Isolated singularities for the exponential type semilinear elliptic equation in ℝ²

Abstract

Abstract. In this article we study positive solutions of the equation −∆u = 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. IntroductionLet Ω ⊂ R 2 be a bounded domain with 0 ∈ Ω. Denote Ω = Ω \ {0}. Let f : (0, ∞) −→ (0, ∞) be a locally Hölder continuous function which is nondecreasing for all large t > 0. In this article we study the following problem:It is well-known from the works of Brezis-Lions [5] that if u solves (P ), then indeed u solves the following problem in the distributional sense in the whole domain Ω:. This leads us to the following two questions: (Q1) Can we find a sharp condition on f that determines whether or not α = 0 in (P α )?(Q2) If α = 0, is it true that u is regular (say, C 2 ) in Ω?We make the following Definition 1.1. We say f is a sub-exponential type function ifWe say f is of super-exponential type if it is not a sub-exponential type function. As a complete answer to question (Q1) we show (Theorem 2.1) that if f is of super-exponential type, then α = 0, and conversely (Theorem 2.2) that (P α ) has solutions for small α > 0 if f is of sub-exponential type.Similarly, we answer question (Q2) by showing that for any f of sub-exponential type, any solution u of (P 0 ) is regular(C 2 ) inside Ω (Theorem 3.1). Conversely, for f of super-exponential type with any prescribed growth at ∞ and behaviour for small t > 0, in Lemma 3.1 and Theorem 3.3 we construct solutions u of (P 0 ) that blow-up only at the origin. To our knowledge, the existence of such singular solutions has not been considered so far for super-exponential type problems. Theorem 3.2 should be contrasted with the results in [2] and [13]. Particularly in [13], the nonlinearity under study is of a model type, viz., f (t) = e t µ , µ > 0. These authors show

Radial singular solutions for the N-Laplace equation with exponential nonlinearities

“…In the present paper, we extend the results contained in [9] to the radial quasilinear case and obtain a complete answer to questions (Q1) and (Q2) above. We also provide a partial answer to (Q3).…”

“…When N = 2 and f has at most a polynomial growth, the first two questions (Q1) and (Q2) were discussed in detail in Brezis-Lions [6] and Lions [18]. For a corresponding discussion involving exponential growth nonlinearities in the case N = 2, we refer to Dhanya-Giacomoni-Prashanth [9].…”

Isolated singularities of polyharmonic operator in even dimension

“…For the dimension N ≥ 3, P.L.Lions [10] found a sharp condition on f that determines whether α is zero or not in the previous expression. In [5], the authors further extended the result for dimension N = 2 by finding the minimal growth rate of the function f which guranteed α to be 0.…”

Isolated Singularities of Polyharmonic Operator in Even Dimension

“…For the dimension N ≥ 3, P.L.Lions [10] found a sharp condition on f that determines whether α is zero or not in the previous expression. In [5], the authors further extended the result for dimension N = 2 by finding the minimal growth rate of the function f which guranteed α to be 0. Taliaferro, in his series of papers (see for e.g.…”