Multiplicity Of Nontrivial Solutions Research Articles

Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i<j\le {N}}\lambda _i(x)\lambda _j(x)\), being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\), with \(\lambda \in \mathbb {R}\), \(\lambda >0\), and \(0<p<\infty \), and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.

Existence and multiplicity results for a coupled system of Kirchhoff type equations

This paper deals with a coupled system of Kirchhoff type equations in R 3 . Under suitable assumptions on the potential functions V(x) and W(x), we obtain the existence and multiplicity of nontrivial solutions when the parameter l is sufficiently large. The method combines the Nehari manifold and the mountain-pass theorem.

Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.

Existence of multiple nontrivial solutions for a Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R3 with potential indefinite in sign and a general nonlinearity. We use the direct variational method and Morse theory to obtain the existence of multiple nontrivial solutions for this system.

Existence and multiplicity results for critical growth polyharmonic elliptic systems

In the present paper, we deal with the existence and multiplicity of nontrivial solutions for a class of polyharmonic elliptic systems with Sobolev critical exponent in a bounded domain. Some new existence and multiplicity results are obtained. Our proofs are based on the Nehari manifold and Ljusternik–Schnirelmann theory. Copyright © 2013 John Wiley & Sons, Ltd.

Multiplicity of positive solutions to weighted nonlinear elliptic system involving critical exponents

This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli- Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of R.

Existence and regularity of solutions to semi-linear Dirichlet problem of infinitely degenerate elliptic operators with singular potential term

In this paper, we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy’s inequality, the existence and regularity of multiple nontrivial solutions have been proved.

Multiple solutions for a superlinear p‐Laplacian equation with concave nonlinearities

AbstractIn this paper, we consider a superlinear p‐Laplacian equation with concave nonlinearities, and by Morse theory we can obtain multiple nontrivial solutions of this equation.

Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.

Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

Nonexistence and multiplicity of nontrivial solutions for some nonuniformly nonlinear systems

The goal of this paper is to study the nonexistence and multiplicity of nontrivial weak solutions for a class of nonlinear systems. The results are proved by Minimum principle and the Mountain pass theorem

Multiple solutions for semilinear resonance elliptic problems with nonconstant coefficients at infinity

In this paper, we study a class of semilinear elliptic equations which are completely resonant with nonconstant coefficients at origin and at infinity. By applying Morse theory and Minimax method, we obtain the existence of multiple nontrivial solutions for the equations.

Multiplicity of solutions for a general p(x)-Laplacian Dirichlet problem

We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Laplacian elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

Multiple nontrivial solutions for critical growth quasilinear elliptic systems

In the present paper, we consider the following critical growth quasilinear elliptic system {−Δpui=1p∗DuiF(u)+λi|ui|q−2uiin Ω,i=1,…,k,ui=0on ∂Ω,i=1,…,k. By exploiting the Nehari manifold and variational methods, we obtain the existence and multiplicity results of nontrivial solutions for the system.

Multiple variational solutions to nonlinear Steklov problems

We consider the problem of finding a harmonic function u in a bounded domain \({\Omega \subset {\mathbb R}^N, N \ge 2,}\) satisfying a nonlinear boundary condition of the form \({\partial_{\nu}u(x)=\lambda\,(\eta(x)u(x)+\mu(x)h(u(x)), x\in \partial \Omega}\) where μ and η are bounded functions and h is a \({\mathcal{C}^1}\) odd function with subcritical growth at infinity and such that lims→∞h′(s) = +∞. By using variants of the mountain pass lemma based on index theory, we discuss existence and multiplicity of non trivial solutions to the problem for every value of λ.

Existence of entire positive solutions for a critical system of nonlinear elliptic equations

In this paper, using the fibering method introduced by Pohozaev, we establish an existence of multiple nontrivial positive solutions for a system of nonlinear elliptic equations in R N with lack of compactness studying the properties of Palais–Smale sequence of the energy functional associated with the system.

A study of nonlinear problems for the [formula omitted]-Laplacian in [formula omitted] via Ricceri’s principle

In this paper, we consider the following nonlinear eigenvalue problems for the p -Laplacian: − div ( a ( x ) | ∇ u | p − 2 ∇ u ) = λ f ( x , u ) + μ g ( x , u ) in R n λ , μ > 0 , lim | x | → ∞ u = 0 , where 1 < p < n , λ , μ > 0 , a is a measurable bounded function, and f and g are nonlinearities having subcritical growth with respect to u . We prove multiple nontrivial solutions using a recent principle of Ricceri (2009) [10]. A regularity result is also established.

Scalar kinks in warped extra dimensions

We study the existence and stability of static kink-like configurations of a 5D scalar field, with Dirichlet boundary conditions, along the extra dimension of a warped braneworld. In the presence of gravity such configurations fail to stabilize the size of the extra dimension, leading us to consider additional scalar fields with the role of stabilization. We numerically identify multiple nontrivial solutions for a given 5D action, made possible by the nonlinear nature of the background equations, which we find is enhanced in the presence of gravity. Finally, we take a first step towards addressing the question of the stability of such configurations by deriving the full perturbative equations for the gravitationally coupled multi-field system.

Multiple Nontrivial Solutions for Neumann Problems Involving the p-Laplacian: a Morse Theoretical Approach

Abstract We consider nonlinear elliptic Neumann problems driven by the p-Laplacian. Using variational techniques together with Morse theory (in particular, critical groups and the Poincaré-Hopf formula), we prove some multiplicity results: either three or four distinct nontrivial solutions are guaranteed, depending on the geometry and smoothness of the nonlinear term.

Multiple solutions for fourth order m-point boundary value problems with sign-changing nonlinearity

Using a fixed point theorem in ordered Banach spaces with lattice structure founded by Liu and Sun, this paper investigates the multiplicity of nontrivial solutions for fourth order m-point boundary value problems with sign-changing nonlinearity. Our results are new and improve on those in the literature.