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In this paper, we consider the following fractional Kirchhoff equation with critical nonlinearity a+b∫R3(−Δ)s2u2dx(−Δ)su+V(x)u=Q(x)f(u)+|u|2s∗−2u,x∈R3,where a,b>0, (−Δ)s is the fractional Laplace operator with s∈(34,1), V vanishes at infinity and 2s∗=63−2s. Under appropriate assumptions on f, we prove the existence of positive solution by using the variational method.

This paper is focused on following time-harmonic Maxwell equation: ∇×(μ−1(x)∇×u)−ω2e(x)u=f(x,u),inΩ,ν×u=0,on∂Ω, where Ω⊂R3 is a bounded Lipschitz domain, ν:∂Ω→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as |u|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor e∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

This paper is concerned with the following Schrodinger elliptic system where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ+V, f(x,t) and g(x,t) are superlinear in t at infinity. By using non-Nehari manifold method developed recently by Tang, we demonstrate the existence of the ground state solutions of Nehari-Pankov type for the above problem with periodic and asymptotically periodic nonlinearity. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the nonlinear Schrodinger–Poisson system −Δu+λV(x)u+K(x)ϕu=f(x,u),x∈R3,−Δϕ=K(x)u2,x∈R3, where λ > 0 is a parameter. We require that V≥0 and has a bounded potential well V−1(0). Combining this with other suitable assumptions on K and f, the existence of nontrivial solutions is obtained by using variational methods. Moreover, the concentration of solutions is also explored on the set V−1(0) as λ→∞. Copyright © 2015 John Wiley & Sons, Ltd.

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang et al. (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems

This paper is concerned with the nonlinear Dirac equations Under suitable assumptions on the nonlinearity, we establish the existence of infinitely many large energy solutions by the generalized variant fountain theorem developed recently by Batkam and Colin. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we deal with existence and multiplicity of solutions of Kirchhoff-type problems Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the preceding problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical point theory. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problems where L(t) and W(t, x) are neither autonomous nor periodic in t. Our approach is variational and it is based on the critical point theory. We prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function. Our theorems appear to be the first such result and our results extend some recent works.

By using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.