scholarly journals Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

2017 ◽  
Vol 37 (4) ◽  
pp. 2207-2226 ◽  
Author(s):  
Li Yin ◽  
◽  
Jinghua Yao ◽  
Qihu Zhang ◽  
Chunshan Zhao ◽  
...  
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Elliptic Systems ◽  
Constant Sign

Multiple solutions for parametric double phase Dirichlet problems

2020 ◽  
pp. 2050006 ◽  
Author(s):  
N. S. Papageorgiou ◽  
C. Vetro ◽  
F. Vetro
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Parameter ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] precisely in terms of the spectrum of the [Formula: see text]-Laplacian.

scholarly journals Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

2013 ◽  
Vol 113 (2) ◽  
pp. 206 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Case ◽  
Critical Value ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Truncation Techniques ◽  
Semilinear Problem ◽  
Concave Term

We consider a nonlinear parametric Dirichlet problem with parameter $\lambda>0$, driven by the $p$-Laplacian and with a concave term $\lambda|u|^{q-2}u$, $1<q<p$ and a Carathéodory perturbation $f(z,\zeta)$ which is asymptotically $(p-1)$-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value $\lambda^*>0$ of the parameter such that for $\lambda\in (0,\lambda^*)$ the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case ($p=2$), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case $\lambda=\lambda^*$, we show that the nonlinear problem ($p\ne 2$) still has two nontrivial constant sign smooth solutions and the semilinear problem ($p=2$) has three nontrivial smooth solutions, two of which have constant sign.

scholarly journals A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM

2008 ◽  
Vol 50 (2) ◽  
pp. 335-349 ◽  
Author(s):  
NIKOLAOS S. PAPAGEORGIOU ◽  
EUGÉNIO M. ROCHA
Keyword(s):  
Dirichlet Problem ◽  
Variable Exponent ◽  
Point Theory ◽  
Critical Groups ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Multiplicity Theorem ◽  
Noncoercive Problems ◽  
Truncation Techniques

AbstractWe consider a nonlinear Dirichlet problem driven by thep(ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term

2016 ◽  
Vol 19 (04) ◽  
pp. 1650043 ◽  
Author(s):  
Hua Chen ◽  
Shuying Tian ◽  
Yawei Wei
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Singular Potential ◽  
Degenerate Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Potential Term ◽  
Degenerate Elliptic

The present paper is concern with the Dirichlet problem for semi-linear corner degenerate elliptic equations with singular potential term. We first give the preliminary of the framework and then discuss the weighted corner type Hardy inequality. By using the variational method, we prove the existence of multiple solutions for the Dirichlet boundary-value problem.

On Dirichlet problem for Beltrami equations in finitely connected domains

2021 ◽  
Vol 34 ◽  
pp. 85-99
Author(s):  
Ihor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Boundary Behavior ◽  
Beltrami Equation ◽  
Elliptic Systems ◽  
Beltrami Equations ◽  
Prime Ends ◽  
Wide Circle ◽  
Continuous Boundary ◽  
Homeomorphic Solution

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

scholarly journals The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO

2019 ◽  
Vol 12 (3) ◽  
pp. 605-720 ◽  
Author(s):  
José María Martell ◽  
Dorina Mitrea ◽  
Irina Mitrea ◽  
Marius Mitrea
Keyword(s):  
Dirichlet Problem ◽  
Half Space ◽  
Elliptic Systems
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