Semilinear elliptic equations and supercritical growth

Budd, C. J.; Norbury, John W.

doi:10.1016/0022-0396(87)90190-2

Cited by 76 publications

(81 citation statements)

References 11 publications

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“…Budd and Norbury in [7] considered (1) for p > N +2 N −2 , and formally derived qualitative properties of this bifurcation branch. In particular, formal asymptotics and numerical computations suggest that the following takes place: Before reaching λ = 0, the curve turns right and then oscillates infinitely many times in the form of an exponentially damped sinusoidal along a line λ = λ * .…”

Section: Introductionmentioning

confidence: 99%

Geometry of phase space and solutions of semilinear elliptic equations in a ball

Dolbeault

Flores

2007

Trans. Amer. Math. Soc.

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Abstract. We consider the problem (1)where B denotes the unit ball in R N , N ≥ 3, λ > 0 and p > 1. Merle and Peletier showed that for p >there is a unique value λ = λ * > 0 such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionallythen for λ close to λ * , a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if λ = λ * . We establish a similar assertion for the problemwhere f (s) = s p + s q , 1 < q < p, and p satisfies the same condition as above.

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Section: Introductionmentioning

confidence: 99%

Geometry of phase space and solutions of semilinear elliptic equations in a ball

Dolbeault

Flores

2007

Trans. Amer. Math. Soc.

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“…We first claim that the maximum of a solution u(x) must occur in [1,4). Suppose the contrary and u(x) attains its maximum at p < 1.…”

Section: Further Examples and Extensionsmentioning

confidence: 95%

“…Hence our result here applies to both (3.1) and (3.2). It is known from a result of Ni and Nussbaum [20] (see also Budd and Norbury [4]) that in the remaining case of a positive constant q(t) and supercritical p, uniqueness is no longer valid. The expression in (3.6) satisfies the V-property instead.…”

Section: Dirichlet Boundary Condition (33)mentioning

confidence: 99%

Uniqueness of Radial Solutions of Semilinear Elliptic Equations

Kwong¹,

Li²

1992

Transactions of the American Mathematical Society

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Abstract.E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one finite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coefficients. One particular example covered is Am + up ± u = 0, with p > 1 . The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong. Yanagida's proof does not extend to solutions of Matukuma's equation satisfying other boundary conditions. We treat these with a completely different method of Kwong and Zhang.

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“…Moreover, if d ≥ 2 and Ω is an annulus, then the solution is unique in the class of positive radial solutions (see [15]). However, there are cases in which the solution is not unique, see; for example, [16,15].…”

Section: Existence Of Positive Classical Solutionsmentioning

confidence: 99%

“…We find an extra difficulty, due to the presence of the finite extinction time. We start by recalling the result of [7], where the following rather peculiar property of the solutions of problem (2.28) is found as a consequence of the global Harnack principle on domains, 16) valid for an R > 0, so small that the box (6.17) but again the box depends on the positivity value of u in the point (t 0 ,x 0 ). It resembles our elliptic Harnack inequality, but again it has to be supported by a positivity result to hold in full generality.…”

Section: Harnack Inequalities For Fde On a Domainmentioning

confidence: 99%

Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

Bonforte

Vázquez

2007

Boundary Value Problems

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We investigate local and global properties of positive solutions to the fast diffusion equation u t = Δu m in the good exponent range (d − 2) + /d < m < 1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space R d , we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of R d with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.

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Semilinear elliptic equations and supercritical growth

Cited by 76 publications

References 11 publications

Geometry of phase space and solutions of semilinear elliptic equations in a ball

Geometry of phase space and solutions of semilinear elliptic equations in a ball

Uniqueness of Radial Solutions of Semilinear Elliptic Equations

Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

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