Journal of Fixed Point Theory and Applications ( IF 1.4 ) Pub Date : 2021-05-20 , DOI: 10.1007/s11784-021-00870-4 Huabo Zhang
In this paper, we are interested in the existence of ground state nodal solutions for the following Schrödinger–Kirchhoff-type Laplacian problems:
$$\begin{aligned} -M\bigg (\int _{{\mathbb {R}}^{3}}|\nabla u|^{2}\mathrm{{d}}x\bigg )\Delta u+V(x)u=|u|^{4}u+ k f(u),\;x\in {\mathbb {R}}^{3}, \end{aligned}$$where \(M(t)=a+bt^\gamma \) with \(0<\gamma <2\), \(a,b>0\) and the nonlinear function \(f\in C({\mathbb {R}},{\mathbb {R}})\). By the nodal Nehari manifold method, for each \(b>0\), we obtain a least energy nodal solution \(u_{b}\) and a ground state solution \(v_b\) of this problems when \(k\gg 1\). Our results improve and extend the known results of the usual case \(\gamma =1\) in the sense that a more wider range of \(\gamma \) is covered.
中文翻译：
Schrödinger–Kirchhoff型拉普拉斯关键问题的基态和节点解
在本文中，我们对以下Schrödinger–Kirchhoff型拉普拉斯问题的基态节点解的存在感兴趣：
$$ \ begin {aligned} -M \ bigg（\ int _ {{{\ mathbb {R}} ^ {3}} | \ nabla u | ^ {2} \ mathrm {{d}} x \ bigg）\ Delta u + V（x）u = | u | ^ {4} u + kf（u），\; x \ in {\ mathbb {R}} ^ {3}，\ end {aligned} $$其中\（M（t）= a + bt ^ \ gamma \）与\（0 <\ gamma <2 \），\（a，b> 0 \）和非线性函数\（f \ in C（{\ mathbb {R}}，{\ mathbb {R}}）\）。由节点的Nehari歧管的方法，对于每个\（B> 0 \），我们得到一个最小能量节点解\（U_ {B} \）和一个基态溶液\（V_B \）的这个问题，当\（K \ gg 1 \）。我们的结果改进和扩展了通常情况\（\ gamma = 1 \）的已知结果，从某种意义上讲，它涵盖了更广泛的\（\ gamma \）范围。
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