Existence Of Multiple Solutions Research Articles

Two positive solutions for quasilinear elliptic equations with singularity and critical exponents

In this paper, we consider the quasilinear elliptic equation with singularity and critical exponents \t\t\t{−Δpu−μ|u|p−2u|x|p=Q(x)|u|p∗(t)−2u|x|t+λu−s,in Ω,u>0,in Ω,u=0,on ∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta_{p}u-\\mu \\frac{ \\vert u \\vert ^{p-2}u}{ \\vert x \\vert ^{p}}=Q(x) \\frac{ \\vert u \\vert ^{p^{*}(t)-2}u}{ \\vert x \\vert ^{t}}+\\lambda u^{-s}, &\\text{in }\\Omega , \\\\ u>0, & \\text{in }\\Omega , \\\\ u=0, &\\text{on }\\partial \\Omega , \\end{cases} $$\\end{document} where Delta_{p}= operatorname {div}(|nabla u|^{p-2}nabla u) is a p-Laplace operator with 1< p< N. p^{*}(t):=frac{p(N-t)}{N-p} is a critical Sobolev–Hardy exponent. We deal with the existence of multiple solutions for the above problem by means of variational and perturbation methods.

Double phase anisotropic variational problems and combined effects of reaction and absorption terms

This paper deals with the existence of multiple solutions for the quasilinear equation −divA(x,∇u)+V(x)|u|α(x)−2u=f(x,u) in RN, which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like |ξ|q(x)−2ξ for small |ξ| and like |ξ|p(x)−2ξ for large |ξ|, where 1<α(⋅)≤p(⋅)<q(⋅)<N. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Rădulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p(⋅) and q(⋅) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Multiple solutions of double phase variational problems with variable exponent

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation-div⁡𝐀⁢(x,∇⁡u)+|u|α⁢(x)-2⁢u=f⁢(x,u) in ℝN,{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator𝐀{\mathbf{A}}in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like|ξ|q⁢(x)-2⁢ξ{|\xi|^{q(x)-2}\xi}for small|ξ|{|\xi|}and like|ξ|p⁢(x)-2⁢ξ{|\xi|^{p(x)-2}\xi}for large|ξ|{|\xi|}, where1&lt;α⁢(⋅)≤p⁢(⋅)&lt;q⁢(⋅)&lt;N{1&lt;\alpha(\,\cdot\,)\leq p(\,\cdot\,)&lt;q(\,\cdot\,)&lt;N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations inℝN\mathbb{R}^{N}via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponentspandqare constant, to the case wherep⁢(⋅){p(\,\cdot\,)}andq⁢(⋅){q(\,\cdot\,)}are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

Existence of Multiple Solutions for a Class Non-uniformly Elliptic Equations with Critical Exponential Growth

This paper deals with the existence and multiplicity of weak solutions to non-uniformly elliptic problem -div(a(x,∇u))+V(x)|u|N-2u=λ(exp(α|u|NN-1)+f(x,u)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\text {div}(a(x,\\nabla u))+V(x)\\vert u\\vert ^{N-2} u=\\lambda \\Bigg (\\exp \\Bigg (\\alpha \\vert u\\vert ^{\\frac{N}{N-1}}\\Bigg )+f(x,u)\\Bigg ), \\end{aligned}$$\\end{document}in mathbb {R}^N, where Nge 2, alpha is some positive constant and lambda is some positive parameter. Our method is mainly based on variational arguments.

Multiple solutions with constant sign for a (p,q)-elliptic system Dirichlet problem with product nonlinear term

In this paper, we consider the existence of multiple solutions of the homogeneous Dirichlet problem for a (p,q)-elliptic system with nonlinear product term as follows: \t\t\t{−Δpu=λα(x)|u|α(x)−2u|v|β(x)+Fu(x,u,v)in Ω,−Δqv=λβ(x)|u|α(x)|v|β(x)−2v+Fv(x,u,v)in Ω,u=0=von ∂Ω.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} {-}\\Delta_{p}u=\\lambda \\alpha (x)\\vert u\\vert ^{ \\alpha (x)-2}u\\vert v\\vert ^{\\beta (x)}+F_{u}(x,u,v)&\\text{in }\\Omega, \\\\ {-}\\Delta_{q}v=\\lambda \\beta (x)\\vert u\\vert ^{ \\alpha (x)}\\vert v\\vert ^{\\beta (x)-2}v+F_{v}(x,u,v)&\\text{in }\\Omega, \\\\ u=0=v&\\text{on }\\partial \\Omega . \\end{cases} $$\\end{document} We emphasize that the potential F(x,u,v) might contain a nonlinear product term which includes F(x,u,v)=vert uvert ^{theta_{1}(x)} vert vvert ^{theta_{2}(x)}ln (1+vert uvert ) ln (1+vert vvert ) as a prototype, and does not require F(x,u,v)rightarrow +infty as vert uvert +vert vvert rightarrow +infty . With novel growth conditions on F(x,u,v), we develop a new method to check the Cerami compactness condition. Through arguments of critical point theory, we prove the existence of multiple constant-sign solutions for our elliptic system without requiring the well-known Ambrosetti–Rabinowitz condition. Moreover, we also give a result guaranteeing the existence of infinitely many solutions.

Existence of multiple solutions for a quasilinear Neumann problem with critical exponent

The main purpose of this paper is to establish the existence and multiplicity of nontrivial solutions for a quasilinear Neumann problem with critical exponent. It is shown, by the methods of the Lions concentration-compactness principle and the mountain pass lemma, that under certain conditions, the existence of nontrivial solutions are obtained.

Multiple solutions for semilinear cone elliptic equations without Ambrosetti–Rabinowitz condition

In this paper we establish the existence of multiple solutions for a class of semilinear cone degenerate elliptic Dirichlet boundary value problems involving subcritical nonlinearity (cone Sobolev exponent) without the Ambrosetti–Rabinowitz condition. The paper uses singular analysis to control the linear part to provide appropriate functional setting that a variation of the Moutain Pass argument can be applied.

Existence of multiple solutions to second-order discrete Neumann boundary value problems

By using the invariant set of descending flow and variational method, we establish the existence of multiple solutions to a class of second-order discrete Neumann boundary value problems. The solutions include sign-changing solutions, positive solutions, and negative solutions. An example is given to illustrate our results.

Existence of solutions for a nonlinear simply supported beam equation

This paper is devotedto prove the existence of one or multiple solutions for a family of nonlinear fourth‐order boundary value problems.We use several fixed point theorems, previously developed by the author and her coauthor, to prove the existence of solutions of some simply supported beam problems. To finish the work, a particular case is studied, and the existence of multiple solutions is proved for 2 different particular nonlinear functions.

On critical Choquard equation with potential well

In this paper we are interested in the following nonlinear Choquard equation $-Δ u+(λ V(x)-β)u = \big(|x|^{-μ}* |u|^{2_{μ}^{*}}\big)|u|^{2_{μ}^{*}-2}u\;\;\;\;\;\;\;\;\;\;\mbox{in}\;\; \mathbb{R}^N,$ where $λ, β∈\mathbb{R}^+$ , $0&lt;μ&lt;N, N≥4, 2_{μ}^{*} = (2N-μ)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V∈ \mathcal{C}(\mathbb{R}^N, \mathbb{R})$ such that $Ω : = \mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. If $β&gt;0$ is a constant such that the operator $-Δ +λ V(x)-β$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $λ$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$ goes to infinity. Furthermore, for any $0&lt;β&lt;β_{1}$ , we are able to prove the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$ is the first eigenvalue of $-Δ$ on $Ω$ with Dirichlet boundary condition.

Multiple existence of solutions for a coupled system involving the distributional Henstock-Kurzweil integral

This paper deals with a coupled system in the sense of distributions (generalized functions). Our main goal is to get the basic multiple existence results via some degree theory arguments. Differently from the literatures, the proof is based on the concept of a general integral named distributional Henstock-Kurzweil integral, which includes the Lebesgue and Henstock-Kurzweil integrals as special cases. Finally, an example is given to illustrate that the presented abstract theory contains some previous results as special cases.

Existence of multiple solutions for a p-Kirchhoff problem with the non-linear boundary condition

ABSTRACTIn this paper, using the Nehari manifold and fibering maps, we study the existence of at least two positive solutions for the following p-Kirchhoff equationwhere is a bounded domain, with a, b, , is the p-Laplacian operator, is a parameter.

Existence of multiple solutions for a quasilinear elliptic problem

In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the $p$-derivative at zero and the $p$-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the $p$-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.

Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the \begin{document} $N$ \end{document} -Laplacian operator.

Applications of variational methods to an anti-periodic boundary value problem of a second-order differential system

In this paper, we discuss the existence of multiple solutions to a second order anti-periodic boundary value problem \[ \ddot {x}(t)+M x(t)+\nabla F(t, x(t))=0\quad\mbox{almost every } t\in [0, T],\\ x(0)=-x(T) \qquad\qquad\qquad\ \, \dot {x}(0)=-\dot {x}(T) \] by using variational methods and critical point theory. Furthermore, we obtain the existence of periodic solutions for corresponding second-order differential systems.

Applications of topolological methods to the semilinear biharmonic problem with different powers

We prove the existence of multiple solutions for the fourth order nonlinear elliptic problem with fully nonlinear term. Our method is based on the critical point theory; the variation of linking method and category theory.

A note on a system with radiation boundary conditions with non-symmetric linearisation

We prove the existence of multiple solutions for a second order ODE system under radiation boundary conditions. The proof is based on the degree computation of \(I-K\), where K is an appropriate fixed point operator. Under a suitable asymptotic Hartman-like assumption for the nonlinearity, we shall prove that the degree is 1 over large balls. Moreover, studying the interaction between the linearised system and the spectrum of the associated linear operator, we obtain a condition under which the degree is \(-1\) over small balls. We thus generalize a result obtained in a previous work for the case in which the linearisation is symmetric.

Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems

In this paper, we deal with the existence, multiplicity and asymptotic behavior of solutions for the following stationary Kirchhoff problems involving the p-Laplacian:−M(∫Ω|∇u|pdx)Δpu=λf(x,u)+|u|p⁎−2uinΩ,u=0 on∂Ω, where Ω is a bounded smooth domain of RN, M:R0+→R0+ is a continuous function satisfying some extra assumptions. By using the mountain pass theorem, the existence of solutions is obtained. By applying an abstract critical point theorem based on the cohomological index, we prove the existence of multiple solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator −‖∇u‖(θ−1)pΔp, where p<N<θp<p⁎. We also investigate the asymptotic behavior of solutions when λ converges to infinity. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero, i.e. the problem is degenerate.

Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods

In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with p-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.

Existence of solutions for nonlinear p-Laplacian difference equations

The aim of this paper is the study of existence of solutions for nonlinear $2n^{\mathrm{th}}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.