scholarly journals Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 83 ◽  
Author(s):  
Mohammed Massar ◽  
EL Miloud Hssini ◽  
Najib Tsouli
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Saddle Point Theorem ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

scholarly journals Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Yu-Cheng An ◽  
Hong-Min Suo
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Blow Up ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Point Theory ◽  
Saddle Point Theorem ◽  
Multiplicity Results ◽  
Near Resonance ◽  
Degenerate Elliptic

We study the degenerate semilinear elliptic systems of the form-div(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ω,u|∂Ω=v|∂Ω=0, whereΩ⊂RN(N≥2)is an open bounded domain with smooth boundary∂Ω, the measurable, nonnegative diffusion coefficientsh1,h2are allowed to vanish inΩ(as well as at the boundary∂Ω) and/or to blow up inΩ¯. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

scholarly journals The Multiplicity of Nontrivial Solutions for a New p x -Kirchhoff-Type Elliptic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chang-Mu Chu ◽  
Yu-Xia Xiao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Existence Of Weak Solutions ◽  
Kirchhoff Problem

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

A SADDLE-POINT THEOREM

1980 ◽  
pp. 123-126
Author(s):  
Peter W. Bates ◽  
Ivar Ekeland
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Saddle Point Theorem

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

A Matching Theorem in GFC-Spaces with Application to Saddle Points

2013 ◽  
Vol 734-737 ◽  
pp. 2867-2870
Author(s):  
Kai Ting Wen ◽  
He Rui Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Saddle Point ◽  
Point Theorem ◽  
Saddle Points ◽  
Minimax Inequality ◽  
Saddle Point Theorem ◽  
Matching Theorem

In this paper, a matching theorem for weakly transfer compactly open valued mappings is established in GFC-spaces. As applications, a fixed point theorem, a minimax inequality and a saddle point theorem are obtained in GFC-spaces. Our results unify, improve and generalize some known results in recent reference.

MOUNTAIN PASS TYPE SOLUTIONS FOR QUASILINEAR ELLIPTIC INCLUSIONS

2002 ◽  
Vol 04 (04) ◽  
pp. 607-637 ◽  
Author(s):  
PH. CLÉMENT ◽  
M. GARCÍA-HUIDOBRO ◽  
R. MANÁSEVICH
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Maximal Monotone ◽  
Mountain Pass ◽  
Inclusion Problem ◽  
Existence Of Weak Solutions ◽  
Smooth Case ◽  
Quasilinear Elliptic

We establish the existence of weak solutions to the inclusion problem [Formula: see text] where Ω is a bounded domain in ℝN, [Formula: see text], and ψ ∊ ℝ × ℝ is a maximal monotone odd graph. Under suitable conditions on ψ, g (which reduce to subcritical and superlinear conditions in the case of powers) we obtain the existence of non-trivial solutions which are of mountain pass type in an appropriate not necessarily reflexive Orlicz Sobolev space. The proof is based on a version of the Mountain Pass Theorem for a non-smooth case.

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