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http://dx.doi.org/10.4134/JKMS.j210099

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)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1461-1484 More about this Journal
Abstract
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.
Keywords
Quasilinear elliptic equations; concave-convex nonlinearities; variational methods; Orlicz-Sobolev spaces;
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