Existence Of Nontrivial Solutions Research Articles

Standing waves for a class of quasilinear Schrödinger equations

We study the existence of standing wave solutions for a class of quasilinear Schrödinger equations with subcritical or critical growthwhere is a given potential, is a parameter. By variational methods, we investigate the range of parameters where the existence of non-trivial solutions can be guaranteed. Especially, by combing Resonance and Hahn–Banach theorems, we improve some previous results on .

Iterative solutions for fractional nonlocal boundary value problems involving integral conditions

This paper is concerned with the following nonlinear fractional boundary value problem: $$\left \{ \textstyle\begin{array}{l} D_{0+}^{\alpha}u(t)+f(t,u(t))=0,\quad t\in(0,1), u ( 0 ) =u' ( 0 ) =0,\qquad D_{0+}^{\beta}u(1)=\int _{0}^{1}D_{0+}^{\beta}u(t)\, dA(t), \end{array}\displaystyle \right . $$ where $2<\alpha\leq3$ , $0<\beta\leq1$ are real numbers and $\int _{0}^{1}D_{0+}^{\beta}u(t)\, dA(t)$ denotes a Riemann-Stieltjes integral. By means of monotone iterative technique and some inequalities associated with the Green function, not only the existence of nontrivial solutions or positive solutions is obtained but also iterative schemes for approximating the solutions are established, which start off with simple functions, which are feasible for computational purposes. An example is also included to illustrate the main results.

Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

In this paper we study the non-linear fractional Schrodinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) in R^{n}, u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.

Existence and concentration of solutions for Schrödinger–Poisson system with steep potential well

This paper is concerned with the nonlinear Schrödinger–Poisson system urn:x-wiley:mma:media:mma3712:mma3712-math-0001 where λ &gt; 0 is a parameter. We require that V≥0 and has a bounded potential well V−1(0). Combining this with other suitable assumptions on K and f, the existence of nontrivial solutions is obtained by using variational methods. Moreover, the concentration of solutions is also explored on the set V−1(0) as λ→∞. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Existence of solutions for a class of quasilinear Schrödinger equations on R ${\mathbb{R}}$

In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrodinger equations in ${\mathbb{R}}$ with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method.

Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations

This paper is concerned with the following fourth-order elliptic equation $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \Delta ^{2}u-\Delta u+V(x)u=|u|^{p-1}u,\,\mathrm{in}\,\mathbb {R}^{N},\\ u\in H^{2}\left( \mathbb {R}^{N}\right) , \end{array} \right. \end{aligned}$$ where \(p\in (2,\,2_{*}-1),\,u{\text {:}}\,\mathbb {R}^{N}\rightarrow \mathbb R.\) Under some appropriate assumptions on potential V(x), the existence of nontrivial solutions and the least energy nodal solution are obtained by using variational methods.

On the characteristic polynomials of linear functional equations

The solutions of a linear functional equation are typically generalized polynomials. The existence of their non-trivial monomial terms strongly depends on the algebraic properties of some related families of parameters. In extremal cases (the parameters are algebraic numbers or the parameters form an algebraically independent system) we have elegant methods to decide the existence of non-trivial solutions. In this paper we are going to extend and unify the treatment of the existence problem by introducing the characteristic polynomials of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted Ls space, the existence of nontrivial solutions is obtained via variational methods.

Multiplicity of nontrivial solutions for a class of nonlinear Kirchhoff-type equations

This paper deals with the existence of nontrivial solutions of the following Kirchhoff-type equation: $- (a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\, dx )\Delta u+V(x)u=f(x,u)$ , in $\mathbb{R}^{N}$ . Under more relaxed assumptions on V and f, some new criteria on the multiplicity of nontrivial solutions are established by combining Morse theory with local linking and Clark’s theorem.

Nontrivial solutions to boundary value problem with general singular differential operator

In this paper, a class of boundary value problems with general singular differential operator is investigated. The nonlinear term in the boundary value problem is sign-changing and may be unbounded from below. By means of the topological degree of a completely continuous field, the existence of nontrivial solutions is obtained. Finally, an example is given to illustrate the application of our main result.

A positive solution for some critical p-Laplacian systems

This paper deals with the existence of a positive solution for two classes of critical quasilinear system. We prove these results by a variant of mountain pass lemma, combining two convergence theorems and two estimate results. Here we avoid the usual compactness arguments (e.g., Palais-Smale condition or Cerami condition) and reveal the potential of some energy level estimates for the existence of nontrivial solutions.

Existence of nontrivial solutions to quasilinear polyharmonic equations with critical exponential growth

AbstractIn this article, we show the existence of a nontrivial solution of the quasilinear polyharmonic equation Δ

Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.

A Matrix Polynomial Spectral Approach for General Joint Block Diagonalization

Joint block diagonalization (JBD) of a given Hermitian matrix set $\{A_i\}_{i=0}^p$ is to find a nonsingular matrix $W$ such that $W^H A_i W$ for $i=0,1,\dots, p$ are all block diagonal matrices with the same prescribed block diagonal structure. General JBD (GJBD) attempts to solve JBD without knowing the resulting block diagonal structure in advance, which is more difficult than JBD. In this paper, we reveal that GJBD of $\{A_i\}_{i=0}^p$ is strongly connected with the spectral information of the corresponding matrix polynomial $P(\lambda)=\sum_{i=0}^p\lambda^i A_i$. Under some conditions, the solutions to GJBD are characterized by the spectral information, and a necessary and sufficient condition is given for the existence of nontrivial solutions to GJBD. In addition, based on the established theory, two feasible numerical methods are proposed to solve GJBD.

Critical elliptic problems in ℝ2 involving resonance in high-order eigenvalues

In this paper we deal with the following class of problems [Formula: see text] where Ω ⊂ ℝ2 is bounded with smooth boundary, g has a unilateral critical behavior of Trudinger–Moser type and λk denotes the kth eigenvalue of [Formula: see text], k ≥ 2. We prove existence of nontrivial solution for this problem using variational methods.

Existence of solution for a three-point boundary value problem for a second-order impulsive differential equation

In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary value problem with impulses. Under certain growth conditions on the nonlinearity function and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of a nontrivial solution are obtained. Furthermore, we illustrate our results with some examples.

Nontrivial Solutions for an Impulsive Differential Equation with Non-separated Boundary Conditions

In this paper we shall discuss the existence of non-trivial solutions for an impulsive differential equation subject to non-separated boundary conditions. Our criteria for the existence theorems will be expressed in terms of the first eigenvalue of the corresponding non-impulsive problem. The main tool here is topological degree theory.

Multiple positive solutions of a class of non autonomous Schrödinger–Poisson systems

For the nonlinear Schrödinger equation coupled with Poisson equation of the version −Δu+u+ϕu=a(x)∣u∣p−2u+λk(x)u in R3 and −Δϕ=u2 in R3, we prove the existence of two positive solutions in H1(R3) when a(x) is sign changing and the linear part is not coercive. We show that the coupled term ϕu is helpful to find multiple positive solutions when a(x) is sign changing, which gives striking contrast to the known result where ϕu is proven to be an obstacle to get the existence of nontrivial solutions. Surprisingly we show that the term ϕu can play the role similar to a sign condition ∫a(x)e1pdx<0, which has turned out to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Alama et al. (1993)).

Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations

We study the Schrödinger equation:Section.Display where is periodic and is periodic in the -variables, is in a gap of the spectrum of the operator and is asymptotically linear as We prove that under some asymptotically linear assumptions for , this equation has a nontrivial solution. Our assumptions for are different from the classical assumptions raised by Li and Szulkin.

Nontrivial solutions of a semilinear elliptic problem with resonance at zero

Abstract In this paper, by Morse theory we obtain the existence of nontrivial solutions for a semilinear elliptic problem with resonance at zero. Note that in our results, except for the growth condition, we do not need any additional global condition on the nonlinearities of this problem.