation

Abstract In this paper, we study the mass constrained problem for the mass-supercritical bi-harmonic Schrödinger equation with power nonlinearities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), where the coefficients of Laplacian and bi-Laplacian have the same sign. First, we construct an unbounded open set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> </mml:mrow> </mml:math> in H 2 that satisfies that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mo>∪</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:math> is a star-shaped domain. Second, we prove the existence of minimizers on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mo>∩</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msubsup> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mtext>for some fixed</mml:mtext> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:math> when the mass does not exceed a certain threshold and the coefficient of the Laplacian is suitably small, meanwhile, the explicit bounds of the Lagrange multiplier are also obtained. Finally, we prove the orbital stability of the local minimizer. This answers the open problem listed in Luo and Yang (2023 Sci. China Math. 66 1237–62). Combined with the existing literature, a systematic answer is given for the mass constrained problem of the arbitrary dimensional bi-harmonic Schrödinger equation with (critical, subcritical and supercritical) power nonlinearities, where the coefficients of Laplacian and bi-Laplacian can have the same or different sign.