Multiple positive solutions for discrete p-Laplacian equations with potential term

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.

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MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

Communications in Contemporary Mathematics ◽

10.1142/s0219199703001105 ◽

2003 ◽

Vol 05 (05) ◽

pp. 737-759 ◽

Cited By ~ 4

Author(s):

NOBUYOSHI FUKAGAI ◽

KIMIAKI NARUKAWA

Keyword(s):

Positive Solutions ◽

Elliptic Problem ◽

Eigenvalue Problems ◽

Multiple Positive Solutions ◽

Combined Effects ◽

Nonlinear Eigenvalue Problems ◽

Quasilinear Elliptic Problem ◽

Quasilinear Elliptic ◽

Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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Unbounded principal eigenfunctions and the logistic equation on RN

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700037229 ◽

2003 ◽

Vol 67 (3) ◽

pp. 413-427 ◽

Cited By ~ 7

Author(s):

Wei Dong ◽

Yihong Du

Keyword(s):

Positive Solution ◽

Positive Solutions ◽

Eigenvalue Problems ◽

Growth Conditions ◽

Logistic Equation ◽

Exponential Rate ◽

Unique Positive Solution ◽

Unbounded Coefficients ◽

Growth Restrictions

We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

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Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

Nonlinear Analysis ◽

10.1016/j.na.2007.10.053 ◽

2008 ◽

Vol 69 (12) ◽

pp. 4286-4300 ◽

Cited By ~ 18

Author(s):

Shouchuan Hu ◽

Nikolas S. Papageorgiou

Keyword(s):

Positive Solutions ◽

Eigenvalue Problems ◽

Multiple Positive Solutions ◽

Nonlinear Eigenvalue Problems ◽

Nonlinear Eigenvalue

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On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/s10231-006-0018-x ◽

2006 ◽

Vol 186 (3) ◽

pp. 539-564 ◽

Cited By ~ 56

Author(s):

Nobuyoshi Fukagai ◽

Kimiaki Narukawa

Keyword(s):

Positive Solutions ◽

Eigenvalue Problems ◽

Multiple Positive Solutions ◽

Quasilinear Elliptic ◽

Elliptic Eigenvalue Problems

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Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0133 ◽

2021 ◽

Vol 11 (1) ◽

pp. 432-453

Author(s):

Qi Han

Keyword(s):

Elliptic Equation ◽

Positive Solutions ◽

Fractional Laplacian ◽

Linear Growth ◽

Critical Sobolev Exponent ◽

Compact Embedding ◽

Multiple Positive Solutions ◽

Measurable Functions ◽

Sobolev Exponent ◽

Laplacian Equations

Abstract In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ) s in ℝ n , for n ≥ 2, such as (0.1) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = K ( x ) f ( u ) + u 2 s ⋆ − 1 . $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q ∈ 2 , 2 s ⋆ $q \in\left[2,2_{s}^{\star}\right)$ with 2 s ⋆ := 2 n n − 2 s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$ being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) ( − Δ ) s u + E ( x ) u + V ( x ) u q − 1 = λ K ( x ) u r − 1 , $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces Ms ;q,p (ℝ n ) as well as their associated compact embedding results.

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