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In this paper, we study the global existence, L estimates and decay estimates of solutions for the quasilinear parabolic system ut = div (|∇ u|mu) + f(u, v), vt = div (|∇ v|mv) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ RN. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation ut = div (|∇ u|mu) + λ |u|α - 1 u.  相似文献
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In this paper, we consider the following quasilinear elliptic exterior problem
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In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u0(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a0>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a0>0 in Ω, and A is bounded in if a(x)?0 in Ω.  相似文献
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In this paper, we study the nonexistence result for the weighted Lane–Emden equation: (0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: (0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, is an upper half space in , and . Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems 0.1 and 0.2 under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.  相似文献
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In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh1(x)|u|m − 2u + h2(x)|u|q − 2u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h1(x),h2(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ0) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h2|u|q − 2u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h1|u|m − 2u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献
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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 < (Np) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (Npd),d = a + 1 − b,ab < 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.  相似文献
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In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the quasilinear p‐Laplacian system with the nonlinear boundary conditions: (0.1) where Ω is a smooth exterior domain in is the outward normal derivative on the boundary Γ = Ω, and . By the Nehari manifold and variational methods, we prove that the problem (0.1) has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ) belongs to a certain subset of . Copyright © 2012 John Wiley & Sons, Ltd.  相似文献
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