TY - JOUR

T1 - Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity

AU - Ishige, Kazuhiro

AU - Okabe, Shinya

AU - Sato, Tokushi

N1 - Funding Information:
The first and the second authors were supported in part by JSPS KAKENHI Grant Number JP19H05599. The second author was supported in part by JSPS KAKENHI Grant Number 20KK0057 and 21H00990.
Publisher Copyright:
© 2022 Kazuhiro Ishige et al., published by De Gruyter.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem -Δu+u=F(u)+κμ in RN, u>0in RN, u(x)→0as |x|→∞, where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ Lc1(RN) 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∗.

AB - In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem -Δu+u=F(u)+κμ in RN, u>0in RN, u(x)→0as |x|→∞, where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ Lc1(RN) 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∗.

KW - exponential nonlinearity

KW - inhomogeneous nonlinear elliptic equation

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U2 - 10.1515/anona-2021-0220

DO - 10.1515/anona-2021-0220

M3 - Article

AN - SCOPUS:85126060767

SN - 2191-9496

VL - 11

SP - 968

EP - 992

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

IS - 1

ER -