Soliton solutions for a class of critical Schrödinger equations with Stein–Weiss convolution parts in $$\mathbb {R}^2$$

Claudianor Oliveira Alves, Liejun Shen
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Abstract

We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts

$$\begin{aligned} -\Delta u+V(x) u+\frac{\kappa }{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2, \end{aligned}$$

where \(\kappa \in \mathbb {R}\backslash \{0\}\) is a parameter, \(\beta >0\), \(0<\mu <2\) with \(0<2\beta +\mu <2\) and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For \(\kappa <0\): (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that \(V\in C^0(\mathbb {R}^2,\mathbb {R}^+)\) and \(\inf _{x \in \mathbb {R}^2}V(x)>0\), which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when \(V\equiv 1\). Moreover, if \(\kappa >0\) is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when \(\kappa \rightarrow 0^+\). It seems that the results presented above are even new for the case \(\kappa =0\).

在 $$\mathbb {R}^2$ 中具有 Stein-Weiss 卷积部分的一类临界薛定谔方程的孤子解
我们考虑在等离子物理学和非线性光学中引入的以下一类具有 Stein-Weiss 卷积部分的准线性薛定谔方程 $$begin{aligned} -\Delta u+V(x) u+\frac{\kappa }{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u)、~x\in \mathbb {R}^2, \end{aligned}$$ 其中 \(\kappa \in \mathbb {R}\backslash \{0\}\) 是一个参数, \(\beta >;0),\(0<\mu <2)与\(0<2\beta +\mu <2),H是h的基元,满足特鲁丁格-莫泽意义上的临界指数增长。对于 \(\kappa <0\):(i) 通过使用变量变化论证和山过定理,我们研究了仅假设 \(V\in C^0(\mathbb {R}^2,\mathbb {R}^+)\) 和 \(\inf _{x \in \mathbb {R}^2}V(x)>0\) 的基态解的存在性,这是对我们最近在 Alves 和 Shen(J Differ Equ 344:352-404, 2023)中提出的问题;(ii) 通过发展一种新型的特鲁丁格-莫泽不等式,当 \(V\equiv 1\) 时,我们通过约束最小化方法建立了一种 Pohoz̆aev 类型的地面解。此外,如果 \(\kappa >0\) 很小,结合山越定理和纳什-莫泽迭代过程,我们得到了非小解的存在,其中还考虑了 \(\kappa\rightarrow 0^+\) 时的渐近行为。对于 \(\kappa =0\)这种情况,上述结果似乎也是新的。
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