Multiple solutions for semilinear discontinuous variational problems with lack of compactness

2021 ◽  
Author(s):  
Claudianor O. Alves ◽  
Jefferson A. dos Santos
Keyword(s):  
Multiple Solutions ◽  
Variational Problems ◽  
Lack Of Compactness

scholarly journals Multiple solutions of double phase variational problems with variable exponent

2020 ◽  
Vol 13 (4) ◽  
pp. 385-401 ◽  
Author(s):  
Xiayang Shi ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš ◽  
Qihu Zhang
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Quasilinear Equation ◽  
Variational Problems ◽  
Quasilinear Elliptic Equations ◽  
Weighting Method ◽  
Double Phase ◽  
Unbounded Potential ◽  
Quasilinear Elliptic

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like {|\xi|^{q(x)-2}\xi} for small {|\xi|} and like {|\xi|^{p(x)-2}\xi} for large {|\xi|}, where {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)<q(\,\cdot\,)<N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in \mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where {p(\,\cdot\,)} and {q(\,\cdot\,)} are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

scholarly journals Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems

2017 ◽  
Vol 39 (1) ◽  
pp. B29-B52 ◽  
Author(s):  
J. H. Adler ◽  
D. B. Emerson ◽  
P. E. Farrell ◽  
S. P. MacLachlan
Keyword(s):  
Multiple Solutions ◽  
Variational Problems ◽  
Nested Iteration

Multiple Solutions of Some Nonlinear Variational Problems

2006 ◽  
Vol 6 (2) ◽  
Author(s):  
A.M. Candela ◽  
G. Palmieri
Keyword(s):  
Multiple Solutions ◽  
Variational Problems ◽  
Multiplicity Results ◽  
Existence And Multiplicity

AbstractThe aim of this paper is to prove some existence and multiplicity results for functionals of type J(u) = ∫

MULTIPLE SOLUTIONS IN A NARROW HORIZONTAL AIR-FILLED ANNULUS

1998 ◽  
Author(s):  
Pierre Cadiou ◽  
Guy Lauriat ◽  
Gilles Desrayaud
Keyword(s):  
Multiple Solutions

An Overview of Risk-Based EEE Counterfeit Part Detection Based on SAE AS6171

2018 ◽  
Author(s):  
Michael H. Azarian
Keyword(s):  
Risk Assessment ◽  
Multiple Solutions ◽  
Test Equipment ◽  
Assessment Methodology ◽  
Test Plan ◽  
Specific Test ◽  
Risk Assessment Methodology

Abstract As counterfeiting techniques and processes grow in sophistication, the methods needed to detect these parts must keep pace. This has the unfortunate effect of raising the costs associated with managing this risk. In order to ensure that the resources devoted to counterfeit detection are commensurate with the potential effects and likelihood of counterfeit part usage in a particular application, a risk based methodology has been adopted for testing of electrical, electronic, and electromechanical (EEE) parts by the SAE AS6171 set of standards. This paper provides an overview of the risk assessment methodology employed within AS6171 to determine the testing that should be utilized to manage the risk associated with the use of a part. A scenario is constructed as a case study to illustrate how multiple solutions exist to address the risk for a particular situation, and the choice of any specific test plan can be made on the basis of practical considerations, such as cost, time, or the availability of particular test equipment.

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