Sign-changing Potential Research Articles

Positive solutions of singular elliptic systems with multiple parameters and Caffarelli–Kohn–Nirenberg exponents

This paper is concerned with the existence of positive solutions for a class of quasilinear singular elliptic systems with Dirichlet boundary condition. By studying the competition between the Caffarelli–Kohn–Nirenberg exponents, the sign-changing potentials and the nonlinear terms, we establish an interval on the range of multiple parameters over which solutions exist in an appropriate weighted Sobolev space. The arguments rely on the method of weak sub- and super-solutions.

On an eigenvalue problem with variable exponents and sign-changing potential

On an eigenvalue problem with variable exponents and sign-changing potential

Nonradial solutions for semilinear Schrödinger equations with sign-changing potential

In this paper, we investigate the existence of infinite nonradial solutions for the Schrodinger equations ( 4 u + b(jxj)u = f(jxj, u), x2 R N , u2 H 1 (R N ),

Multiple solutions for superlinear Schrödinger–Poisson system with sign-changing potential and nonlinearity

In this paper, we study the following Schrödinger–Poisson system {−Δu+V(x)u+λϕ(x)u=f(x,u),in R3,−Δϕ=u2,in R3, where the nonlinearity f and the potential V are allowed to be sign-changing. Under some appropriate assumptions on V and f, we obtain the existence of two different solutions of the system via the Ekeland’s variational principle and the Mountain Pass Theorem.

Infinitely many standing wave solutions for the nonlinear Klein–Gordon–Maxwell system with sign-changing potential

In this paper, we study the infinitely many solutions for the nonlinear Klein–Gordon–Maxwell system {−Δu+V(x)u−(2ω+ϕ)ϕu=f(x,u),x∈R3,Δϕ=(ω+ϕ)u2,x∈R3, where ω>0 is a constant, u, ϕ:R3→R, the potential V(x) is allowed to be sign-changing, and the primitive of the nonlinearity f is of super-linear growth near infinity. Our super-linear conditions are weaker than the usual super-linear conditions.

Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential

In this paper, we study the following quasilinear Schrödinger equation of the form−Δu+V(x)u−Δ(u2)u=g(x,u),x∈RN, where the potential V(x) is allowed to be sign-changing, and the primitive of the nonlinearity g(x,u) is of superlinear growth at infinity in u and is also allowed to be sign-changing. We obtain the existence of infinitely many nontrivial solutions by using dual approach and Mountain Pass Theorem.

Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential

In this paper, we study the existence and multiplicity of solutions for the Schrodinger–Poisson equations $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\lambda V(x)u+K(x)\phi u=f(x,u)\ \ \ \ \ &{} \ \text{ in }\mathbb {R}^3,\\ -\Delta \phi =K(x)u^2\ \ \ \ \ \ &{} \ \text{ in } \mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(\lambda >0\) is a parameter, the potential \(V\) may change sign and \(f\) is either superlinear or sublinear in \(u\) as \(|u|\rightarrow \infty \).

Sign-changing solutions for schrödinger equations with vanishing and sign-changing potentials

In this article, we study the existence of sign-changing solutions for the following Schrödinger equation −Δu+λV(x)u=K(x)|u|p−2u x∈ℝN, u→0 as |x| →+∞, where N ≥ 3, λ > 0 is a parameter, 2 >p >2NN−2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where $${\Omega \subset \mathbb{R}^N,N \geq 3}$$ , is a bounded domain containing the origin with smooth boundary $${\partial \Omega ; h_i, i = 1, 2}$$ , are nonhomogeneous potentials; $${(F_u, F_v) = \nabla F}$$ stands for the gradient of a sign-changing C 1-function $${F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}$$ in the variable $${{w = (u, v) \in \mathbb{R}^2}}$$ ; and λ and μ are parameters.

Existence of infinitely many solutions for [formula omitted]-Laplacian equations in [formula omitted

Based on the cohomological linking method for cones and a new direct sum decomposition of W1,p(RN), we study the existence and multiplicity of solutions of a p-Laplacian equation in RN with sign-changing potential and subcritical p-superlinear nonlinearity. Our assumptions are mild and different from those studied previously.

Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations {−△u+V(x)u=f(x,u),x∈RN,u∈H1(RN), where the potential V is allowed to be sign-changing, and the primitive of the nonlinearity f is of super-quadratic growth near infinity in u and is also allowed to be sign-changing. Our super-quadratic growth conditions weaken the Ambrosetti–Rabinowitz type condition.

Nonlinear Schrödinger Equations with Sign-Changing Potential

Abstract In this paper, we show that the following system −Δu + λV(x)u = g(x, v), −Δv + λV(x)v = f (x, u), x ∈ ℝN (0.1) possesses at least one non-trivial solution pair (u, v) for λ > 0 large enough, where f (x, t), g(x, t) are continuous functions on ℝN × ℝ and super-linear at t = 0 as well as at t = +∞, V(x) is allowed to be sign-changing. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition.

On comparison principles for the periodic Hill's equation

In this work, we make an exhaustive study of the properties of the Green's function related to the periodic boundary value problem L α x ≡ x ″ + a ( t ) x = 0 , x ( 0 ) = x ( T ) , x ′ ( 0 ) = x ′ ( T ) , with a sign-changing potential a(t). Moreover, we obtain new explicit criteria that ensures that the maximum or anti-maximum principle holds for this equation. The given criteria complement previous results in the literature.

Existence of nontrivial solutions for p-Laplacian equations in [formula omitted

In this paper, we consider a p-Laplacian equation in R N with sign-changing potential and subcritical p-superlinear nonlinearity. By using the cohomological linking method for cones, an existence result is obtained. We also give a result on the existence of periodic solutions for one-dimensional p-Laplacian equations which can be proved by the same method.

Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations −Δu+V(x)u=f(x,u), x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing.

PERIODIC SOLUTIONS OF SINGULAR DIFFERENTIAL EQUATIONS WITH SIGN-CHANGING POTENTIAL

AbstractIn this paper we study the existence of positive periodic solutions to second-order singular differential equations with the sign-changing potential. Both the repulsive case and the attractive case are studied. The proof is based on Schauder’s fixed point theorem. Recent results in the literature are generalized and significantly improved.

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the particular eigenvalue given by the infimum of the associated Rayleigh quotient when the variable potential lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.

Asymptotically linear Schrödinger equations with sign-changing potential

Abstract In this paper, we show that the following system: (0.1) − Δ u + V ( x ) u = g ( x , v ) , − Δ v + V ( x ) v = f ( x , u ) , x ∈ R N possesses at least one non-trivial solution pair ( u , v ) , where f ( x , t ) , g ( x , t ) are continuous functions on R N × R and asymptotically linear as t → + ∞ , and V ( x ) is allowed to be sign-changing.

A generalized anti-maximum principle for the periodic one-dimensional [formula omitted]-Laplacian with sign-changing potential

It is known that the anti-maximum principle holds for the quasilinear periodic problem ( | u ′ | p − 2 u ′ ) ′ + μ ( t ) ( | u | p − 2 u ) = h ( t ) , u ( 0 ) = u ( T ) , u ′ ( 0 ) = u ′ ( T ) , if μ ≥ 0 in [ 0 , T ] and 0 < ‖ μ ‖ ∞ ≤ ( π p / T ) p , where π p = 2 ( p − 1 ) 1 / p ∫ 0 1 ( 1 − s p ) − 1 / p d s , or p = 2 and 0 < ‖ μ ‖ α ≤ inf { ‖ u ′ ‖ 2 2 ‖ u ‖ α 2 : u ∈ W 0 1 , 2 [ 0 , T ] ∖ { 0 } } for some α , 1 ≤ α ≤ ∞ . In this paper we give sharp conditions on the L α -norm of the potential μ ( t ) in order to ensure the validity of the anti-maximum principle even in the case where μ ( t ) can change its sign in [ 0 , T ] .

Bound states for a stationary nonlinear Schrödinger-Poisson system with sign-changing potential in ℝ3

We study the following Schrödinger-Poisson system(pλ){−Δu+V(x)u+λϕ(x)u = Q(x)uP,x∈ℝ3,−Δϕ = u2, lim |x|→+∞ϕ(x) = 0,u>0,where λ ≥ 0 is a parameter, 1 <p < +∞, V(x) and Q(x) are sign-changing or non-positive functions in L∞(ℝ3). When V(x) ≡ Q(x) ≡ 1, D. Ruiz [19] proved that (Pλ) with p ∈ (2, 5) has always a positive radial solution, but (Pλ) with p ∈ (1,2] has solution only if λ > 0 small enough and no any nontrivial solution if λ ≥ ¼. By using sub-supersolution method, we prove that there exists λ0 > 0 such that (Pλ) with p ∈ (l,+∞) has always a bound state (H1(ℝ3) solution) for λ ∈ [0, λ0) and certain functions V(x) and Q(x) in L∞(ℝ3). Moreover, for every λ ∈ [0, λ0), the solutions uλ of (Pλ) converges, along a subsequence, to a solution of (P0) in H1 as λ → 0.