Infinitely many high energy solutions for fractional Schrödinger equations with magnetic field

In this paper we investigate the existence of infinitely many solutions for nonlocal Schrödinger equation involving a magnetic potential $$ (-\triangle )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert \bigr)u, \quad\text{in } {\mathbb {R}}^{N}, $$(−△)Asu+V(x)u=f(x,|u|)u,in RN, where $s\in (0,1)$s∈(0,1) is fixed, $N>2s$N>2s, $V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{+}$V:RN→R+ is an electric potential, the magnetic potential $A:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$A:RN→RN is a continuous function, and $(-\triangle )_{A}^{s}$(−△)As is the fractional magnetic operator. Under suitable assumptions for the potential function V and nonlinearity f, we obtain the existence of infinitely many nontrivial high energy solutions by using the variant fountain theorem.

Existence and Multiplicity of Solutions for Sublinear Schrödinger Equations with Coercive Potentials

In this study, we consider the following sublinear Schrödinger equations −Δu+Vxu=fx,u,for x∈ℝN,ux⟶0,asu⟶∞, where fx,u satisfies some sublinear growth conditions with respect to u and is not required to be integrable with respect to x. Moreover, V is assumed to be coercive to guarantee the compactness of the embedding from working space to LpℝN for all p∈1,2∗. We show that the abovementioned problem admits at least one solution by using the linking theorem, and there are infinitely many solutions when fx,u is odd in u by using the variant fountain theorem.

Homoclinic solutions for fractional Hamiltonian systems with indefinite conditions

In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems {-tD??(-?D?tu(t))-L(t)u(t) + ?W(t,u(t)) = 0, u ? H?(R,Rn), (FHS) where ? ? (1/2,1), t ? R, u ? Rn, L ? C(R,Rn2) is a symmetric matrix for all t ? R, W ? C1(R x Rn,R) and ?W(t,u) is the gradient of W(t,u) at u. The novelty of this paper is that, when L(t) is allowed to be indefinite and W(t,u) satisfies some new superquadratic conditions, we show that (FHS) possesses infinitely many homoclinic solutions via a variant fountain theorem. Recent results are generalized and significantly improved.

Existence of Multiple Solutions for Fourth-Order Elliptic Problem

By using the variant fountain theorem, we study the existence of multiple solutions for a class of superquadratic fourth-order elliptic problem with Navier boundary value condition.

Infinitely Many Weak Solutions of thep-Laplacian Equation with Nonlinear Boundary Conditions

We study the followingp-Laplacian equation with nonlinear boundary conditions:-Δpu+μ(x)|u|p-2u=f(x,u)+g(x,u), x∈Ω,|∇u|p-2∂u/∂n=η|u|p-2uandx∈∂Ω, whereΩis a bounded domain inℝNwith smooth boundary∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) andf,gdo not need to satisfy the(P.S)or(P.S*)condition.

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.