论文标题
重新审视的大规模超批评问题
A mass supercritical problem revisited
论文作者
论文摘要
在任何维度$ n \ geq1 $中，对于给定的质量$ m> 0 $，我们以$ l^2 $约束来重新访问非线性标量字段方程： $$-ΔU= f（u）-μu，\ quad u \ in H^1（\ Mathbb {r}^n）\ quad \ text {with} \ quad \ quad \ | u \ |^2_ 2_ {l^2（\ Mathbb {r}^n）} = m。 $$，其中$μ\ in \ mathbb {r} $将作为拉格朗日乘数出现。只有假设非线性$ f $是连续的并且满足弱质量超临界条件，我们表明了地面状态的存在，并揭示了基态能源$ e_m $的基本行为，为$ m> 0 $。特别是，在寻找基础状态时要克服紧凑性问题，我们会提出强大的论点，我们认为这些论点将允许在一般的质量超临界环境中处理其他$ l^2 $约束问题。在相同的假设下，我们还为任何$ n \ geq2 $获得了无限的许多径向解决方案，并在$ n \ geq4 $时建立了非放置签名解决方案的存在和多重性。最后，我们提出了两个开放问题。
In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -Δu=f(u)-μu, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $μ\in\mathbb{R}$ 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 $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.
