Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity
| Autor: | Ishige Kazuhiro, Okabe Shinya, Sato Tokushi |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Advances in Nonlinear Analysis, Vol 11, Iss 1, Pp 968-992 (2022) |
| Druh dokumentu: | article |
| ISSN: | 2191-9496 2191-950X |
| DOI: | 10.1515/anona-2021-0220 |
| Popis: | In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P)−Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞,- \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0}\mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*. |
| Databáze: | Directory of Open Access Journals |
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