Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

• Published in: Advances in nonlinear analysis Vol. 12; no. 1; pp. 2149 - 2167
• Main Authors: Liu, Cuiling, Zhang, Xingyong
• Format: Journal Article
• Language: English
• Published: De Gruyter 02.03.2023
• Subjects: 35J50; 35J62; 35J92; condition; locally super-; locally super-(m 1, m 2) condition; mountain pass theorem; Orlicz-Sobolev spaces; symmetric mountain pass theorem
• ISSN: 2191-950X; 2191-950X
• DOI: 10.1515/anona-2022-0289

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space . We assume that the nonlinear term satisfies the locally super- condition, that is, for a.e. , where is a domain in , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, for a.e. . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [ , J. Dynam. Differ. Equ. (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [ , J. Nonlinear Sci. Appl. (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.