Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case
Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case
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DOI：
10.1515/anona-2020-0129
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发表时间：
2021-06
影响因子：
4.2
通讯作者：
Yong Ma;Ying Wang;C. Ledesma
中科院分区：
文献类型：
--
作者：
Yong Ma;Ying Wang;C. Ledesma
Abstract Our purpose of this paper is to study positive solutions of Lane-Emden equation −Δu=VupinRN∖{0} $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$(0.1) perturbed by a non-homogeneous potential V when p∈[pc,N+2N−2), $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where pc is the Joseph-Ludgren exponent. When p∈(NN−2,pc), $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −Δ u = up in ℝN ∖ {0} by authors in [9]. While the fast decaying solution wk is unstable for p∈(pc,N+2N−2), $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of −Δu=uN+2N−2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝN and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.
Abstract Our purpose of this paper is to study positive solutions of Lane-Emden equation −Δu=VupinRN∖{0} $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$(0.1) perturbed by a non-homogeneous potential V when p∈[pc,N+2N−2), $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where pc is the Joseph-Ludgren exponent. When p∈(NN−2,pc), $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −Δ u = up in ℝN ∖ {0} by authors in [9]. While the fast decaying solution wk is unstable for p∈(pc,N+2N−2), $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of −Δu=uN+2N−2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝN and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.