Elliptic equations in R2 with nonlinearities in the critical growth range

Figueiredo, Djairo G. de; Miyagaki, Olı́mpio H.; Ruf, Bernhard

doi:10.1007/bf01205003

Cited by 366 publications

(73 citation statements)

References 4 publications

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“…Multiplicity results for solutions of nonlinear boundary value problems, or systems, in bounded domains of R 2 in presence of an exponential nonlinearity have been faced many times (see [2,4,5] for bifurcation results related to (1) in the whole of R 2 , [1] and [19] for uniqueness results in the ball, [6] for multiplicity results for more general operators). In particular, we focus on the recent paper [21]: denote by λ 1 < λ 2 ≤ λ 3 .…”

Section: Introductionmentioning

confidence: 99%

Four nontrivial solutions for subcritical exponential equations

Mugnai

2008

Calc. Var.

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Section: Introductionmentioning

confidence: 99%

Four nontrivial solutions for subcritical exponential equations

Mugnai

2008

Calc. Var.

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“…In order to prove that a Palais-Smale sequence converges to a solution of problem (P) we need the following convergence Lemma. We refer to Lemma 2.1 in [15] for a proof.…”

Section: Existence Of Solutionmentioning

confidence: 99%

“…The critical exponent problems with exponential type nonlinearities, motivated by Moser-Trudinger inequality [30], in the limiting cases was initially studied by Adimurthi [2] and later by several authors [15,31,32] and [28]. These problems with singular exponential growth nonlinearities for Laplacian and n-Laplacian was studied in [3], where an interpolation inequality of Hardy and Moser-Trudinger inequality (see [4] also) is proved for W 1,n 0 (Ω).…”

Section: Introductionmentioning

confidence: 99%

Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

Mishra¹,

Goyal²,

Sreenadh³

2016

CPAA

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In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity −M Ω |∇ m u| n m dx ∆ m n m u = f (x,u) |x| α in Ω, u = ∇u = • • • = ∇ m−1 u = 0 on ∂Ω, where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < α < n, n ≥ 2m ≥ 2 and f (x, u) behaves like e |u| n n−m as |u| → ∞. Using mountain pass structure and the concentration compactness principle, we show the existence of a nontrivial solution. In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using the Nehari manifold technique, we show the existence and multiplicity of nonnegative solutions.

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“…In this article, we consider the following class of quasilinear problem:

\{\begin{matrix} left - ε^{N} Δ_{N} u + (1 + μ A false(x false)) {false| u false|}^{N - 2} u = f (u) in R^{N} (N \geq 2), \\ left u > 0 in R^{N}, \end{matrix} false(P_{μ, ε} false)

where

{normalΔ}_{N} u = d i v ({(||, \nabla, u)}^{N - 2} \nabla u)

is the N ‐Laplacian operator, μ and ε are positive parameters. We assume that the nonlinear term f is a function with exponential critical growth in the sense of Pohozaev–Trudinger–Moser inequality (see and ), more precisely: ( H 0 )There exists

α_{0} > 0

such that

\underset{|s| \to \infty}{trueprefixlim} \frac{(|| f (s))}{e^{α {|s|}^{\frac{N}{N - 1}}}} = \{\begin{matrix} left 0, & left if & left α > α_{0}, \\ left + \infty, & left if & left α < α_{0} . \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Multiplicity results for a class of quasilinear equations with exponential critical growth

Alves

Freitas

2017

Mathematische Nachrichten

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In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equationswhere ∆ N is the N-Laplacian operator, N ≥ 2, f is a function with exponential critical growth, µ and ǫ are positive parameters and A is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments and LusternikSchnirelman category theory .

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Elliptic equations in R2 with nonlinearities in the critical growth range

Cited by 366 publications

References 4 publications

Four nontrivial solutions for subcritical exponential equations

Four nontrivial solutions for subcritical exponential equations

Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

Multiplicity results for a class of quasilinear equations with exponential critical growth

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