Nonexistence results on the space or the half space of \begin{document}$ -\Delta u+\lambda u = |u|^{p-1}u $\end{document} via the Morse index

Abdelbaki Selmi; ,Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Tunis, Département de Mathématiques, , Faculté des Sciences de Bizerte, Zarzouna, 7021 Bizerte, Tunisia; Abdellaziz Harrabi; Cherif Zaidi; ,Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Kairouan, Département de Mathématiques, , Institut Superieur des Mathématiques Appliquées et de l'Informatique; ,Faculté des Sciences, Département de Mathématiques, , B.P 1171 Sfax 3000, Université de Sfax, Tunisia

doi:10.3934/cpaa.2020124

In this paper we consider the following semi-linear elliptic problem with Dirichlet boundary conditions. Here N ≥ 2, p > 1 and λ is a positive real parameter. The main goal of this work is to analyze the influence of the linear term λu, in order to classify regular stable solutions possibly unbounded and sign-changing. Our analysis reveals the nonexistence of nontrivial stable solutions (respectively solutions which are stable outside a compact set) for all p > 1 (respectively for all p ≥ N +2

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... −2 , or 1 < p < N +2 N −2 and |u| p−1 < λ(p+1) 2 ). Inspired by [6, 9, 16 , 23], we establish a monotonicity formula to discuss the supercritical case. Regarding the case O = R N , we obtain a complete classification which states that problem (P ) has regular solutions which are stable outside a compact set if and only if p ∈ (1, ∞) and N = 2; or p ∈ (1, N +2 N −2 ) and N ≥ 3. 2010 Mathematics Subject Classification. Primary: 35J60, 35J65; Secondary: 58E05. 2839 2840 ABDELBAKI SELMI, ABDELLAZIZ HARRABI AND CHERIF ZAIDI (0 < α < 1) and satisfies lim |x |→+∞ ϕ(x ) = +∞.

doi:10.3934/cpaa.2020124 fatcat:4pee6gvs3rajvgon2brcq6cqbm

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Abdelbaki Selmi, ,Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Tunis, Département de Mathématiques, , Faculté des Sciences de Bizerte, Zarzouna, 7021 Bizerte, Tunisia, Abdellaziz Harrabi, Cherif Zaidi, ,Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Kairouan, Département de Mathématiques, , Institut Superieur des Mathématiques Appliquées et de l'Informatique, ,Faculté des Sciences, Département de Mathématiques, , B.P 1171 Sfax 3000, Université de Sfax, Tunisia. "Nonexistence results on the space or the half space of \begin{document}$ -\Delta u+\lambda u = |u|^{p-1}u $\end{document} via the Morse index." Communications on Pure and Applied Analysis 19.5 (2020) 2839-2852

Nonexistence results on the space or the half space of \begin{document}$ -\Delta u+\lambda u = |u|^{p-1}u $\end{document} via the Morse index

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