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We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.
We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.
We investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation
for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.
Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is performed. The method of solution utilizes the uniform asymptotic formulas for Green’s and Neumann functions recently obtained by V. Maz’ya and A. Movchan. The theoretical results of the paper are illustrated by numerical simulations.
Assuming $T_{0}$ to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$ and prove stability of square root domains, that is,
which is most suitable for partial differential equation applications. We apply this approach to elliptic second-order partial differential operators of the form
in $L^{2}({\rm\Omega})$ on certain open sets ${\rm\Omega}\subseteq \mathbb{R}^{n}$, $n\in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial {\rm\Omega}$, under general hypotheses on the (typically, non-smooth, unbounded) coefficients and on $\partial {\rm\Omega}$.
We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities ${\it\epsilon}$ have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of ${\it\epsilon}$ is of the order ${\it\eta}^{2}$, where ${\it\eta}>0$ is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the “homogenization limit”, as ${\it\eta}\rightarrow 0$, and derive the limit system of Maxwell equations in $\mathbb{R}^{3}$. Our results extend a number of conclusions of a paper by Zhikov [On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J.16(5) (2004), 719–773] to the case of the full system of Maxwell equations.
In the present paper we deal with the existence of multiple solutions for a quasilinear elliptic problem involving an arbitrary perturbation. Our approach, based on an abstract result of Ricceri, combines truncation arguments with Moser-type iteration technique.
In this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.
The aim of the paper is to establish estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake domains, as well as uniform estimates for the solutions of the Dirichlet problems on pre-fractal approximating domains.
in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.
In this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:
where $f(x,u)$ is asymptotically linear with respect to $u$, $V(x)$ is 1-periodic in each of $x_{1},x_{2},\dots ,x_{N}$ and $\sup [{\it\sigma}(-\triangle +V)\cap (-\infty ,0)]<0<\inf [{\it\sigma}(-\triangle +V)\cap (0,\infty )]$. We develop a direct approach to find ground state solutions of Nehari–Pankov type for the above problem. The main idea is to find a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold ${\mathcal{N}}^{-}$ by using the diagonal method.
In this paper we prove existence and qualitative properties of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Schrödinger equation with the Poisson equation. We use a contraction map approach together with estimates of the Bessel potential used to rewrite the system in an integral form.
We provide an existence result of radially symmetric, positive, classical solutions for a nonlinear Schrödinger equation driven by the infinitesimal generator of a rotationally invariant Lévy process.
In this paper, we prove the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the $Q$ curvature.
which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function $t\mapsto f(x, t)/|t|^{p-1}$. In particular, both $tf(x, t)$ and $tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions.
We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.
We prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potential
and study the stability and oscillation properties of these solutions. We also show that if the equation on ℝN has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝN a family of the exterior Dirichlet problems in ℝN \ K admits a continuum of stable slow-decay infinite-energy solutions.
Let $\Omega $ be a bounded open interval, and let $p\gt 1$ and $q\in (0, p- 1)$. Let $m\in {L}^{{p}^{\prime } } (\Omega )$ and $0\leq c\in {L}^{\infty } (\Omega )$. We study the existence of strictly positive solutions for elliptic problems of the form $- (\vert {u}^{\prime } \mathop{\vert }\nolimits ^{p- 2} {u}^{\prime } ){\text{} }^{\prime } + c(x){u}^{p- 1} = m(x){u}^{q} $ in $\Omega $, $u= 0$ on $\partial \Omega $. We mention that our results are new even in the case $c\equiv 0$.
Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.