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Existence and multiplicity of solutions for equations of p(x)-laplace type in rN without AR-condition
- Existence and multiplicity of solutions for equations of p(x)-laplace type in rN without AR-condition
- Kim J.-M.; Kim Y.-H.; Lee J.
- Ewha Authors
- Issue Date
- Journal Title
- Differential and Integral Equations
- Differential and Integral Equations vol. 31, pp. 435 - 464
- Khayyam Publishing
- SCIE; SCOPUS
- Document Type
- We are concerned with the following elliptic equations with variable exponents −div(ϕ(x,∇u)) + V(x)
p(x)−2u = λf(x,u) in RN, where the function ϕ(x,v) is of type
p(x)−2v with continuous function p : RN → (1,∞), V : RN → (0,∞) is a continuous potential function, and f : RN×R → R satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of λ’s for which our problem admits a nontrivial solution with simple assumptions in some sense. © 2018 Khayyam Publishing. All Rights Reserved.
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