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We demonstrate that the phase space of the Cauchy-Dirichlet problem for the equation u_t - xΔu_t = vΔu - K(u) + f is a simple Banach C~∞-manifold.
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Boundary value problems for degenerate semilinear elliptic pseudodifferential operators are considered. Using the apparatus of the theory of pseudodifferential operators, function spaces introduced in [4-8], and the Rabinowitz con...
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Boundary value problems for degenerate semilinear elliptic pseudodifferential operators are considered. Using the apparatus of the theory of pseudodifferential operators, function spaces introduced in [4-8], and the Rabinowitz construction [12] based on the Borsuk theorem (see [11]), we prove the existence of solutions of the problem in suitable function spaces.
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The paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal, where imbedding of the energy space into L~p is not compact. In absence of invariant transformations involved in conventiona...
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The paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal, where imbedding of the energy space into L~p is not compact. In absence of invariant transformations involved in conventional concentration-compactness argument, the paper develops convergence reasoning based on the fractal's self-similarity.
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Let Omega be an open bounded domain in R-N( N = 3) with smooth boundary a Omega, 0 epsilon a Omega. We are concerned with the critical Neumann problem {-Delta u-u u/vertical bar x vertical bar 2 + lambda u = Q (x) vertical bar u v...
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Let Omega be an open bounded domain in R-N( N = 3) with smooth boundary a Omega, 0 epsilon a Omega. We are concerned with the critical Neumann problem {-Delta u-u u/vertical bar x vertical bar 2 + lambda u = Q (x) vertical bar u vertical bar(2*-2)u in Omega, au/av =0 on a Omega where 0 < u < u = (N-2/2)(2), 2* = 2N/N-2, lambda > 0 and Q(x) is a positive continuous function on ohm. Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions on Q, mu, we, by means of a variational method, prove that there exists lambda 0 = lambda 0(mu) > 0 such that for every lambda > lambda 0, problem (*) has a positive solution and a pair of sign-changing solutions.
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We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the up...
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We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. In these spaces of fractional order we obtain a weighted Sobolev embedding and a nonlinear estimate. Using these, we establish the decay estimate of the solution for large time provided the power of nonlinearity is greater than a critical value.
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This paper is concerned with the existence and nonexistence of positive solutions for the nonlinear integral equations with weights related to the sharp Hardy-Littlewood-Sobolev (hereinafter referred to as HLS) inequality on bound...
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This paper is concerned with the existence and nonexistence of positive solutions for the nonlinear integral equations with weights related to the sharp Hardy-Littlewood-Sobolev (hereinafter referred to as HLS) inequality on bounded domains of the Heisenberg group H-n: f(q-1)(xi) = integral(Omega) G(xi)f(eta)G(eta)/vertical bar eta(-1) xi vertical bar(Q-alpha) d eta + lambda integral(Omega) f(eta)/vertical bar eta(-1) xi vertical bar(Q-alpha-beta) d eta, xi is an element of(Omega) over bar, where q > 1, 0 < alpha < Q, 0 < beta < Q - alpha, Q = 2n + 2 is the homogeneous dimension of H-n, lambda is an element of R, Omega subset of H-n is a smooth bounded domain and G(xi) is nonnegative continuous in (Omega) over bar.
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We study the existence, smoothing properties and the long time behaviour for a class of nonlinear Cauchy problems in infinite dimensions under the assumption of F-Sobolev inequalities.
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In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in...
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In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in $\mathbb{R}^N$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].
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In this paper, we study the local behavior of a positive singular solution u near its singular points of the following equation:{Deltau(x)+d(x, Z)(2N)u n+2/n-2 = 0 in Omega\Z,{u(x)>0 and uis an element of C-2 in Omega\Z,where N is...
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In this paper, we study the local behavior of a positive singular solution u near its singular points of the following equation:{Deltau(x)+d(x, Z)(2N)u n+2/n-2 = 0 in Omega\Z,{u(x)>0 and uis an element of C-2 in Omega\Z,where N is a positive integer, Omega is a bounded open domain in R-n, Z is a finite set of points, and d(x, Z) denotes the distance between x and Z. (C) 2004 Elsevier Inc. All rights reserved.
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Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in R~n (n ≥ 3), we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain Ω, μ_s(Ω): = inf{∫...
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Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in R~n (n ≥ 3), we show that the value and the attainability of the best Hardy-Sobolev constant on a smooth domain Ω, μ_s(Ω): = inf{∫_Ω |▽u|~2 dx; u ∈ H_0~1(Ω) and ∫_Ω (|u|~2~*(s))/(|x|~s) = 1} when 0 < s < 2, 2~*(s) = (2(n-s))/(n-2), and when 0 is on the boundary Ω are closely related to the properties of the curvature of Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form: -Δu = (u~(p-1))/(|x|~s) + f(x,u) in Ω is contained in R~n, where f is a lower order perturbative term at infinity and f(x,0) = 0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.
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