Existence of solution for a class of quasilinear elliptic problem without Δ2-condition

The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain. The nonlinear term [Formula: see text] is a continuous function which is superlinear at the origin and infinity. The function [Formula: see text] is an [Formula: see text]-function where the well-known [Formula: see text]-condition is not assumed. Then the Orlicz–Sobolev space [Formula: see text] may be non-reflexive. As a main model, we have the function [Formula: see text]. Here, we consider some situations where it is possible to work with global minimization, local minimization and mountain pass theorem. However, some estimates employed here are not standard for this type of problem taking into account the modular given by the [Formula: see text]-function [Formula: see text].

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Quasilinear elliptic problems with concave–convex nonlinearities

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500504 ◽

2017 ◽

Vol 19 (06) ◽

pp. 1650050 ◽

Cited By ~ 5

Author(s):

M. L. M. Carvalho ◽

Edcarlos D. da Silva ◽

C. Goulart

Keyword(s):

Elliptic Problem ◽

Ground State ◽

Nonlinear Term ◽

Laplacian Operator ◽

Main Difficulty ◽

Quasilinear Elliptic Problem ◽

Existence And Multiplicity ◽

Quasilinear Elliptic Problems ◽

Nehari Method ◽

Quasilinear Elliptic

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the [Formula: see text]-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the [Formula: see text]-Laplacian operator is not homogeneous and the nonlinear term is indefinite.

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Regularity of Entropy Solutions of Quasilinear Elliptic Problems Related to Hardy-Sobolev Inequalities

Advanced Nonlinear Studies ◽

10.1515/ans-2006-0404 ◽

2006 ◽

Vol 6 (4) ◽

Author(s):

Boumediene Abdellaoui ◽

Eduardo Colorado ◽

Manel Sanchón

Keyword(s):

Bounded Domain ◽

Entropy Solution ◽

Elliptic Problems ◽

Entropy Solutions ◽

Sobolev Inequalities ◽

Smooth Bounded Domain ◽

Quasilinear Elliptic Problems ◽

Quasilinear Elliptic

AbstractThis article is concerned with the regularity of the entropy solution ofwhere Ω is a smooth bounded domain Ω of ℝ

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Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

Advances in Nonlinear Analysis ◽

10.1515/anona-2021-0203 ◽

2021 ◽

Vol 11 (1) ◽

pp. 482-502

Author(s):

Zeyi Liu ◽

Lulu Tao ◽

Deli Zhang ◽

Sihua Liang ◽

Yueqiang Song

Keyword(s):

Heisenberg Group ◽

Mountain Pass Theorem ◽

Poisson System ◽

Heisenberg Groups ◽

Euclidean Case ◽

Smooth Bounded Domain ◽

Critical Point Theorem ◽

Critical Nonlinearities ◽

Existence And Multiplicity ◽

Non Local

Abstract In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

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Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Abstract and Applied Analysis ◽

10.1155/2013/834918 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Jincheng Huang

Keyword(s):

Boundary Conditions ◽

Exterior Domain ◽

Nonlinear Boundary ◽

Nonlinear Boundary Conditions ◽

Multiplicity Results ◽

Exterior Problems ◽

Existence And Multiplicity ◽

Nonnegative Solutions ◽

Quasilinear Elliptic Problems ◽

Quasilinear Elliptic

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

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Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.526.177 ◽

2014 ◽

Vol 526 ◽

pp. 177-181

Author(s):

Yuan Li ◽

Ai Hui Sheng

Keyword(s):

Dirichlet Problem ◽

Variational Methods ◽

Nonlinear Term ◽

Mountain Pass Theorem ◽

Nehari Manifold ◽

Nontrivial Solutions ◽

Perturbation Theorem ◽

Existence And Multiplicity ◽

Manifold Theory ◽

The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

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Quasilinear elliptic problems with critical exponents and Hardy terms in ℝN

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091508000187 ◽

2010 ◽

Vol 53 (1) ◽

pp. 175-193 ◽

Cited By ~ 1

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.

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Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000475 ◽

2016 ◽

Vol 146 (2) ◽

pp. 435-448 ◽

Cited By ~ 16

Author(s):

Juntao Sun ◽

Tsung-fang Wu

Keyword(s):

Variational Principle ◽

Mountain Pass Theorem ◽

Type Equation ◽

Bounded Domains ◽

Type Problem ◽

Smooth Bounded Domain ◽

Kirchhoff Type ◽

Existence And Multiplicity ◽

Kirchhoff Type Problem ◽

Image Position

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

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QUASILINEAR ELLIPTIC EQUATIONS OF THE HENON-TYPE: EXISTENCE OF NON-RADIAL SOLUTIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199709003594 ◽

2009 ◽

Vol 11 (05) ◽

pp. 783-798 ◽

Cited By ~ 2

Author(s):

P. C. CARRIÃO ◽

D. G. DE FIGUEIREDO ◽

O. H. MIYAGAKI

Keyword(s):

Elliptic Equations ◽

Elliptic Problems ◽

Quasilinear Elliptic Equations ◽

Open Ball ◽

Radial Solutions ◽

Existence And Multiplicity ◽

Quasilinear Elliptic Problems ◽

Quasilinear Elliptic

In this work, we prove results on existence and multiplicity of non-radial solutions for a class of singular quasilinear elliptic problems of the form [Formula: see text] where B = {x ∈ ℝN: |x| < 1} (N ≥ 3) is a unit open ball centered at the origin, -∞ < a < (N - p)/p, β > 0 and [Formula: see text].

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A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0056 ◽

2020 ◽

Vol 9 (1) ◽

pp. 1351-1382 ◽

Cited By ~ 1

Author(s):

Salvador López-Martínez

Keyword(s):

Break Point ◽

Singular Case ◽

Multiplicity Results ◽

Existence And Uniqueness Results ◽

Existence And Multiplicity ◽

Uniqueness Results ◽

Quasilinear Elliptic Problems ◽

Bounded Smooth ◽

Quasilinear Elliptic

Abstract In this paper we deal with the elliptic problem $$\begin{array}{} \begin{cases} \displaystyle-{\it\Delta} u=\lambda u+\mu(x)\frac{|\nabla u|^q}{u^\alpha}+f(x)\quad &\text{ in }{\it\Omega}, \\ u \gt 0 \quad &\text{ in }{\it\Omega}, \\ u=0\quad &\text{ on }\partial{\it\Omega}, \end{cases} \end{array} $$ where Ω ⊂ ℝN is a bounded smooth domain, 0 ≨ μ ∈ L∞(Ω), 0 ≨ f ∈ Lp0(Ω) for some p0 > $\begin{array}{} \frac{N}{2} \end{array}$, 1 < q < 2, α ∈ [0 1] and λ ∈ ℝ. We establish existence and multiplicity results for λ > 0 and α < q – 1, including the non-singular case α = 0. In contrast, we also derive existence and uniqueness results for λ > 0 and q – 1 < α ≤ 1. We thus complement the results in [1, 2], which are concerned with α = q – 1, and show that the value α = q – 1 plays the role of a break point for the multiplicity/uniqueness of solution.

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