Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

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

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“…By elliptic regularity, we can show that w λ belongs to C 2,γ (Ω) for some γ ∈ (0, 1). This proves (1)- (2).…”

Section: The Exponential Type Semilinear Elliptic Equation In Rsupporting

confidence: 48%

“…From properties (5) and (6) in Theorem 4.1, we get two different situations determined by the asymptotic behaviour of f . For the detailed microscopic blow-up analysis of u λ see [2] and [4], where more general cases are considered.…”

Section: Proof Of Claimmentioning

confidence: 99%

Isolated singularities for the exponential type semilinear elliptic equation in ℝ²

Dhanya

Giacomoni²,

Prashanth

2009

Proc. Amer. Math. Soc.

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“…[1][2][3]7])) as it deals with general bounded sequences rather than with critical sequences of specific functionals. Provided that a suitable extension of transformations (1.6) to non-radial functions is found, an analysis based such transformations appears to be more promising than the study of blow-ups based on the ination on the linear scale.…”

mentioning

confidence: 99%

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

Adimurthi

Tintarev

2010

Nonlinear Differ. Equ. Appl.

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Abstract. The paper studies quasilinear elliptic problems in the Sobolev spaces W 1,p (Ω), Ω ⊂ R N , with p = N , that is, the case of Pohozhaev-Trudinger-Moser inequality. Similarly to the case p < N where the loss of compactness in W 1,p (R N ) occurs due to dilation operators u → t (N −p)/p u(tx), t > 0, and can be accounted for in decompositions of the type of Struwe's "global compactness" and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W 1,N 0 over a ball in R N . We give a one-parameter scale of Hardy-Sobolev functionals, a "p = N "-counterpart of the Hölder interpolation scale, for p > N, between the Hardy functional |u| p |x| p dx and the Sobolev functional |u| pN/(N −mp) dx. Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W 1,N -norms under HardySobolev constraints. Mathematics Subject Classification (2000). Primary 35J20, 35J35, 35J60; Secondary 46E35, 47J30, 58E05.

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“…In fact the existence and multiplicity question seems much more difficult than its critical Sobolev exponent counterpart. Some results are known: From the result in [3], it follows that there is a λ 0 > 0 such that a solution to (1.1) exists whenever 0 < λ < λ 0 (this is in fact true for a larger class of nonlinearities with critical exponential growth). By construction this solution falls, as λ → 0, into the bubbling category (1.4) with k = 1.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 95%

New solutions for Trudinger–Moser critical equations in R2

Pino

Musso

Ruf

2010

Journal of Functional Analysis

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Let Ω be a bounded, smooth domain in R 2 . We consider critical points of the Trudinger-Moser type functional J λ (u) = 1 2 Ω |∇u| 2 − λ 2 Ω e u 2 in H 1 0 (Ω), namely solutions of the boundary value problem u + λue u 2 = 0 with homogeneous Dirichlet boundary conditions, where λ > 0 is a small parameter. Given k 1 we find conditions under which there exists a solution u λ which blows up at exactly k points in Ω as λ → 0 and J λ (u λ ) → 2kπ. We find that at least one such solution always exists if k = 2 and Ω is not simply connected. If Ω has d 1 holes, in addition d + 1 bubbling solutions with k = 1 exist. These results are existence counterparts of one by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217-269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth, including this one as a prototype case.

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Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

Cited by 23 publications

References 14 publications

Isolated singularities for the exponential type semilinear elliptic equation in ℝ²

Isolated singularities for the exponential type semilinear elliptic equation in ℝ²

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

New solutions for Trudinger–Moser critical equations in R2

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