Nonexistence Of Nontrivial Solutions Research Articles

Ground state for relativistic nonlinear Schrödinger equations involving general nonlinear term

In this paper, the existence and nonexistence of positive solutions for a relativistic nonlinear Schrödinger equation with vanish potential and general nonlinear term were obtained. First, we reduce this equation to a semilinear elliptic equation by a change. Then, using mountain pass theorem and Strauss compactness lemma, we get the existence of positive radial ground state solutions for this equation. Moreover, the nonexistence of nontrivial solutions for this equation was gotten by Pohozaev identity.

Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t| p–2 t at infinity

Abstract We consider the following p-harmonic problem Δ ( | Δ u | p − 2 Δ u ) + m | u | p − 2 u = f ( x , u ) , x ∈ R N , u ∈ W 2 , p ( R N ) , $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2p ≥ 4 and lim t → ∞ f ( x , t ) | t | p − 2 t = l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

Nonexistence of Sign-Changing Solutions for Some Elliptic Inequalities in Bounded Domains

Nonlinear elliptic inequalities in bounded domains and systems of such inequalities involving terms depending differently on the positive and negative parts of the sought function are considered. Sufficient conditions for the nonexistence of nontrivial solutions to the inequalities and systems under study in corresponding functional classes are obtained.

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.

Stable Solutions of $-\Delta u+\lambda u=|u|^{p-1}u $ in Strips

This paper is devoted to study the following equation $-\Delta u+\lambda u= |u|^{p-1}u \;\;\text{in}\;\Omega $ , with homogeneous Dirichlet or Neumann boundary conditions where $p>1$ , $\lambda >0$ , $\Omega =\mathbb{R}^{n-k}\times \omega $ , $n\geq 2$ , $k\geq 1$ , and $\omega $ is a smoothly bounded domain of $\mathbb{R}^{k}$ . We prove Liouville-type theorems for $C^{2}$ solutions which are stable or stable outside a compact set of $\Omega $ . We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for $p\in [p_{s}(n), p_{s}(n-k)]$ , where $p_{s}(n)$ is the classical Sobolev exponent in dimension $n$ . Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of $\Omega $ for all $p>1$ .

Sobolev hyperbola for the periodic parabolic Lane–Emden system

Consider the periodic parabolic Lane–Emden type system , in a bounded domain of with periodic coefficients and , subject to homogeneous Dirichlet boundary condition. It is known that for the elliptic system , in the 3-dimensional (3D) space, there exists a Sobolev hyperbola , which characterizes the existence and nonexistence of nontrivial solutions. In this paper, we show that such a Sobolev hyperbola is also critical to the existence of periodic solutions for the periodic Lane–Emden type system. Moreover, different from the elliptic system, there exists another cure pq = 1, which is singular for the existence of periodic solutions.

Positive solutions of a superlinear kirchhoff type equation in [formula omitted] (N ≥ 4)

In this paper, we study a class of superlinear Kirchhoff type equations with steep potential well:{−(a∫RN|∇u|2dx+b)Δu+λV(x)u=f(x,u)inRN,u∈H1(RN),where N ≥ 4, a, b > 0, and the parameter λ > 0. By proposing new techniques and introducing new superlinear hypotheses on f, we establish the existence of two positive solutions, one of which will blow up as the nonlocal term vanishes. Some existing results are improved and generalized in the literature. Moreover, the nonexistence of nontrivial solution is also obtained.

Conformal identity of the Chern–Simons gauged wave equations and its applications

The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As applications of it, we show some decay estimates and the nonexistence of nontrivial solutions.

Nonexistence of Nontrivial Solutions with Decay Order for a Biharmonic P-Laplacian Equation and System

We use the Morawetz multiplier to show that there are no nontrivial solutions of certain decay order for a biharmonic equation with a p-Laplacian term and a system of coupled biharmonic equations with p-Laplacian terms in the entire Euclidean space.

On the indefinite Kirchhoff type equations with local sublinearity and linearity

We consider a class of indefinite Kirchhoff-type equations:where , , and . We require that f(x, u) is “local" sublinear at the origin and “local" linear at infinity on u, respectively. Using mountain pass theorem and Ekeland variational principle, we obtain the existence and multiplicity of nontrivial solutions. In particular, the existence criterion of three nontrivial solutions is established. Furthermore, the nonexistence of nontrivial solutions is explored as well.

On the nonlinear Schrödinger–Poisson systems with sign-changing potential

In this paper, we study a nonlinear Schrodinger–Poisson system $$\left\{ \begin{array}{ll} -\Delta u+V_{\lambda }( x) u+\mu K( x) \phi u=f (x, u) & \text{in}\;\mathbb{R}^{3}, \\ -\Delta \phi =K ( x ) u^{2} & \text{in}\;\mathbb{R}^{3},\end{array}\right.$$ where \({\mu > 0}\) is a parameter, \({V_{\lambda }}\) is allowed to be sign-changing and f is an indefinite function. We require that \({V_{\lambda }:=\lambda V^{+}-V^{-}}\) with V+ having a bounded potential well Ω whose depth is controlled by λ and \({V^{-} \geq 0}\) for all \({x\in \mathbb{R} ^{3}}\). Under some suitable assumptions on K and f, the existence and the nonexistence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is explored as well.

Existence and Nonexistence for Elliptic Equation with Cylindrical Potentials, Subcritical Exponent and Concave Term

In this paper, we consider the existence and nonexistence of non-trivial solutions to elliptic equations with cylindrical potentials, concave term and subcritical exponent. First, we shall obtain a local minimizer by using the Ekeland’s variational principle. Secondly, we deduce a Pohozaev-type identity and obtain a nonexistence result.

Nonexistence of solutions for singular quasilinear elliptic equation in $\mathbb{R}^N$

In this paper, we study the nonexistence of solutions for singular quasilinear elliptic equation div(|x| ap|—u|p 2—u)+V (x)|u|p 2u = h(x)|u|s 2u+H(x)|u|r 2u, x 2 RN , (P) where 1 1. When the weight functions V (x),h(x) and H(x) are radially symmetric with respect to x in RN , we establish a variational identity for solution of (P) and then obtain some sufficient conditions on the nonexistence of nontrivial solutions for (P). When V (x)⌘ 0 and the functions h(x),H(x) are not necessarily radially symmetric with respect to x in RN , we get conditions sufficient to ensure the nonexistence of nonnegative entire solutions for (P) via the test function method developed by Mitidieri and Pohozaev.

Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications

The paper is devoted to prove maximum principles for a certain func- tionals defined for solution of the fourth order nonlinear elliptic equation. The max- imum principle so obtained is used to prove the nonexistence of nontrivial solutions of the fourth order nonlinear elliptic equation with some zero boundary conditions. Hopf's maximum principle is the main ingredient.

Nontrivial solutions for a class of critical elliptic equations of Caffarelli–Kohn–Nirenberg type

In this paper, we deal with a class of critical elliptic equations related to Caffarelli–Kohn–Nirenberg inequalities. The novelty is the presence of a critical exponent and singular coefficients which may be sign-changing. Existence and non-existence of nontrivial solutions are established by means of the Mountain Pass Theorem and variational identity.

Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in

We study the following Schrödinger-Poisson system: , , , where are positive radial functions, , , and is allowed to take two different forms including and with . Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the plane where the system has no nontrivial solutions.

Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.

Existence of positive radial solutions for a nonlinear elliptic problem in annulus domains

In this paper, we study a nonlinear elliptic problem in an annulus domain. For which, the nonexistence of nontrivial solutions has been obtained by some authors in star-shaped domain. Although we note that it may admit non-trivial solutions if the domain is non-star shaped. With the use of a variational method, we establish the existence of positive radial solutions in an annulus domain. On the basis of this, we also extend this result to the periodic problem of the corresponding degenerate parabolic problem. Copyright © 2012 John Wiley & Sons, Ltd.

Solutions of the Allen-Cahn equation which are invariant under screw-motion

We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of \({\mathbb R^{3}}\).