Superlinear Elliptic Problems Research Articles

Five nontrivial solutions of superlinear elliptic problem

In this paper, we consider the following superlinear elliptic problem(P){−Δu=λ|u|p−2u+f(x,u),inΩ,u=0,on∂Ω, where λ>0 and 2<p<2+δ for some δ>0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u=0 is a local minimizer of the associated energy functional of equation (P).Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ=1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ.

Infinitely many radial solutions of superlinear elliptic problems with dependence on the gradient terms in an annulus

In this paper, we are concerned with elliptic problems {−Δu=f(u)+g(|x|,u,x|x|⋅∇u),x∈Ω,u|∂Ω=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta u= f(u)+ g( \\vert x \\vert ,u,\\frac{x}{ \\vert x \\vert }\\cdot \ abla u),&x\\in \\Omega , \\\\ u|_{\\partial \\Omega}=0, \\end{cases} $$\\end{document} where Omega ={xin mathbb{R}^{N}:R_{1}<|x|<R_{2}} is an annular domain, N>2, 0< R_{1}<R_{2}<infty , 4(R_{2}-R_{1})(N-1)leq R_{1}. f:mathbb{R} to mathbb{R} is continuous and satisfies lim_{|xi |to infty}frac{f(xi )}{xi}=infty . g:[R_{1},R_{2}] times mathbb{R}^{2}to mathbb{R} is continuous, |g(r,xi _{0}, xi _{1})|leq C+beta |xi _{0}| for some C>0, 0<beta < frac{1}{4(R_{2}-R_{1})^{2}}. We obtain infinitely many radial solutions with prescribed nodal properties using bifurcation techniques.

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions. Superlinear elliptic problems can be expected to have multiple positive solutions by some case. Conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the one, and then deduce the uniqueness result for a positive solution. An application of our results to the logistic boundary condition arising from population genetics is given.

Fourth-order elliptic problems involving concave-superlinear nonlinearities

The existence of solutions for a huge class of superlinear elliptic problems involving fourth-order elliptic problems defined on bounded domains under Navier boundary conditions is established. To this end we do not apply the well-known Ambrosetti-Rabinowitz condition. Instead, we assume that the nonlinear term is nonquadratic at infinity. Furthermore, the nonlinear term is a concave-superlinear function which can be indefinite in sign. In order to apply variational methods we employ some delicate arguments recovering some kind of compactness.

Fourth-order superlinear elliptic problems interacting with high eigenvalues

Fourth-order superlinear elliptic problems interacting with high eigenvalues

Positive solutions of superlinear elliptic problems with discontinuous non-linearities

Рассматривается эллиптическая краевая задача с однородным граничным условием Дирихле, параметром и разрывной нелинейностью. Положительный параметр входит в нелинейность мультипликативно, при этом изучаемая задача имеет нулевое решение при любом значении параметра. Нелинейность на бесконечности имеет суперлинейный рост. Топологическим методом доказывается существование положительных решений исследуемой задачи. Библиография: 30 наименований.

Positive solutions of superlinear elliptic problems with discontinuous non-linearities

We consider an elliptic boundary-value problem with a homogeneous Dirichlet boundary condition, a parameter and a discontinuous non-linearity. The positive parameter appears as a multiplicative term in the non-linearity, and the problem has a zero solution for any value of the parameter. The non-linearity has superlinear growth at infinity. We prove the existence of positive solutions by a topological method.

Nonradial solutions of elliptic weighted superlinear problems in bounded symmetric domains

The present work has two objectives. First, we prove that a weighted superlinear elliptic problem has infinitely many nonradial solutions in the unit ball. Second, we obtain the same conclusion in annuli for a more general nonlinearity which also involves a weight. We use a lower estimate of the energy level of radial solutions with k−1 zeros in the interior of the domain and a simple counting. Uniqueness results due to Tanaka [14, 2008] and [13, 2007] are very useful in our approach.

Existence and multiplicity for a superlinear elliptic problem under a non-quadradicity condition at infinity

In this article, we study the existence and multiplicity of solutions of the boundary-value problem $$\displaylines{ -\Delta u = f(x,u), \quad \text{in } \Omega, \cr u = 0, \quad \text{on } \partial\Omega, }$$ where \(\Delta\) denotes the N-dimensional Laplacian, \(\Omega\) is a bounded domain with smooth boundary, \(\partial\Omega\), in \(\mathbb{R}^N\) \((N\geq 3)\), and f is a continuous function having subcritical growth in the second variable. Using infinite-dimensional Morse theory, we extended the results of Furtado and Silva [9] by proving the existence of a second nontrivial solution under a non-quadradicity condition at infinity on the non-linearity. Assuming more regularity on the non-linearity f, we are able to prove the existence of at least three nontrivial solutions.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/60/abstr.html

Some superlinear problems with nonlocal diffusion coefficient

We study a superlinear elliptic problem with a non-local diffusion coefficient. We show that there exists a drastic change on the structure of the set of positive solutions when the non-local coefficient grows fast enough to infinity. We combine mainly sub-super and bifurcation methods to obtain our results.

On the fractional Lazer–McKenna conjecture with superlinear potential

The goal of this paper is to study the following non-local superlinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} (-\Delta )^s u=|u|^p-\sigma \phi _1 &{}\hbox {in } \Omega ,\\ u=0 &{}\hbox {in } {\mathbb {R}}^N{\setminus }\Omega ,\\ u>0 &{}\hbox {in }\Omega , \end{array}\right. \end{aligned}$$ where $$(-\Delta )^s$$ is the fractional Laplace operator, $$\Omega \subset \mathbb R^N$$ is an open domain with Lipschitz boundary, $$\sigma >0$$ , $$p\in (1, 2^*_s-1)$$ with $$2^{*}_{s}=\frac{2N}{N-2s}$$ and $$\phi _1$$ is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary condition. We prove the non-local version of a conjecture due to Lazer and McKenna (Proc R Soc Edinb Sect A 95:275–283, 1983) by constructing solutions with sharp peaks near local maximum points of $$\phi _1$$ .

TWO-SIDED ESTIMATES FOR POSITIVE SOLUTIONS OF SUPERLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

We give two-sided estimates for positive solutions of the superlinear elliptic problem$-\unicode[STIX]{x1D6E5}u=a(x)|u|^{p-1}u$with zero Dirichlet boundary condition in a bounded Lipschitz domain. Our result improves the well-knowna priori$L^{\infty }$-estimate and provides information about the boundary decay rate of solutions.

A priori bounds for superlinear elliptic equations with semidefinite nonlinearity

We derive a priori bounds for positive solutions of the superlinear elliptic problems −Δu=a(x)up on a bounded domain Ω in RN, where a(x) is Hölder continuous in Ω. Our main motivation is to study the case where a(x)≥0, a(x)≢0 and a(x) has some zero sets in Ω. We show that, in this case, the scaling arguments reduce the problem of a priori bounds to the Liouville-type results for the equation −Δu=A(x′)up in RN, where A is the continuous function defined on the subspace Rk with 1≤k≤N and x′∈Rk. We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.

Sign Changing Solutions for Quasilinear Superlinear Elliptic Problems

Results on existence and multiplicity of solutions for a nonlinear elliptic problem driven by the $\Phi$-Laplace operator are established. We employ minimization arguments on suitable Nehari manifolds to build a negative and a positive ground state solutions. In order to find a nodal solution we employ additionally the well known Deformation Lemma and Topological Degree Theory.

Multiplicity results for a nonlinear elliptic problem involving the fractional laplacian

In this paper, we consider a class of superlinear elliptic problems involving fractional Laplacian (−Δ)s/2u = δf (u) in a bounded smooth domain with zero Dirichlet boundary condition. We use the method on harmonic extension to study the dependence of the number of sign-changing solutions on the parameter A.

Gaps of consecutive eigenvalues of Laplace operator and the existence of multiple solutions for superlinear elliptic problem

This paper is mainly related to multiple nontrivial solutions of the elliptic boundary value problem{−Δu=|u|p−2u+f(x,u),x∈Ω,u=0,x∈∂Ω, for p∈(2,2⁎). It is reasonable to guess that for dim⁡Ω≥2 above problem possesses infinitely many distinct solutions since this is proved to be true for ODE. However, so far one does not even know if there exists a fourth nontrivial solution. By using a new homological linking theorem, Morse theory, and some precise estimates we disclose the relationship among the gaps of consecutive eigenvalues of Laplace operator, growth trend of nonlinear terms and the existence of multiple solutions of superlinear elliptic boundary value problem. Moreover, as p is close to 2, we get the fourth nontrivial solution under appropriate hypotheses, where f(x,u) satisfies Ambrosetti–Rabinowitz condition.

A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems

Abstract We consider the semilinear elliptic problem where a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..

Superlinear elliptic problems under the non-quadraticity condition at infinity

We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.

An asymmetric superlinear elliptic problem at resonance

In this article, we study the existence and multiplicity of solutions to the problem {−Δu=g(x,u), in Ω;u=0, on ∂Ω, where Ω is a bounded domain in RN(N⩾2) with smooth boundary, and g:Ω¯×R→R is a differentiable function. We will assume that g(x,s) has a resonant behavior for large negative values of s and that a Landesman–Lazer type condition is satisfied. We also assume that g(x,s) is superlinear, but subcritical, for large positive values of s. We prove the existence and multiplicity of solutions for problem (1.1) by using minimax methods and infinite-dimensional Morse theory.

ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Abstract. This paper deals with the superlinear elliptic problem with-out Ambrosetti and Rabinowitz type growth condition of the form:−div(1 + |∇u| p(x) √ 1+|∇u| 2p(x) )|∇u| p(x)−2 ∇u= λf(x,u), a.e. in Ω,u = 0, on ∂Ω,where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, λ > 0 isa parameter. The purpose of this paper is to obtain the existence resultsof nontrivial solutions for every parameter λ. Firstly, by using the moun-tain pass theorem a nontrivial solution is constructed for almost everyparameter λ > 0. Then we consider the continuation of the solutions.Our results are a generalization of that of Manuela Rodrigues. 1. IntroductionDuring the last fifteen years, variational problems and partial differentialequations with various types of nonstandard growth conditions have becomeincreasingly popular. This is partly due to their frequent appearance in ap-plications such as the modeling of electrorheological fluids [1, 12] and imageprocessing [2], but these problems are very interesting from a purely mathe-matical point of view as well.In this paper, we consider the following nonlinear eigenvalue problems forp(x)-Laplacian-like operators originated from a capillary phenomena of thefollowing form:(P)−div(1+