Mountain Pass Theorem

2022 ◽  
pp. 187-195
Keyword(s):  
Mountain Pass Theorem ◽  
Mountain Pass

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.

The existence of a positive solution to asymptotically linear scalar field equations

2000 ◽  
Vol 130 (1) ◽  
pp. 81-105 ◽  
Author(s):  
G. Li ◽  
H.-S. Zhou
Keyword(s):  
Positive Solution ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Field Equations ◽  
Usual Condition ◽  
Image Position ◽  
Scalar Field Equations ◽  
Constraint Method ◽  
Linear Scalar

We consider the following elliptic equation: where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

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 .

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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