Sign-changing Potential Research Articles

Properties of the positive solutions of fractional p&q-Laplace equations with a sign-changing potential

Properties of the positive solutions of fractional p&q-Laplace equations with a sign-changing potential

Multiple Standing Waves of Matrix Nonlinear Schrödinger Equations with Sign-Changing Potentials

Multiple Standing Waves of Matrix Nonlinear Schrödinger Equations with Sign-Changing Potentials

Multiplicity of solutions for a singular system with sign-changing potential

Abstract This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d {\mathbb{R}^{d}} . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.

Solutions for gauged nonlinear Schrödinger equations on $ {\mathbb R}^2 $ involving sign-changing potentials

This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function $ V $ is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential $ V $ and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory.

Infinitely many solutions for a Schrödinger equation with sign-changing potential and nonlinear term

Infinitely many solutions for a Schrödinger equation with sign-changing potential and nonlinear term

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac–Coulomb operators

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on C^\infty_c(\mathbb{R}^3\setminus\{0\}, \mathbb{C}^4) . In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137

Negative eigenvalues of non-local Schrödinger operators with sign-changing potentials

Simon’s results on the negative spectrum of recurrent Schrödinger operators ( d = 1 , 2 d=1,2 ) are extended to a wider class of potentials and to non-local operators. An example of L 1 − L^1- potental is constructed for which the essential spectrum of two dimensional Schrödinger operator covers the whole axis. Some counterexamples are provided for transient operators ( d ≥ 3 d\geq 3 ) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can’t be improved significantly.

Multiplicity of solutions for the stationary quantum Zakharov system with combined nonlinearities

In this article, we prove the existence and multiplicity of nontrivial solutions for the stationary quantum Zakharov system with a sign-changing potential and a general nonlinearity involving a combination of a superlinear f ( x , u ) source term and a sublinear term ( K ( x ) | u | q − 2 u , K ∈ L 2 2 − q ( R 3 ) , where 1 < q < 2 ) .

Infinitely many high energy solutions for nonlocal fourth-order equation with sign-changing potential

This paper is devoted to a class of general nonlocal fourth-order elliptic equation where is the biharmonic operator, are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature.

The 2-dimensional nonlinear Schrödinger–Maxwell system

In this paper we carry on the study of a system recently introduced by the first author as the planar version of the well known electrostatic Schrödinger–Maxwell equations.In the positive potential case, we exhibit situations where the existence of solutions depends on the strength of the coupling, being this one modulated by a parameter.We also present some results in the case of a sign-changing potential.

Positive solutions of an elliptic equation involving a sign-changing potential and a gradient term

The objective of this paper is to investigate the elliptic singular Laplacian equation Δu−|∇u|q+up−u−δ=0 in RN, where N≥1, 1&lt;q&lt;p and δ&gt;2. Our main contributions consist of establishing the existence of an entire strictly positive solution and analyzing certain properties of its asymptotic behavior, particularly when it exhibits monotonicity.

Existence and multiplicity results for a class of Kirchhoff–Choquard equations with a generalized sign-changing potential

In the present work, we are concerned with the Kirchhoff–Choquard-type equation -M\big(\|\nabla u\|_2^2\big)\Delta u+Q(x)u+\mu\big(V\big(|\cdot|\big)\ast u^2\big)u = f(u)\quad\text{in }\mathbb{R}^2, for M:\mathbb{R}\rightarrow\mathbb{R} given by M(t)=a+bt , \mu&gt;0 , V a sign-changing and possible unbounded potential, Q a continuous external potential, and a nonlinearity f with exponential critical growth. We prove existence and multiplicity of solutions in the nondegenerate case and guarantee the existence of solutions in the degenerate case.

Nontrivial solutions for Klein–Gordon–Maxwell systems with sign-changing potentials

This paper is concerned with the nonlinear Klein–Gordon–Maxwell systems. Unlike all known results in the literature, the Schrödinger operator -Delta +V is allowed to be indefinite and the weaker superlinear conditions are imposed instead of the common 4-superlinear conditions on f. By combining a local linking argument and Morse theory, we obtain that the system admits a nontrivial solution.

Positive and Sign-changing Solutions for Critical Schrödinger–Poisson Systems with Sign-changing Potential

Positive and Sign-changing Solutions for Critical Schrödinger–Poisson Systems with Sign-changing Potential

Positive solutions of fractional Schrödinger-Poisson systems involving critical nonlinearities with potential

In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. By means of the Nehari manifold and Ljusternik-Schnirelmann category, we show how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions, and present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$. The new feature which distinguishes this paper from other related works lies in the fact that we focus on the case of $1 < q < 2$, which has not been presented in the literature.

Multiplicity and concentration of solutions for a fractional Schrödinger–Poisson system with sign-changing potential

This paper is concerned with the following fractional Schrödinger–Poisson system: where is a parameter, , , , is allowed to be sign-changing and f is an indefinite function. We require that with having a potential well Ω whose depth is controlled by λ and for all . Under some suitable assumptions on and , we prove that the above system has at least two nontrivial solutions. Moreover, the phenomenon of concentration of the solutions is also explored.

On nontrivial solutions of nonlinear Schr\xf6dinger equations with sign-changing potential

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.

Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

In this paper, we discuss the existence of multiple solutions of the generalized quasilinear Schrödinger equation of Kirchhoff-type where b>0 is a parameter, . In preceding approaches, the order of the nonlinear term about t at infinity is usually supposed to be higher than . But in this paper, this condition is not necessary. Under weaker condition , where is the primitive function of , , we get the existence of infinitely many solutions of the equation. Our work can be regarded as the extension of those such as [J. Zhang et al. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl Math Comput. 2014;242:491–499] and related works.

Existence and concentration of solutions to the Gross–Pitaevskii equation with steep potential well

In this paper, we consider the following Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases: 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda Q(x)u+\omega _{1}|u|^{2}u+\omega _{2}(K\star |u|^{2})u=0, &{} x\in {\mathbb {R}}^{3},\\ \displaystyle u>0,\quad u\in H^{1}({\mathbb {R}}^{3}),\\ \end{array}\right. } \end{aligned}$$ where $$\lambda >0$$ , $$Q\in C({\mathbb {R}}^{3},{\mathbb {R}})$$ is a potential well, $$\star $$ denotes the convolution, $$K(x)=\frac{1-3\cos ^{2}\theta }{|x|^{3}}$$ and $$\theta =\theta (x)$$ is the angle between the dipole axis determined by the vector x and the vector (0, 0, 1). When $$(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$$ lies in the defined unstable regime, under some suitable conditions on Q, the existence and concentration of nontrivial solutions to problem (0.1) are proved by using variational methods. In particular, the trapping potential is allowed to be sign-changing. Moreover, when $$(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$$ lies in the stable regime, we show that problem (0.1) with small bounded sign-changing potential has only trivial solutions.

Multiple solutions with compact support for a quasilinear Schrödinger equation with sign-changing potentials

ABSTRACT In this paper, we study the existence of compactly supported solutions for the Schrödinger equations with indefinite potentials where , 1<p<2, 0<q<p−1, and change sign in . Firstly, we establish the existence of infinitely many weak solution. Next, we study the compactness of support of classical solutions for the above equation. This paper is a continuation of the recent work established by [Bedoui N, Ounaies H. Qualitative properties and support compactness of solutions for quasilinear Schrödinger equation with sign changing potentials. Nonlinear Anal. 2020;198:111843.].