Existence Of Multiple Nontrivial Solutions Research Articles

A bifurcation and multiplicity result for a critical growth elliptic problem

We consider a Brézis–Nirenberg type critical growth p-Laplacian problem involving a parameter μ>0 in a smooth bounded domain Ω. We prove the existence of multiple nontrivial solutions if either μ or the volume of Ω is sufficiently small. The proof is based on an abstract critical point theorem that only assumes a local (PS) condition. Our results are new even in the semilinear case p=2.

Multiplicity of solutions for anisotropic Dirichlet problem with variable exponent

We establish some results on the existence of multiple nontrivial solutions for a general anisotropic 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.

Nontrivial solutions for a general p(x)-Laplacian Robin problem

We establish the existence of multiple nontrivial solutions for a class of $p(x)$-Laplacian Robin problem. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined withadequate variational methods and a variant of the Mountain Pass lemma.

Infinitely many solutions for fractional elliptic systems involving critical nonlinearities and Hardy potentials

This paper is dedicated to studying the existence of multiple nontrivial solutions for a class of singular fractional elliptic systems involving critical nonlinearities and Hardy potentials in RN. Based upon the Caffarelli–Silvestre extension method and the Krasnoselskii genus theory, together with the symmetric criticality principle of Palais, we establish several existence results of infinitely many solutions under certain appropriate hypotheses on the weights and the parameters.

Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

Existence of solutions for a coupled Schrödinger equations with critical exponent

<abstract><p>In this paper we study the existence of multiple nontrivial solutions of the coupled Schrödinger system with external sources terms as perturbations. This type of the system arises from Bose-Einstein condensate. As these external sources terms are nonlinear functions and small in some sense, we use fibre map to divide the Nehari manifold into threes parts, and then prove the existence of a nontrivial ground state solution and a bound state solution.</p></abstract>

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

We study the existence of multiple nontrivial solutions for nonlinear fourth order discrete boundary value problems. We first establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are positive, negative, and sign-changing, respectively. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending flow.

Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

Abstract In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrodinger system with quadratic and cubic nonlinearity. This kind of system of Schrodinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.

Remarks on infinitely many solutions for a class of Schrödinger equations with sublinear nonlinearity

In this paper, we study the existence of multiple nontrivial solutions for the following semilinear Schrödinger equation: where the potential V is continuous and allowed to be sign‐changing and f(x,u) satisfies more general sublinear growth conditions than those in previous studies. The first result of this paper obtains the existence of a nontrivial solution of (1). The second establishes an existent result on infinitely many nontrivial solutions of (1) by using a variant fountain theorems.

Existence of multiple non-trivial solutions for a nonlocal problem

In this paper, we are concerned with the following general nonlocal problem\begin{equation*}\begin{cases}-\mathcal{L}_K u=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega,\end{cases}\end{equation*}where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian\begin{equation*}\begin{cases}(-\Delta)^su=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega.\end{cases}\end{equation*}

Application of Morse Theory for Some Damped Dirichlet Nonlinear Impulsive Differential Equations

In this paper, we consider the multiplicity of solutions for some damped nonlinear impulsive differential equations by using Morse theory in combination with the minimax arguments. Under some assumptions, we get some new results on the existence of multiple nontrivial solutions for the problems. Thus, we improve and extend recent results.

The existence of multiple solutions for the Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory on [formula omitted

In this paper, we prove the existence of multiple solutions for the following Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory {−Δu+a(x)u−(2ω+ϕ)ϕu=λk(x)|u|q−2u+g(x)|u|p−2u,x∈R3,Δϕ+βΔ4ϕ=4π(ω+ϕ)u2,x∈R3, where 1<q<2<p<6. Under appropriate assumptions on a, k, g and λ, the existence of multiple nontrivial solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory.

Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem

In this paper, we study the following Kirchhoff type problem with critical growth − M ∫ Ω | ∇ u | 2 d x △ u = λ | u | 2 u + | u | 4 u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R 3 , M ∈ C ( R + , R ) and λ > 0 . We prove the existence of multiple nontrivial solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator − M ( ∫ Ω | ∇ u | 2 d x ) △ .

Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space

Abstract In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of C 1 {C^{1}} -functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator u ↦ div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) {u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})} . The solvability of a general non-potential system is also established.

Multiple positive solutions for a coupled nonlinear Hartree type equations with perturbations

In this paper we study the existence of multiple nontrivial solutions of coupled nonlinear Hartree equations with perturbations. This type of equations was used to describe the multi-component Bose–Einstein condensates and the basic quantum chemistry model of small number of electrons interacting with static nuclei. By combining the Nehari constraint and minimax methods, we prove the existence of two nontrivial solutions of the equations.

Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions

This paper deals with a $p$-Kirchhoff type problem involvingsign-changing weight functions. It is shown that under certainconditions, by means of variational methods, the existence ofmultiple nontrivial nonnegative solutions for the problem with thesubcritical exponent are obtained. Moreover, in the case of criticalexponent, we establish the existence of the solutions and prove thatthe elliptic equation possesses at least one nontrivial nonnegativesolution.

Multiplicity results for a class of boundary value problems with impulsive effects

In this paper we study a class of boundary value problems with impulsive effects. Under some natural assumptions, by using Morse theory in combination with the minimax arguments, we establish some new results on the existence of multiple nontrivial solutions for the problems. Recent results from the literature are improved and extended.

Existence and multiplicity of solutions for asymptotically linear Schrödinger–Kirchhoff equations

The purpose of this work is to study a Schrödinger–Kirchhoff equation in R3 with the nonlinearity asymptotically linear and the potential indefinite in sign. By variational methods, we obtain the existence of multiple nontrivial solutions for this problem.

Nontrivial solutions of boundary value problems for second-order functional differential equations

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.