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POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR -LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS
Journal of the Australian Mathematical Society ( IF 0.5 ) Pub Date : 2019-07-29 , DOI: 10.1017/s1446788719000260 J. C. DE ALBUQUERQUE , JOÃO MARCOS DO Ó , EDCARLOS D. SILVA
Journal of the Australian Mathematical Society ( IF 0.5 ) Pub Date : 2019-07-29 , DOI: 10.1017/s1446788719000260 J. C. DE ALBUQUERQUE , JOÃO MARCOS DO Ó , EDCARLOS D. SILVA
We study the existence of positive ground state solutions for the following class of $(p,q)$ -Laplacian coupled systems $$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ where $1<p\leq q<N$ . Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$ , where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$ . Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.
中文翻译:
一类涉及薛定谔方程的超线性-拉普拉斯耦合系统的正基态
我们研究以下类别的正基态解的存在$(p,q)$ - 拉普拉斯耦合系统$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p -2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^ {\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^ {q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u |^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$ 在哪里$1<p\leq q<N$ . 这里的系数$\unicode[STIX]{x1D706}(x)$ 耦合项与条件的电位有关$|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[ STIX]{x1D6FD}/q}$ , 在哪里$\unicode[STIX]{x1D6FF}\in (0,1)$ 和$\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$ . 使用基于 Nehari 流形最小化的变分方法,我们为一大类非线性项和势建立了正基态解的存在。
更新日期:2019-07-29
中文翻译:
一类涉及薛定谔方程的超线性-拉普拉斯耦合系统的正基态
我们研究以下类别的正基态解的存在