Related papers: Dual variational methods for time-harmonic nonline…
We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$…
We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in…
Based on the analysis of biquaternion quadratic forms of field, it is shown that Maxwell equations arise as a consequence of the principle of conservation of the energy-momentum flow of field in space-time. It turns out that this principle…
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of…
This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of…
We consider Maxwell theory on a non-spin manifold. Depending on the choice of statistics for line operators, there are three non-anomalous theories and one anomalous theory with different symmetry fractionalizations. We establish the…
In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding…
This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficient. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of…
In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in R^3. This representation leads to a…
In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the…
We present a method for proving the existence of solutions to a class of one dimensional variational problems. The method is demonstrated by two examples of optimal interpolation problems which are motivated by engineering applications. In…
We obtain a new quantitative deformation lemma, and then gain a new mountain pass theorem. More precisely, the new mountain pass theorem is independent of the functional value on the boundary of the mountain, which improves the well known…
We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned}…
The Maxwell equations for the spherical components of the electromagnetic fields outside sources do not separate into equations for each component alone. We show, however, that general solutions can be obtained by separation of variables in…
Numerical discretization of the large-scale Maxwell's equations leads to an ill-conditioned linear system that is challenging to solve. The key requirement for successive solutions of this linear system is to choose an efficient solver. In…
In this paper, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions type concentration-compactness principle and its variant at infinity for the…
The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quaternion) form. It was shown that the plane…
We show existence and uniqueness results for nonlinear parabolic equations in noncylindrical domains with possible jumps in the time variable
In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schr\"odinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schr\"odinger-Poisson system \begin{equation}\nonumber…
We are concerned with super-Liouville equations on the two sphere, which have variational structure with a strongly-indefinite functional. We prove the existence of non-trivial solutions by combining the use of Nehari manifolds, balancing…