Existence and concentration of solutions for singularly perturbed doubly nonlocal elliptic equations

2018 ◽  
Vol 22 (01) ◽  
pp. 1850074
Author(s):  
Dengfeng Lü ◽  
Shuangjie Peng
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Local Minimum ◽  
External Potential ◽  
Singularly Perturbed ◽  
Penalization Method ◽  
Concentration Of Solutions ◽  
Minimum Points

We consider the existence and concentration of positive solutions to a singularly perturbed doubly nonlocal elliptic equation [Formula: see text] where [Formula: see text] is a parameter, [Formula: see text] are constants, [Formula: see text] and [Formula: see text] is an external potential, [Formula: see text] and [Formula: see text]. Under some suitable assumptions on [Formula: see text] and [Formula: see text], by using the penalization method, we prove that for [Formula: see text] small enough there exists a family of positive solutions which concentrate on the local minimum points of the potential [Formula: see text] as [Formula: see text].

Positive solutions of elliptic equations involving supercritical growth

1991 ◽  
Vol 118 (1-2) ◽  
pp. 49-62 ◽  
Author(s):  
F. Merle ◽  
L. A. Peletier
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Singular Solution ◽  
Supremum Norm ◽  
Radial Solutions ◽  
Positive Radial Solutions

SynopsisPositive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity. It is shown that they converge, uniformly away from the origin, as well as in H1, to the unique singular solution.

On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem

1997 ◽  
Vol 127 (4) ◽  
pp. 691-701 ◽  
Author(s):  
E. N. Dancer ◽  
Juncheng Wei
Keyword(s):  
Boundary Layer ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Singularly Perturbed ◽  
Convex Domains ◽  
Existence Of Positive Solutions

We discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign. In particular, we obtain solutions with both a boundary layer and a sharp interior peak.

Non-homogeneous elliptic equations in exterior domains

2006 ◽  
Vol 136 (1) ◽  
pp. 139-147 ◽  
Author(s):  
J. do Ó ◽  
S. Lorca ◽  
J. Sánchez ◽  
P. Ubilla
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domains ◽  
Existence And Multiplicity ◽  
Negative Function ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Typical Model ◽  
Homogeneous Elliptic Equation

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method

2015 ◽  
Vol 146 (1) ◽  
pp. 23-58 ◽  
Author(s):  
Claudianor O. Alves ◽  
Minbo Yang
Keyword(s):  
Real Function ◽  
Local Minimum ◽  
Laplacian Operator ◽  
Positive Parameter ◽  
Choquard Equation ◽  
Primitive Function ◽  
Continuous Real Function ◽  
Penalization Method ◽  
Concentration Of Solutions ◽  
Image Position

We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equationwhere Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.

scholarly journals A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

2020 ◽  
Vol 88 (2) ◽  
pp. 507-534
Author(s):  
Daniele Cassani ◽  
Youjun Wang ◽  
Jianjun Zhang
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Real Function ◽  
Critical Growth ◽  
External Potential ◽  
Singularly Perturbed ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Unified Approach

AbstractIn this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations, $$-\varepsilon^2\Delta u+V(x)u\mp\varepsilon^{2+\gamma}u\Delta u^2=h(u),\ \ x\in \mathbb{R}^N, $$ - ε 2 Δ u + V ( x ) u ∓ ε 2 + γ u Δ u 2 = h ( u ) , x ∈ R N , where $$N\geqslant3, \varepsilon > 0, V(x)$$ N ⩾ 3 , ε > 0 , V ( x ) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter $$\gamma>0$$ γ > 0 . Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution $$u_{\varepsilon,\gamma}$$ u ε , γ concentrating, as $$\varepsilon\rightarrow 0$$ ε → 0 , around minima points of the potential.

Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents

2010 ◽  
Vol 140 (3) ◽  
pp. 617-633 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Multiple Positive Solutions ◽  
Singular Elliptic Equations ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , $1\le q<2$, $0\le\mu<\bar{\mu}=(N-2)^2/4$, 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on $\bar{\varOmega}$, that change sign on Ω.

scholarly journals Multiple Positive Solutions for Semilinear Elliptic Equations in Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation in , , where , if , if ), , satisfy suitable conditions, and may change sign in .

Elliptic equations with critical nonlinearities and Hardy terms

2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mohammed Bouchekif ◽  
Yasmina Nasri
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Critical Nonlinearities ◽  
Existence Of Positive Solutions ◽  
Inverse Square Potentials

AbstractUsing variational methods, we prove the existence of positive solutions to an elliptic equation involving critical nonlinearities and multiple inverse square potentials.

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