Isolated singularities of polyharmonic operator in even dimension
TL;DR: In this paper, the authors considered the problem of finding the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma), where a is a nonnegative measurable function in some Lebesgue space.
Abstract: We consider the equation Delta(2)u = g(x, u) >= 0 in the sense of distribution in Omega' = Omega\textbackslash {0} where u and -Delta u >= 0. Then it is known that u solves Delta(2)u = g(x, u) + alpha delta(0) - beta Delta delta(0), for some nonnegative constants alpha and beta. In this paper, we study the existence of singular solutions to Delta(2)u = a(x) f (u) + alpha delta(0) - beta Delta delta(0) in a domain Omega subset of R-4, a is a nonnegative measurable function in some Lebesgue space. If Delta(2)u = a(x) f (u) in Omega', then we find the growth of the nonlinearity f that determines alpha and beta to be 0. In case when alpha = beta = 0, we will establish regularity results when f (t) 0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N >= 5) with a specific weight function a(x) = |x|(sigma). Later, we discuss its analogous generalization for the polyharmonic operator.
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TL;DR: In this article, the authors studied nonnegative classical solutions of the polyharmonic inequality and gave necessary and sufficient conditions on integers n⩾2 and m⩽1 such that these solutions u satisfy a pointwise a priori bound as x→0.
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TL;DR: In this paper, the authors studied the growth near the origin of C 2 positive solutions u(x) of where n is a punctured neighborhood of the origin, a ∈ [0, 1] is a constant, an
Abstract: In this paper we continue our study began in [10] of the growth near the origin of C 2 positive solutions u(x) of where Ω ⊂ n is a punctured neighborhood of the origin, a ∈ [0, 1) is a constant, an...
7 citations
15 Jul 2009
TL;DR: In this article, the authors studied positive solutions of the equation -Δu = f (u) in a punctured domain Ω' = Ω \ {0} in ℝ 2 and showed sharp conditions on the nonlinearity f(t) that enables them to extend such a solution to the whole domain and also preserve its regularity.
Abstract: In this article we study positive solutions of the equation -Δu = f (u) in a punctured domain Ω' = Ω \ {0} in ℝ 2 and show sharp conditions on the nonlinearity f(t) that enables us to extend such a solution to the whole domain Ω and also preserve its regularity. We also show, using the framework of bifurcation theory, the existence of at least two solutions for certain classes of exponential type nonlinearities.
6 citations
TL;DR: In this article, the Dirichlet problem admits a solution in some particular cases of the nonlinearities f and g. No assumptions on the sign of the functions f and G are required.
Abstract: We give general existence results of solutions ( u , v ) to the Dirichlet problem (P) { − Δ u = f ( x , u , v ) + c δ 0 , − Δ v = g ( x , u , v ) + d δ 0 in D ′ ( B ) , u = v = 0 on ∂ B , where B is the unit ball centered at zero in R N , N ≥ 3 , δ 0 is the Dirac mass at 0 and c , d are nonnegative constants. No assumptions on the sign of the functions f and g are required. We also characterize the set of ( c , d ) such that problem (P) admits a solution in some particular cases of the nonlinearities f and g .
2 citations