Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2020-10-14 , DOI: 10.1007/s00033-020-01414-5 Marcelo F. Furtado , Edcarlos D. Silva , Uberlandio B. Severo
We establish the existence and multiplicity of solutions for Kirchhoff elliptic problems of type
$$\begin{aligned} -m\left( \mathop \int \limits _{\mathbb {R}^3} |\nabla u|^2 \mathrm{{d}}x\right) \Delta u = f(x,u), \quad x \in \mathbb {R}^3, \end{aligned}$$where \(m:\mathbb {R}_+\rightarrow \mathbb {R}\) is continuous, positive and satisfies appropriate growth and/or monotonicity conditions. We consider the cases that f is asymptotically \(3-\)linear or \(3-\)superlinear at infinity, in an appropriated sense. By using variational methods, we obtain our results under crossing assumptions of the functions m and f with respect to limit eigenvalues problems. In the model case \(m(t)=a+bt\), we also prove a concentration result for some solutions when \(b\rightarrow 0^+\).
中文翻译：
渐近线性或超线性非线性的Kirchhoff椭圆问题
我们建立了Kirchhoff椭圆型问题的解的存在性和多重性
$$ \ begin {aligned} -m \ left（\ mathop \ int \ limits _ {\ mathbb {R} ^ 3} | \ nabla u | ^ 2 \ mathrm {{d}} x \ right）\ Delta u = f（x，u），\ quad x \ in \ mathbb {R} ^ 3，\ end {aligned} $$其中\（m：\ mathbb {R} _ + \ rightarrow \ mathbb {R} \）是连续的，正的，并且满足适当的增长和/或单调性条件。在适当的意义上，我们考虑f在无穷大处渐近\（3- \）线性或\（3- \）超线性的情况。通过使用变分方法，我们在函数m和f的交叉假设下就极限特征值问题获得了结果。在模型情况\（m（t）= a + bt \）中，当\（b \ rightarrow 0 ^ + \）时，我们还证明了某些解的集中结果。
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