A mass supercritical problem revisited

Abstract

In any dimension $N\ge 1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$-\Delta u=f(u)-\mu u,u\in H^1({\mathbb{R}}^N)\text{with}\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2=m.$$ where $\mu \in \mathbb{R}$ 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 and reveal the basic behavior of the ground state energy $E_m$ 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 $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\ge 2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\ge 4$. Finally we propose two open problems.

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Cited by in corpus (13)

• Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth
• Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation
• On global minimizers for a mass constrained problem
• Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems
• Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
• Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions
• Normalized solutions for a Choquard equation with exponential growth in ${\mathbb{R}}^2$
• Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities
• Positive solutions with prescribed mass for a planar Choquard equation with critical growth
• Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities
• Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation
• Normalized ground states for NLS equations with mass critical nonlinearities
• Normalized ground state solutions for the fractional Sobolev critical NLSE with an extra mass supercritical nonlinearity