scholarly journals On a class of Kirchhoff problems via local mountain pass

2020 ◽  
pp. 1-43
Author(s):  
Vincenzo Ambrosio ◽  
Dušan Repovš
Keyword(s):  
Continuous Function ◽  
Small Parameter ◽  
Positive Solutions ◽  
Local Minimum ◽  
Mountain Pass ◽  
Multiplicity Result ◽  
Positive Potential ◽  
Subcritical Growth ◽  
Made In

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( u ) + γ u 5 in  R 3 , u ∈ H 1 ( R 3 ) , u > 0 in  R 3 , where ε > 0 is a small parameter, a , b > 0 are constants, γ ∈ { 0 , 1 }, V is a continuous positive potential with a local minimum, and f is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud. 14 (2014), 483–510; J. Differ. Equ. 252 (2012), 1813–1834; J. Differ. Equ. 253 (2012), 2314–2351).

Multiplicity and concentration results for a class of critical fractional Schrödinger–Poisson systems via penalization method

2018 ◽  
Vol 22 (01) ◽  
pp. 1850078 ◽  
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Type System ◽  
Laplacian Operator ◽  
Positive Potential ◽  
Penalization Method ◽  
Subcritical Growth ◽  
Fractional Laplacian Operator ◽  
Poisson Type

We deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Poisson-type system with critical growth: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], is the fractional Laplacian operator, [Formula: see text] is a continuous positive potential and [Formula: see text] is a superlinear continuous function with subcritical growth. Using penalization techniques and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value.

scholarly journals A multiplicity result for a (p, q)-Schrödinger–Kirchhoff type equation

2021 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Teresa Isernia
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Category Theory ◽  
Type Equation ◽  
Multiplicity Result ◽  
Positive Potential ◽  
Subcritical Growth ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Penalization Techniques

AbstractIn this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

Solitary Waves for a Class of Quasilinear Schrödinger Equations Involving Vanishing Potentials

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
João Marcos do Ó ◽  
Elisandra Gloss ◽  
Cláudia Santana
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Solitary Waves ◽  
Iteration Scheme ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonnegative Continuous Function ◽  
Vanishing Potentials

AbstractIn this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations−Δu + V(x)u − [Δ(uwhere h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.

scholarly journals Existence and concentration of positive solutions for a critical p&q equation

2021 ◽  
Vol 11 (1) ◽  
pp. 243-267
Author(s):  
Gustavo S. Costa ◽  
Giovany M. Figueiredo
Keyword(s):  
Small Parameter ◽  
Positive Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Lack Of Compactness ◽  
Critical Problems ◽  
Compactness Principle

Abstract We show existence and concentration results for a class of p&q critical problems given by − d i v a ϵ p | ∇ u | p ϵ p | ∇ u | p − 2 ∇ u + V ( z ) b | u | p | u | p − 2 u = f ( u ) + | u | q ⋆ − 2 u in R N , $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1,p (ℝ N ) ∩ W 1,q (ℝ N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N, N ≥ 2 and q * = Nq/(N − q). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.

Concentration of positive solutions for a class of fractional p-Kirchhoff type equations

2020 ◽  
pp. 1-51 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Teresa Isernia ◽  
Vicenţiu D. Radulescu
Keyword(s):  
Positive Solutions ◽  
Local Minimum ◽  
Existence Result ◽  
Laplacian Operator ◽  
Multiplicity Result ◽  
Kirchhoff Type ◽  
Small Positive Parameter ◽  
Continuous Potential ◽  
Image Position ◽  
Kirchhoff Type Problems

Abstract We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems: $$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$ where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp \in (\frac {3}{2}, 3)$ , $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian operator, f: ℝ → ℝ is a superlinear continuous function with subcritical growth and V: ℝ3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f(u) = uq−1 + γur−1, where γ > 0 is sufficiently small, and the powers q and r satisfy 2p < q < p* s  ⩽ r. The main results are obtained by using some appropriate variational arguments.

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

scholarly journals Generalized Emden-Fowler equations of subcritical growth

1993 ◽  
Vol 54 (2) ◽  
pp. 254-262 ◽  
Author(s):  
Wolfgang Rother
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Operator ◽  
Positive Solutions ◽  
Subcritical Growth ◽  
Existence Of Positive Solutions

AbstractThe existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.

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