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Multi-bump type nodal solutions for a logarithmic Schrödinger equation with deepening potential well
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-03-16 , DOI: 10.1007/s00033-021-01504-y
Chao Ji

In this paper, we are concerned with the existence and multiplicity of multi-bump type nodal solutions for the following logarithmic Schrödinger equation

$$ \left\{ \begin{array}{ll} -\Delta u+ \lambda V(x)u=u \log u^2, &{}\quad \text{ in } \quad {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), \\ \end{array} \right. $$

where \(N \ge 1\), \(\lambda >0\) is a real parameter and the nonnegative continuous function \(V: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) has a potential well \(\Omega : =\text {int}\, V^{-1}(0)\) which possesses k disjoint bounded components \(\Omega =\bigcup _{j=1}^{k}\Omega _{j}\). Using the variational methods, we prove that if the parameter \(\lambda >0\) is large enough, then the equation has at least \(2^{k}-1\) multi-bump type nodal solutions.



中文翻译:

具有势阱加深对数Schrödinger方程的多凸点型节点解

在本文中,我们关注以下对数Schrödinger方程的多凸点型节点解的存在性和多重性

$$ \ left \ {\ begin {array} {ll}-\ Delta u + \ lambda V(x)u = u \ log u ^ 2,&{} \ quad \ text {in} \ quad {\ mathbb {R }} ^ {N},\\ u \ in H ^ 1({\ mathbb {R}} ^ {N}),\\ \ end {array} \ right。$$

其中\(N \ ge 1 \)\(\ lambda> 0 \)是实参,非负连续函数\(V:{\ mathbb {R}} ^ {N} \ rightarrow {\ mathbb {R} } \)有一个势阱\(\ Omega:= \文本{int} \,V ^ {-1}(0)\)具有k个不相交的有界分量\(\ Omega = \ bigcup _ {j = 1} ^ {k} \ Omega _ {j} \)。使用变分方法,我们证明如果参数\(\ lambda> 0 \)足够大,则该方程至少具有\(2 ^ {k} -1 \)多凸点类型节点解。

更新日期:2021-03-17
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