Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain |
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Authors: | Caisheng Chen Lifang Shao |
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Institution: | College of Science, Hohai University, , Nanjing 210098, China |
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Abstract: | In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ?Ω, 1 < p < N,0 ≤ a < (N ? p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N ? pd),d = a + 1 ? b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions uk under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | ? bq, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions uk with 1 < r < p < q and J(uk) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1 . Copyright © 2013 John Wiley & Sons, Ltd. |
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Keywords: | singular quasilinear elliptic equation variational methods concave and convex nonlinearities exterior domain concentration– compactness principle |
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