scholarly journals Multiplicity of positive solutions for (p,q)-Laplace equations with two parameters

2021 ◽  
pp. 2150008
Author(s):  
Vladimir Bobkov ◽  
Mieko Tanaka
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Critical Curves ◽  
Laplace Equations ◽  
Multiplicity Of Positive Solutions ◽  
Two Parameters ◽  
First Eigenfunction ◽  
Special Classes

We study the zero Dirichlet problem for the equation [Formula: see text] in a bounded domain [Formula: see text], with [Formula: see text]. We investigate the relation between two critical curves on the [Formula: see text]-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighborhoods of the point [Formula: see text], where [Formula: see text] is the first eigenfunction of the [Formula: see text]-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents [Formula: see text] and [Formula: see text].

scholarly journals On Dirichlet problem for fractional p-Laplacian with singular non-linearity

2016 ◽  
Vol 8 (1) ◽  
pp. 52-72 ◽  
Author(s):  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega} , {n>sp} , {s\in(0,1)} , {\lambda>0} , {0<q\leq 1} and {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

scholarly journals Generalized Picone inequalities and their applications to (p,q)-Laplace equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1030-1044
Author(s):  
Vladimir Bobkov ◽  
Mieko Tanaka
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Positive Solutions ◽  
First Eigenvalue ◽  
Resonance Case ◽  
The First Eigenvalue ◽  
Laplace Equations ◽  
Existence And Nonexistence ◽  
Laplace Type ◽  
Picone Inequality

Abstract We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation -\hspace{-0.25em}{\text{&#x0394;}}_{p}u-{\text{&#x0394;}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain \text{&#x03A9;}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where {\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250021 ◽  
Author(s):  
FRANCISCO ODAIR DE PAIVA
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Solution Method ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions

This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem [Formula: see text] where Ω is a bounded domain of ℝN, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)uq + b(x)up and c(x)(1 + u)p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.

scholarly journals Positive Solutions for Second-Order Nonlinear Ordinary Differential Systems with Two Parameters

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Lan Sun ◽  
Yukun An ◽  
Min Jiang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Second Order ◽  
Differential Systems ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Two Parameters

By using fixed-point theorem and under suitable conditions, we investigate the existence and multiplicity positive solutions to the following systems: , where are four positive constants and , , and . We derive two explicit intervals of and , such that the existence and multiplicity of positive solutions for the systems is guaranteed.

On Schrödinger–Poisson systems involving concave–convex nonlinearities via a novel constraint approach

2020 ◽  
pp. 2050048
Author(s):  
Juntao Sun ◽  
Tsung-Fang Wu
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Potential Well ◽  
Concave And Convex Nonlinearities ◽  
Multiplicity Of Positive Solutions ◽  
Two Parameters ◽  
Ps Condition ◽  
Constraint Approach

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: [Formula: see text] where [Formula: see text] are two parameters, [Formula: see text], [Formula: see text] is a potential well, [Formula: see text] and [Formula: see text]. Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on [Formula: see text] due to [Formula: see text]. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.

scholarly journals Positive Solutions for Nonlinear Singular Differential Systems Involving Parameter on the Half-Line

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Lishan Liu ◽  
Ying Wang ◽  
Xinan Hao ◽  
Yonghong Wu
Keyword(s):  
Positive Solutions ◽  
Half Line ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Differential Systems ◽  
Second Order Differential Equations ◽  
The Fixed Point Theorem ◽  
Multiplicity Of Positive Solutions ◽  
Two Parameters ◽  
Special Space

By using the upper-lower solutions method and the fixed-point theorem on cone in a special space, we study the singular boundary value problem for systems of nonlinear second-order differential equations involving two parameters on the half-line. Some results for the existence, nonexistence and multiplicity of positive solutions for the problem are obtained.

scholarly journals On sign-changing solutions for (p,q)-Laplace equations with two parameters

2016 ◽  
Vol 8 (1) ◽  
pp. 101-129 ◽  
Author(s):  
Vladimir Bobkov ◽  
Mieko Tanaka
Keyword(s):  
Dirichlet Problem ◽  
One Dimension ◽  
Linking Theorem ◽  
Nodal Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Domains ◽  
Laplace Equations ◽  
Alpha Beta ◽  
Two Parameters ◽  
Sign Changing Solutions

Abstract We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous {(p,q)} -Laplace equations {-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where {p\neq q} . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the {(\alpha,\beta)} -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.

A result of multiplicity of solutions for a class of quasilinear equations

2012 ◽  
Vol 55 (2) ◽  
pp. 291-309 ◽  
Author(s):  
Claudianor O. Alves ◽  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Category Theory ◽  
Nonlinear Term ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Subcritical Growth ◽  
Minimax Methods ◽  
Multiplicity Of Positive Solutions

AbstractWe establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem−Lpu+ |u|p−2u=h(u)in Ωλ,u= 0 on ∂Ωλ, where Ωλ= λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter,Lpu≐ Δpu+ Δp(u2)uand the nonlinear termh(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

scholarly journals Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoxiao Su ◽  
Ruyun Ma
Keyword(s):  
Dirichlet Problem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Second Order ◽  
Topological Method ◽  
Multiple Positive Solutions ◽  
Nonlinear Difference Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractWe consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , where $\lambda ,\mu \geq 0$ λ , μ ≥ 0 , $\mathbb{T}=\{2,\ldots ,N-1\}$ T = { 2 , … , N − 1 } with $N>3$ N > 3 , $\phi (s)=s/\sqrt{1-s^{2}}$ ϕ ( s ) = s / 1 − s 2 . The function $f:=\lambda a(t,s)+\mu b(t,s)$ f : = λ a ( t , s ) + μ b ( t , s ) is either sublinear, or superlinear, or sub-superlinear near $s=0$ s = 0 . Applying the topological method, we prove the existence of either one or two, or three positive solutions.

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