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September/October 2021 Global multiplicity of solutions for a quasilinear elliptic equation with concave and convex nonlinearities
Siyu Chen, Carlos Alberto Santos, Minbo Yang, Jiazheng Zhou
Adv. Differential Equations 26(9/10): 425-458 (September/October 2021). DOI: 10.57262/ade026-0910-425

Abstract

We consider the modified elliptic problem $$ -\Delta u-u\Delta u^2= a(x) u^\alpha+ \lambda b(x)u^{\beta}~ \mbox{in}~ \Omega, $$ with $u(x)=0$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^N$ is a regular domain and $N\geq3$, $0 < a(x)\in C(\Omega)\cap L^\infty(\Omega),b(x)\in C(\Omega)\cap L^\infty(\Omega)$, $0 < \alpha < 1 < \beta < \infty$ and $\lambda > 0$ is a parameter. By using sub- and super-solutions methods and variational methods, we establish the existence of two nontrivial solutions for the modified equation with appropriate exponents $\alpha,\beta$ and potentials $a(x), b(x)$.

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Siyu Chen. Carlos Alberto Santos. Minbo Yang. Jiazheng Zhou. "Global multiplicity of solutions for a quasilinear elliptic equation with concave and convex nonlinearities." Adv. Differential Equations 26 (9/10) 425 - 458, September/October 2021. https://doi.org/10.57262/ade026-0910-425

Information

Published: September/October 2021
First available in Project Euclid: 12 August 2021

Digital Object Identifier: 10.57262/ade026-0910-425

Subjects:
Primary: 35J10 , 35J62 , 35J75 , 35P15

Rights: Copyright © 2021 Khayyam Publishing, Inc.

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Vol.26 • No. 9/10 • September/October 2021
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