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

김윤호|박기섭|김인현|Chenshuo Li
대한수학회 (The Korean Mathematical Society)|2021.11|대한수학회지 (Journal of the KMS)|제58권 제6호|pp. 1461-1484|DOI 10.4134/JKMS.j210099

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참고문헌

키워드

variational methodsOrlicz-Sobolev spacesQuasilinear elliptic equationsconcave-convex nonlinearities
주제분류수학

초록

We deal with the following elliptic equations: { − div ( φ ′ ( | ∇ z | 2 ) ∇ z ) + V ( x ) | z | α − 2 z = λ ρ ( x ) | z | r − 2 z + h ( x , z ) , z ( x ) → 0 , a s | x | → ∞ , 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.

Abstract

We deal with the following elliptic equations: { − div ( φ ′ ( | ∇ z | 2 ) ∇ z ) + V ( x ) | z | α − 2 z = λ ρ ( x ) | z | r − 2 z + h ( x , z ) , z ( x ) → 0 , a s | x | → ∞ , 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.

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