A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
J. García-Melián ◽  
L. Iturriaga ◽  
H. Ramos Quoirin
Keyword(s):  
Continuous Function ◽  
Positive Solution ◽  
Critical Points ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
A Priori ◽  
A Priori Bounds ◽  
Abstract Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

AbstractWe consider the semilinear elliptic problemwhere a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..

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 .

scholarly journals Existence and bifurcation for some elliptic problems on exterior strip domains

2006 ◽  
Vol 2006 ◽  
pp. 1-26
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Positive Constant ◽  
Elliptic Problems ◽  
Unique Positive Solution ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Strip Domain

We consider the semilinear elliptic problem−Δu+u=λK(x)up+f(x)inΩ,u>0inΩ,u∈H01(Ω), whereλ≥0,N≥3,1<p<(N+2)/(N−2), andΩis an exterior strip domain inℝN. Under some suitable conditions onK(x)andf(x), we show that there exists a positive constantλ∗such that the above semilinear elliptic problem has at least two solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no solution ifλ>λ∗. We also obtain some bifurcation results of the solutions atλ=λ∗.

Multiplicity of non-radial solutions of critical elliptic problems in an annulus

2005 ◽  
Vol 135 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Djairo Guedes de Figueiredo ◽  
Olímpio Hiroshi Miyagaki
Keyword(s):  
Critical Points ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Critical Sobolev Exponent ◽  
Radial Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

By looking for critical points of functionals defined in some subspaces of , invariant under some subgroups of O (N), we prove the existence of many positive non-radial solutions for the following semilinear elliptic problem involving critical Sobolev exponent on an annulus, where 2* − 1 := (N + 2)/(N − 2) (N ≥ 4), the domain is an annulus and f : R+ × R+ → R is a C1 function, which is a subcritical perturbation.

scholarly journals Singular Limits for 2-Dimensional Elliptic Problem with Exponentially Dominated Nonlinearity and a Quadratic Convection Term

2013 ◽  
Vol 21 (1) ◽  
pp. 19-50
Author(s):  
Sami Baraket ◽  
Imen Bazarbacha ◽  
Saber Kharrati ◽  
Taieb Ouni
Keyword(s):  
Elliptic Problem ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Gradient Term ◽  
Two Dimensional ◽  
Convection Term ◽  
Linear Gradient ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Singular Limits

Abstract We study existence of solutions with singular limits for a two-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a quadratic convection non linear gradient term, imposing Dirichlet boundary condition. This paper extends previous results obtained in [1], [3], [4] and some references therein for related issues.

scholarly journals Existence of solutions for some degenerate semilinear elliptic equations with measure data

Mathematica ◽  
2021 ◽  
Vol 63 (86) (1) ◽  
pp. 23-31
Author(s):  
Badajena Arun Kumar ◽  
Pradhan Shesadev
Keyword(s):  
Weak Solution ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Semilinear Elliptic Equations ◽  
Measure Data ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We study the existence of a weak solution for a certain degenerate semilinear elliptic problem.

scholarly journals Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

2018 ◽  
Vol 9 (1) ◽  
pp. 108-123 ◽  
Author(s):  
Claudianor O. Alves ◽  
Grey Ercole ◽  
M. Daniel Huamán Bolaños
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.

Bifurcation results for semilinear elliptic problems in RN

2004 ◽  
Vol 134 (1) ◽  
pp. 11-32 ◽  
Author(s):  
Marino Badiale ◽  
Alessio Pomponio
Keyword(s):  
Critical Points ◽  
Elliptic Problem ◽  
Reduction Method ◽  
Elliptic Problems ◽  
Semilinear Elliptic Problem ◽  
Finite Dimensional ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

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