Existence of Multiple Positive Solutions for N-Laplacian in a Bounded Domain in ℝN

2005 ◽  
Vol 5 (1) ◽  
Author(s):  
S. Prashanth ◽  
K. Sreenadh
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Exponential Type ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Multiple Positive Solutions ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic

AbstractLet Ω be a bounded domain in ℝIn this article we show the existence of at least two positive solutions for the following quasilinear elliptic problem with an exponential type nonlinearity:We use Monotonicity and Variational methods to obtain this multiplicity result.

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

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
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.

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

2015 ◽  
Vol 17 (02) ◽  
pp. 1450029 ◽  
Author(s):  
Silvia Cingolani ◽  
Giuseppina Vannella ◽  
Daniela Visetti
Keyword(s):  
Riemannian Manifold ◽  
Positive Solutions ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Quasilinear Equations ◽  
Metric Tensor ◽  
Poincaré Polynomial ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Problem ◽  
Quasilinear Elliptic

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of [Formula: see text] nonconstant solutions u ∈ H1,p(M) to the quasilinear problem [Formula: see text] for ε > 0 small enough, where 2 ≤ p < n, p < q < p*, p* = np/(n - p) and [Formula: see text] is the p-laplacian associated to g of u (note that Δ2,g = Δg) and [Formula: see text] denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).

scholarly journals EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL HARDY–SOBOLEV EXPONENTS AND SIGN-CHANGING WEIGHT FUNCTION

2012 ◽  
Vol 17 (3) ◽  
pp. 330-350 ◽  
Author(s):  
Nemat Nyamoradi
Keyword(s):  
Weight Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Sobolev Exponent ◽  
Quasilinear Elliptic

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Quasilinear elliptic problems with critical exponents and Hardy terms in ℝN

2010 ◽  
Vol 53 (1) ◽  
pp. 175-193 ◽  
Author(s):  
Dongsheng Kang
Keyword(s):  
Variational Methods ◽  
Elliptic Problem ◽  
Critical Exponents ◽  
Ground State ◽  
Elliptic Problems ◽  
Ground State Solutions ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic

AbstractWe deal with a singular quasilinear elliptic problem, which involves critical Hardy-Sobolev exponents and multiple Hardy terms. Using variational methods and analytic techniques, the existence of ground state solutions to the problem is obtained.

scholarly journals AN EIGENVALUE PROBLEM FOR A NON-BOUNDED QUASILINEAR OPERATOR

2004 ◽  
Vol 47 (2) ◽  
pp. 353-363 ◽  
Author(s):  
José Carmona ◽  
Antonio Suárez
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Bifurcation Theory ◽  
Bifurcation Parameter ◽  
Quasilinear Elliptic Problem ◽  
Existence Of Positive Solutions ◽  
Positive Eigenfunction ◽  
Quasilinear Elliptic ◽  
Range Of Values

AbstractIn this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with an operator that is not necessarily bounded. For that, we use the bifurcation theory and obtain the existence of positive solutions for a range of values of the bifurcation parameter.AMS 2000 Mathematics subject classification: Primary 35J60; 35J25. Secondary 35D05

Existence of positive solutions for a class of p-Laplacian superlinear semipositone problems

2015 ◽  
Vol 145 (5) ◽  
pp. 925-936 ◽  
Author(s):  
M. Chhetri ◽  
P. Drábek ◽  
R. Shivaji
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
A Priori ◽  
Degree Theory ◽  
Connected Component ◽  
Quasilinear Elliptic Problem ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Quasilinear Elliptic

We consider a quasilinear elliptic problem of the formwhere λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

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