Sign-changing Potential Research Articles

Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials

In this paper we consider a mean field problem on a compact surface without boundary in presence of conical singularities. The corresponding equation, named after Liouville, appears in the Gaussian curvature prescription problem in Geometry, and also in the Electroweak Theory and in the abelian Chern–Simons–Higgs model in Physics. Our contribution focuses on the case of sign-changing potentials, and gives results on compactness, existence and multiplicity of solutions.

Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential

We investigate a class of generalized quasilinear Schrodinger equations \begin{document}$ -{\rm div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u=f(x,u) \mbox{ in }\mathbb{R}^{N},$ \end{document} where \begin{document}$ g(u):\mathbb{R}\to\mathbb{R}^{+} $\end{document} is a nondecreasing function with respect to \begin{document}$ |u| $\end{document} , the potential \begin{document}$ V(x) $\end{document} and the primitive of the nonlinearity \begin{document}$ f(x,u) $\end{document} are allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions. The proof is based on a change of variable as well as symmetric Mountain Pass Theorem.

Positive and nodal solutions for a nonlinear Schrödinger–Poisson system with sign-changing potentials

We use a variational approach to deal with the system −Δu+V(x)u+K(x)ϕu=a(x)|u|p−1u,−Δϕ=K(x)u2,x∈R3, with 3<p<5. Under some mild conditions on the sign-changing potentials V and a we obtain two nonzero solutions. One of them is nonnegative and the other one changes sign.

Nonhomogeneous eigenvalue problem with indefinite weight

This paper, following the theory of partial differential equations on the Orlicz–Sobolev spaces, is mainly concerned with the nonhomogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. The results show that the spectrum of such problems contains a continuous family of eigenvalues.

Multiple solutions for nonhomogeneous schrödinger-poisson equations with sign-changing potential

In this article, we study the following nonhomogeneous Schrodinger-Poisson equations {-Δu+λV(x)u+K(x)ϕu=f(x,u)+g(x),x∈R3,-Δϕ=K(x)u2,x∈R3,where λ > 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be sign-changing.

On a class of fractional Schrödinger equations in with sign-changing potential

We are interested in finding solutions to a class of problems involving the fractional Laplacian operator. Specifically, we study the equation where , denotes the fractional Laplacian of order s, , V(x) is a continuous and unbounded potential which may change sign, and the nonlinearity is a continuous function which may be unbounded in x since its growth is controlled by V(x) and has subcritical growth in in the sense of the Sobolev embedding. Assuming suitable conditions under V(x) and and applying a approach variational, we prove the existence of the multiplicity of solution for this equation.

Existence of multiple solutions for modified Schrödinger–Kirchhoff–Poisson type systems via perturbation method with sign-changing potential

In this paper, we prove the existence of positive solutions and negative solutions for the following modified Schrödinger–Kirchhoff–Poisson type systems {−(a+b∫R3∣∇u∣2)Δu+V(x)u+ϕu−12uΔ(u2)=f(x,u),inR3,−Δϕ=u2,inR3, where a>0, b≥0, V and f are continuous functions and V(x) is allowed to be sign-changing. Under some certain assumptions on V and f, we prove the existence of nontrivial nonnegative solutions, nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method.

New existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff problems involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following fractional p-Laplacian equation of Schrodinger–Kirchhoff type with sign-changing potential $$\begin{aligned} M\left( \int \int _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\Delta )^s_p u+V(x)|u|^{p-2}u=f(x,u)+ g(x),\quad x\in {\mathbb {R}}^N, \end{aligned}$$ where \((-\Delta )^s_p\) denotes the fractional p-Laplacian of order \(s\in (0,1)\), \(2\le p<\infty \), \(ps<N\), V(x) is allowed to be sign-changing and \(g : {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a perturbation. Under some certain assumptions on f which is much weaker than those in Pucci et al. (Calc. Var. 54 : 2785-2806 2015), by using variation methods, we obtain infinitely many solutions for this equations. Our results generalize and extend some existing results.

Infinitely many solutions for super-quadratic Kirchhoff-type equations with sign-changing potential

This paper is concerned with the following Kirchhoff type equation − a + b ∫ R 3 | ∇ u | 2 d x △ u + V ( x ) u = f ( x , u ) , x ∈ R 3 , u ( x ) → 0 , | x | → ∞ , where a > 0 , b ≥ 0 , V ∈ C ( R 3 , R ) , f ∈ C ( R 3 × R , R ) , V ( x ) and f ( x , u ) u are both allowed to be sign-changing. Using a weaker assumption lim | t | → ∞ ∫ 0 t f ( x , s ) d s | t | 3 = ∞ uniformly in x ∈ R 3 , instead of the common assumption lim | t | → ∞ ∫ 0 t f ( x , s ) d s | t | 4 = ∞ uniformly in x ∈ R 3 , we establish the existence of infinitely many high energy solutions of the above problem, where some new tricks are introduced to overcome the competing effect of nonlocal term. Our result unifies both asymptotically cubic or super-cubic cases, which generalizes and improves the existing ones.

Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential

In the paper, we study a class of biharmonic equations with p -Laplacian and singular sign-changing potential as follows: Δ 2 u − β Δ p u + V λ ( x ) u = 0 in R N , u ∈ H 2 ( R N ) , where N ≥ 5 , β < 0 , Δ p u = div ( | ∇ u | p − 2 ∇ u ) with p ≥ 2 and V λ ( x ) = λ a ( x ) − b ( x ) with λ > 0 . Under some suitable assumptions on a and b , we obtain the existence of nontrivial solutions for λ large enough.

High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential

In the present paper, we consider the following modified quasilinear fourth-order elliptic equation (1.1){Δ2u−Δu+V(x)u−12Δ(u2)u=f(x,u),forx∈RN,u(x)∈H2(RN), where Δ2:=Δ(Δ) is the biharmonic operator, V∈C(RN,R) and f∈C(RN×R,R), N≤5, are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods.

Infinitely many small solutions for a sublinear Schrödinger–Poisson system with sign-changing potential

In this paper, we study the existence of infinitely many small solutions for a class of sublinear Schrödinger–Poisson system with sign-changing potential. By using a dual approach, we prove that the problem has infinitely many small solutions. As a main novelty with respect to some previous results, we do not require any growth condition on the nonlinear term.

Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials

We consider the following quasilinear Schrodinger system in $${\mathbb{R}^N}$$ with $${N\geq3}$$ : P $$\left\{\begin{array}{l}\sum_{i,j=1}^{N}D_j(a_{ij}(u)D_i u)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(u) D_i u D_j u-A(x) u+F_u(u,v)=0 \\ \sum_{i,j=1}^{N}D_j(a_{ij}(v)D_iv)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(v) D_i v D_j v-B(x)v+F_v(u,v)=0,\end{array} \right.$$ where $${D_i=\frac{\partial}{\partial x_i},\ \ D_s a_{ij}(s)=\frac{d}{ds}a_{ij}(s)}$$ , $${F(u,v)}$$ is the coupling term, $${A(x)}$$ and $${B(x)}$$ are finite and sign-changing potential functions. Using an approximation scheme and $${q}$$ -Laplacian regularization, we prove the existence of infinitely many solutions for system $${(P)}$$ .

Multiple solutions for a nonlinear Schrödinger–Poisson system with sign-changing potential

This paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear Schrödinger–Poisson system. Under some suitable conditions, some criteria on the existence of nontrivial solutions, including a ground state solution, two solutions and a sequence of nontrivial solutions {uk} with ‖uk‖→0 as k→+∞, is established by applying the Morse theory and variational methods. Some known results in the literatures are extended and improved.

New Existence of Solutions for the Fractional p-Laplacian Equations with Sign-Changing Potential and Nonlinearity

In the present paper, we consider the fractional p-Laplacian equation $$(-\Delta)_{p}^{s}u + V(x)|u|^{p-2}u = f(x, u),\quad \forall \in R^{N},$$ (1.1) where \({p \geq 2, N \geq 2}\), \({0 < s < 1}\), \({V \in C(R^N, R)}\) and \({f \in C(R^N \times R, R)}\) are allowed to be sign-changing. In such a double sign-changing case, a new result on the existence of nontrivial solutions for Eq. (1.1) is obtained via variational methods, which is even new for p = 2.

On a class of superlinear (p,q)-Laplacian type equations on [formula omitted

Starting from a new sum decomposition of W1,p(RN)∩W1,q(RN) and using a variational approach, we investigate the existence of multiple weak solutions of a (p,q)-Laplacian equation on RN, for 1<q<p<N, with a sign-changing potential and a Carathéodory reaction term satisfying the celebrated Ambrosetti–Rabinowitz condition. Our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p-Laplacian.

Positive solutions for semilinear elliptic systems with sign-changing potentials

Abstract In this paper, we study the existence of positive solutions of the Dirichlet problem -Δu = λ p(x)f(u; v) ; -Δv = λ q(x)g(u; v); in D, and u = v = 0 on ∂∞D, where D ⊂ Rn (n ≥ 3) is an C1,1-domain with compact boundary and λ &gt; 0. The potential functions p; q are not necessarily bounded, may change sign and the functions f; g : ℝ2 → ℝ are continuous with f(0; 0) &gt; 0, g(0; 0) &gt; 0. By applying the Leray- Schauder fixed point theorem, we establish the existence of positive solutions for λ sufficiently small.

Multiplicity for a 4-sublinear Schrödinger–Poisson system with sign-changing potential via Morse theory

In this paper, we study a 4-sublinear Schrödinger–Poisson system with sign-changing potential. Under some suitable assumptions, the existence of two nontrivial solutions are obtained by using the Morse theory. Our result improves the recent ones of Chen and Zhang (2014) [6].

Weak solutions for a fractional p-Laplacian equation with sign-changing potential

In this paper, we use some critical point theorems to discuss the existence of weak solutions for the fractional p-Laplacian equation in where , , is a parameter, is the fractional p-Laplacian and is a Carathéodory function.

Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.