scholarly journals A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OFp-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES

2009 ◽  
Vol 51 (3) ◽  
pp. 513-524 ◽  
Author(s):  
NGUYEN THANH CHUNG ◽  
QUỐC ANH NGÔ
Keyword(s):  
Bounded Domain ◽  
Multiplicity Result ◽  
Positive Parameter ◽  
Existence And Multiplicity ◽  
Carathéodory Function ◽  
Image Position

AbstractUsing variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the formwhere Ω is a bounded domain inN,N≧ 3,fis a sign-changing Carathéodory function on Ω × [0, +∞) and λ is a positive parameter.

Existence of solution for quasilinear equations involving local conditions

2019 ◽  
Vol 150 (6) ◽  
pp. 3074-3086
Author(s):  
Patricio Cerda ◽  
Leonelo Iturriaga
Keyword(s):  
Continuous Function ◽  
Nonlinear Term ◽  
Quasilinear Equation ◽  
Existence Of Solution ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Local Conditions ◽  
Existence Of Weak Solutions ◽  
Carathéodory Function ◽  
Image Position

AbstractIn this paper, we study the existence of weak solutions of the quasilinear equation \begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

2020 ◽  
pp. 1-38 ◽  
Author(s):  
Qilin Xie ◽  
Xu Zhang
Keyword(s):  
Variational Method ◽  
Critical Frequency ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Classical State ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

Abstract In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

An indefinite and critical concave—convex-type equation

2013 ◽  
Vol 143 (1) ◽  
pp. 169-184 ◽  
Author(s):  
Humberto Ramos Quoirin ◽  
Pedro Ubilla
Keyword(s):  
Bounded Domain ◽  
Type Equation ◽  
Nehari Manifold ◽  
Real Parameter ◽  
Linear Case ◽  
Energy Estimates ◽  
Multiplicity Result ◽  
Image Position

We analyse the multiplicity of non-negative solutions for the critical concave—convex-type equationwhere Ω is a bounded domain of ℝN, 1 < r < p and λ > 0, and μ is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when λ is positive and sufficiently small and μ is in a right neighbourhood of μ1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2.

Half-eigenvalues of elliptic operators

2002 ◽  
Vol 132 (6) ◽  
pp. 1439-1451 ◽  
Author(s):  
Bryan P. Rynne
Keyword(s):  
Differential Operator ◽  
Bounded Domain ◽  
Trivial Solution ◽  
Partial Differential Operator ◽  
Elliptic Operators ◽  
Second Order ◽  
Partial Differential ◽  
Carathéodory Function ◽  
Image Position ◽  
Semilinear Problem

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains

1987 ◽  
Vol 105 (1) ◽  
pp. 205-213 ◽  
Author(s):  
Donato Fortunato ◽  
Enrico Jannelli
Keyword(s):  
Boundary Value Problem ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Boundary Value ◽  
Sobolev Embedding ◽  
Infinitely Many Solutions ◽  
Positive Parameter ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Image Position

SynopsisWe consider the boundary value problemwhere Ω ⊂ ℝn is a bounded domain, n≧3, 2* = 2n/(n − 2) is the critical exponent for the Sobolev embedding and λ is a real positive parameter. We prove the existence of infinitely many solutions of (*) when Ω exhibits suitable symmetries.

Multiplicity of solutions to the weighted critical quasilinear problems

2012 ◽  
Vol 55 (1) ◽  
pp. 181-195 ◽  
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Bounded Domain ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Positive Parameter ◽  
Symmetric Mountain Pass Theorem ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Quasilinear Problems ◽  
Image Position

AbstractWe consider a class of critical quasilinear problemswhere 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

scholarly journals UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

2013 ◽  
Vol 55 (2) ◽  
pp. 399-409 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problems \[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \] where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, λ is a positive parameter and f:(0,∞) → (0,∞) is sublinear at ∞ and is allowed to be singular at 0.

A multiplicity result including a sign-changing solution for an inhomogeneous Neumann problem with critical exponent

2007 ◽  
Vol 137 (2) ◽  
pp. 333-347 ◽  
Author(s):  
Norimichi Hirano ◽  
Naoki Shioji
Keyword(s):  
Critical Exponent ◽  
Neumann Problem ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Normal Derivative ◽  
Multiplicity Result ◽  
Image Position

We study the existence of multiple solutions for the problemand we show that at least one of them is sign changing. Here Ω is a bounded domain in ℝN with N ⩾ 5 whose boundary is of class C2, ∂/∂ν is the outward normal derivative, and f ∈ LN/2(Ω) whose L2N/(N+2)(Ω) norm is sufficiently small.

An existence result on positive solutions for a class of semilinear elliptic systems

2004 ◽  
Vol 134 (1) ◽  
pp. 137-141 ◽  
Author(s):  
D. D. Hai ◽  
R. Shivaji
Keyword(s):  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Existence Result ◽  
Elliptic Systems ◽  
Positive Parameter ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Sign Conditions ◽  
Image Position

Consider the system where λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

2015 ◽  
Vol 58 (2) ◽  
pp. 461-469 ◽  
Author(s):  
D. D. HAI ◽  
R. C. SMITH
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Positive Parameter ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
Formula Group ◽  
Image Position

AbstractWe prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

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