DOI QR코드

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MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR SCHRÖDINGER TYPE EQUATIONS WITH THE CONCAVE-CONVEX NONLINEARITIES

  • Kim, In Hyoun (Department of Mathematics Incheon National University) ;
  • Kim, Yun-Ho (Department of Mathematics Education Sangmyung University) ;
  • Li, Chenshuo (Questrom School of Business Boston University) ;
  • Park, Kisoeb (Department of IT Convergence Software Seoul Theological University)
  • 투고 : 2021.02.05
  • 심사 : 2021.08.04
  • 발행 : 2021.11.01

초록

We deal with the following elliptic equations: $\{-div({\varphi}^{\prime}(\left|{\nabla}z\right|^2){\nabla}z)+V(x)\left|z\right|^{{\alpha}-2}z={\lambda}{\rho}(x)\left|z\right|^{r-2}z+h(x,z),\\z(x){\rightarrow}0,\;as\;\left|x\right|{\rightarrow}{\infty},$ in ℝN , where N ≥ 2, 1 < p < q < N, 1 < α ≤ p*q'/p', α < q, 1 < r < min{p, α}, φ(t) behaves like tq/2 for small t and tp/2 for large t, and p' and q' the conjugate exponents of p and q, respectively. Here, V : ℝN → (0, ∞) is a potential function and h : ℝN × ℝ → ℝ is a Carathéodory function. The present paper is devoted to the existence of at least two distinct nontrivial solutions to quasilinear elliptic problems of Schrödinger type, which provides a concave-convex nature to the problem. The primary tools are the well-known mountain pass theorem and a variant of Ekeland's variational principle.

키워드

과제정보

The first author was supported by the Incheon National University Research Grant in 2017. The authors gratefully thank to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

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