# American Institute of Mathematical Sciences

• Previous Article
Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle
• DCDS Home
• This Issue
• Next Article
The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
August  2018, 38(8): 3831-3850. doi: 10.3934/dcds.2018166

## Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

 1 Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile 2 Departamento de Matemática, Instituto de Ciêencias Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

* Corresponding author: E. Massa

Received  August 2017 Revised  April 2018 Published  May 2018

Fund Project: The first author gratefully acknowledges financial support from Fondecyt grants 1161635, 1171532 and 1171691
The author E. Massa was supported by: grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grants #308354/2014-1, #303447/2017-6, CNPq/Brazil.

In this paper we consider the equation $(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$ with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and $f, g$ are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form $λ>\overline λ$ and $0 < μ< \overlineμ(λ)$, then we provide further conditions under which, respectively, the bound $\overlineμ(λ)$ is either necessary, or can be removed.

Citation: Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166
##### References:

show all references

##### References:
 [1] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [2] Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 [3] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [4] Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $N-$Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306 [5] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [6] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294 [7] Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360 [8] Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 [9] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [10] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [11] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [12] Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495 [13] Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326 [14] Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180 [15] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 [16] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403 [17] Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007 [18] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [19] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 [20] Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

2019 Impact Factor: 1.338