Search results for " solution"

Existence of two positive solutions for anisotropic nonlinear elliptic equations

2021

This paper deals with the existence of nontrivial solutions for a class of nonlinear elliptic equations driven by an anisotropic Laplacian operator. In particular, the existence of two nontrivial solutions is obtained, adapting a two critical point results to a suitable functional framework that involves the anisotropic Sobolev spaces.

First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities

2022

AbstractWe study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \in {\mathbb {C}}$$ λ , μ ∈ C , $$\lambda \ne 0$$ λ ≠ 0 , and $$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$ u ( 0 , · ) , w ∈ L loc 1 ( R N , C ) . We consider both the cases where $$\mu =0$$ μ = 0 and $$\mu \ne 0$$ μ ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In ea…

Multiple solutions with sign information for a (p,2)-equation with combined nonlinearities

2020

We consider a parametric nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a Laplacian (a (p,2)-equation) and with a reaction which has the competing effects of two distinct nonlinearities. A parametric term which is (p−1)-superlinear (convex term) and a perturbation which is (p−1)-sublinear (concave term). First we show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, all with sign information. Then by strengthening the regularity of the two nonlinearities we produce two more nodal solutions, for a total of seven nontrivial smooth solutions all with sign informations. Our proofs use critical point theory, critical gro…

Existence and multiplicity of solutions for non linear elliptic Dirichlet systems

2012

The existence and multiplicity of solutions for systems of nonlinear elliptic equations with Dirichlet boundary conditions is investigated. Under suitable assumptions on the potential of the nonlinearity, the existence of one, or two, or three solutions is established. Our approach is based on variational methods.

Existence Results for Periodic Boundary Value Problems with a Convenction Term

2020

By using an abstract coincidence point theorem for sequentially weakly continuous maps the existence of at least one positive solution is obtained for a periodic second order boundary value problem with a reaction term involving the derivative \(u'\) of the solution u: the so called convention term. As a consequence of the main result also the existence of at least one positive solution is obtained for a parameter-depending problem.

Multiple solutions for a mixed boundary value problem

2010

Positive and nodal solutions for nonlinear nonhomogeneous parametric neumann problems

2020

We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter λ > 0. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution.

On a regularized approach for the method of fundamental solution

2018

The method of fundamental solution is a boundary meshless method recently adopted in the framework of non-invasive neu- roimaging techniques. The method approximates the solution of a BVP by a linear combination of fundamental solutions of the governing PDE. A crucial feature of the method is the placement of the fictitious boundary to avoid the singularities of fundamental solutions. In this paper we report on our experiences with a regularized MFS method in the neuroimaging context.

A meshfree approach for brain activity source modeling

2015

Weak electrical currents in the brain flow as a consequence of acquisition, processing and transmission of information by neurons, giving raise to electric and magnetic fields, which are representable by means of quasi-stationary approximation of the Maxwell’s equations. Measurements of electric scalar potential differences at the scalp and magnetic fields near the head constitute the input data for, respectively, electroencephalography (EEG) and magnetoencepharography (MEG), which allow for reconstructing the cerebral electrical currents and thus investigating the neuronal activity in the human brain in a non-invasive way. This is a typical erectromagnetic inverse problem, since measuremen…

EFFICACY OF ARGENTUM-QUARTZ SOLUTION IN THE TREATMENT OF PERIANAL FISTULAS: A PRELIMINARY STUDY

2015

Objective: Nowadays, an optional and effective medical surgery remains the gold standard for perianal fistulas. Hereby we reported preliminary rsults in favor of using Argentum-Quartz solution for both primary and recurrent perianal fisrtulas. Methods: Three patients with intersphimncter and extrasphinteric fistulas were enrolled. Argentum-Quartz solution was administrated twice a week for sa period of 4 weeks, followed by a pause of 8 days and then another 4 weeks of treatment, totally 16 administrations. After treatment, all patients were monitored for 4-months follow-up. Results: Complete closures of 2 extrasphinteric fistulas and a partial closure with absence of inflammation and supera…