Multiplicity Of 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.

Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with φp-Laplacian

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φp-Laplacian for the discrete 2n-th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied.

Multiple Nontrivial Solutions for Biharmonic Systems Involving Rellich-type Potentials and Homogeneous Terms

Multiple Nontrivial Solutions for Biharmonic Systems Involving Rellich-type Potentials and Homogeneous Terms

Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition

Abstract In this article, we study a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity ( − Δ ) s u = λ u + f ( x , u ) , in Ω , u = 0 , in R N \ Ω . \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{\left(-\Delta )}^{s}u=\lambda u+f\left(x,u),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right.\end{array}\right. Multiplicity of nontrivial solutions is obtained when the parameter is near the eigenvalue of the fractional Laplace operator without Ambrosetti and Rabinowitz condition for the nonlinearity. Our methods are the combination of minimax method, bifurcation theory, and Morse theory.

Multiple nontrivial periodic solutions to a second-order partial difference equation

<abstract><p>In this article, applying variational technique as well as critical point theory, we establish a series of criteria to ensure the existence and multiplicity of nontrivial periodic solutions to a second-order nonlinear partial difference equation. Our results generalize some known results. Moreover, numerical stimulations are presented to illustrate applications of our major findings.</p></abstract>

Existence and multiplicity of nontrivial solutions for a class of Kirchhoff-Schrödinger-Poisson systems

We consider a class of Kirchhoff-Schrödinger-Poisson systems. By using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem of Rabinowitz, we prove the existence and multiplicity of nontrivial solutions. Our conditions weaken the Ambrosetti Rabinowitz type condition.

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 and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs

In this paper, we investigate the existence and multiplicity of nontrivial solutions for poly-Laplacian system on a finite graph G=(V, E), which is a generalization of the Yamabe equation on a finite graph. When the nonlinear term F satisfies the super-(p, q)-linear growth condition, by using the mountain pass theorem we obtain that the system has at least one nontrivial solution, and by using the symmetric mountain pass theorem, we obtain that the system has at least dim W nontrivial solutions, where W is the working space of the poly-Laplacian system. We also obtain the corresponding result for the poly-Laplacian equation. In some sense, our results improve some results in (Grigor’yan et al. in J. Differ. Equ. 261(9):4924–4943, 2016).

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.

Results on multiple nontrivial solutions to partial difference equations

<abstract><p>In this paper, we consider the existence and multiplicity of nontrivial solutions to second order partial difference equation with Dirichlet boundary conditions by Morse theory. Given suitable conditions, we establish multiple results that the problem admits at least two nontrivial solutions. Moreover, we provide five examples to illustrate applications of our theorems.</p></abstract>

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>

Existence and multiplicity of solutions for Kirchhoff–Schrödinger–Poisson system with critical growth

This paper is concerned with the following Kirchhoff–Schrödinger–Poisson system: [Formula: see text] where constants [Formula: see text], [Formula: see text] and [Formula: see text] are the parameters. Under some appropriate assumptions on [Formula: see text], [Formula: see text] and [Formula: see text], we prove the existence and multiplicity of nontrivial solutions for the above system via variational methods. Some recent results from the literature are greatly improved and extended.

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.

Multiple periodic solutions for an asymptotically linear wave equation withx-dependent coefficients

In this paper, we investigate the existence of multiple nontrivial periodic solutions for an asymptotically linear wave equation with x-dependent coefficients. Such a mathematical model can be derived from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By constructing a suitable function space, we characterize the problem as a variational problem, and then we prove that there are at least three nontrivial periodic solutions via variational methods.

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.

Uniqueness and positivity issues in a quasilinear indefinite problem

We consider the problem $$\begin{aligned} (P_\lambda )\quad -\Delta _{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\ge 0\quad \text{ in } \Omega \end{aligned}$$under Dirichlet or Neumann boundary conditions. Here \(\Omega \) is a smooth bounded domain of \({\mathbb {R}}^{N}\) (\(N\ge 1\)), \(\lambda \in {\mathbb {R}}\), \(1<q<p\), and \(a\in C({\overline{\Omega }})\) changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for \(\lambda <0\) the functional $$\begin{aligned} I_{\lambda }(u):=\int _{\Omega }\left( \frac{1}{p}|\nabla u|^{p}-\frac{\lambda }{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \end{aligned}$$defined in \(X=W_{0}^{1,p}(\Omega )\) or \(X=W^{1,p}(\Omega )\), has exactly one nonnegative global minimizer, and this one is the only solution of \((P_{\lambda })\) being positive in \(\Omega _{a}^{+}\) (the set where \(a>0\)). In particular, this problem has at most one positive solution for \(\lambda <0\). Under some condition on a, the above uniqueness result fails for some values of \(\lambda >0\) as we obtain, besides the ground state solution, a second solution positive in \(\Omega _{a}^{+}\). We also provide conditions on \(\lambda \), a and q such that these solutions become positive in \(\Omega \), and analyze the formation of dead cores for a generic solution.

Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

In this paper, we study the following nonlocal problem − a − b ∫ Ω ∇ u 2 d x Δ u = λ u + f x u p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b &gt; 0 are constants, 1 &lt; p &lt; 2 , λ &gt; 0 , f ∈ L ∞ Ω is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3 . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f ∞ is small enough.

GLOBAL BIFURCATION RESULT FOR DISCRETE BOUNDARY VALUE PROBLEM INVOLVING THE MEAN CURVATURE OPERATOR

<abstract> In this paper, by applying bifurcation technique, we obtain that there are two distinct unbounded continua $ \mathcal{C}_k^+ $ and $ \mathcal{C}_k^- $ for a class of discrete Dirichlet problem involving the mean curvature operator which bifurcate from intervals of the line of trivial solutions. Under some suitable conditions on nonlinear term near at the origin, we will show the existence and multiplicity of nontrivial solutions. </abstract>