Multiplicity of positive solutions for p-Laplace equation with superlinear-type nonlinearity

Prashanth, S.; Sreenadh, K.

doi:10.1016/j.na.2003.10.026

Cited by 12 publications

(12 citation statements)

References 9 publications

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“…When p = 2, the related results have been obtained by [11][12][13][14][15][16] (including bounded domains with zero Dirichlet condition or R N ). Our existence 2 Boundary Value Problems results extend that of Brezis and Kamin (see [11,Theorem 1]) for semilinear problem, and complement results in [3][4][5][6][7][8][9][10].…”

supporting

confidence: 82%

“…Problem (1) for bounded domains with zero Dirichlet condition has been extensively studied (even for more general sublinear functions). We refer in particular to [3][4][5][6][7][8][9][10] (see also the references therein). When p = 2, the related results have been obtained by [11][12][13][14][15][16] (including bounded domains with zero Dirichlet condition or R N ).…”

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confidence: 99%

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Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Yang

2007

Boundary Value Problems

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We consider the problem −div(|∇u|In this work, we are interested in studying the existence of solutions to the following quasilinear equation:is not identically zero. We will assume throughout the paper that a(x) ∈ C(R N ). Equations of the above form are mathematical models occuring in studies of the p-Laplace equation, generalized reaction-diffusion theory [1], non-Newtonian fluid theory, and the turbulent flow of a gas in porous medium [2]. In the non-Newtonian fluid theory, the quantity p is characteristic of the medium. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. Problem (1) for bounded domains with zero Dirichlet condition has been extensively studied (even for more general sublinear functions). We refer in particular to [3][4][5][6][7][8][9][10] (see also the references therein). When p = 2, the related results have been obtained by [11][12][13][14][15][16] (including bounded domains with zero Dirichlet condition or R N ). Our existence

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confidence: 82%

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confidence: 99%

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Yang

2007

Boundary Value Problems

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“…This was the approach taken in [26]. In this case, due to well-known symmetry results (see [10,12]), every solution of (1.3) posed on Ω = B 1 (0) is radially symmetric about the origin and radially nonincreasing.…”

Section: Introductionmentioning

confidence: 99%

“…[26] Let Ω = B 1 (0), 1 < p < N, 0 < q < p − 1. Then there exists Λ > 0 such that (1.3) admits at least two solutions for all λ ∈ (0, Λ) and no solution for λ > Λ. Additionally, if 1 < p < 2, then (1.3) admits exactly two solutions for all small λ > 0.…”

Section: Introductionmentioning

confidence: 99%

“…Then, (i) there exists Λ > 0 such that (P λ ) admits at least two solutions for all λ ∈ (0, Λ), say u λ and v λ , one solution for λ = Λ and no solutions for λ > Λ; To our knowledge, even when N = 2 only few uniqueness results are known concerning the class of problems like (P λ ) and these are for the cases where f (t) is C 1 near t = 0 and grows like a power function (see Corollary 2.32 and Section 2.4 in [25] and [3]) or behaves like te t (see [2] and the extension for more general nonlinearities in [29]). At this point, we want to stress that Pohozaev Identities which are used to prove uniqueness or exact number of solutions when N 3 (see for instance [5,16,26,28]) do not work when N = 2. Therefore, the case N = 2 is of independent interest and we restate the above uniqueness theorem in this case as:…”

Section: Introductionmentioning

confidence: 99%

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A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type

Giacomoni¹,

Prashanth²,

Sreenadh³

2007

Journal of Differential Equations

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Let Ω ⊂ R N , N 2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ > 0:where f (t) is a nonlinearity that grows like e t N/N−1 as t → ∞ and behaves like t α , for some α ∈ (0, N − 1), as t → 0 + . More precisely, we require f to satisfy assumptions (A1)-(A5) in Section 1. With these assumptions we show the existence of Λ > 0 such that (P λ ) admits at least two solutions for all λ ∈ (0, Λ), one solution for λ = Λ and no solution for all λ > Λ. We also study the problem (P λ ) posed on the ball B 1 (0) ⊂ R N and show that the assumptions (A1)-(A5) are sharp for obtaining global multiplicity. We use a combination of monotonicity and variational methods to show multiplicity on general domains and asymptotic analysis of ODEs for the case of the ball.

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Lyapunov‐type inequalities for singular elliptic partial differential equations

Kumar

Tyagi

2020

Math Methods in App Sciences

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We establish Lyapunov‐type inequalities for a class of singular elliptic partial differential equations. As an application of Lyapunov‐type inequalities, we obtain lower bounds for the first eigenvalue of the associated singular elliptic partial differential equations. We also make an attempt to answer a question raised by de Nápoli and Pinasco in J. Funct. Anal. 270 (2016), no. 6, 1995–2018.

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Multiplicity of positive solutions for p-Laplace equation with superlinear-type nonlinearity

Cited by 12 publications

References 9 publications

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type

Lyapunov‐type inequalities for singular elliptic partial differential equations

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