Exact multiplicity for some quasilinear elliptic Dirichlet problems where the non-linearity changes sign

2005 ◽  
Vol 61 (7) ◽  
pp. 1135-1155 ◽  
Author(s):  
Zongming Guo
Keyword(s):  
Dirichlet Problems ◽  
Exact Multiplicity ◽  
Quasilinear Elliptic

scholarly journals Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongliang Gao ◽  
Jing Xu
Keyword(s):  
Positive Solutions ◽  
Dirichlet Problems ◽  
Precise Description ◽  
One Dimensional ◽  
Curvature Equation ◽  
Time Map ◽  
The One ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity ◽  
Prime 2

AbstractIn this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , where λ and L are positive parameters, $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , and $f(u)>0$ f ( u ) > 0 for $0< u< L$ 0 < u < L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies $f''(u)>0$ f ″ ( u ) > 0 and $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) for $0< u< L$ 0 < u < L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

scholarly journals Perturbation of Global Solution Curves for Semilinear Problems

2003 ◽  
Vol 3 (2) ◽  
Author(s):  
Philip Korman ◽  
Yi Li ◽  
Tiancheng Ouyang
Keyword(s):  
Global Solution ◽  
Positive Solutions ◽  
Turning Points ◽  
Dirichlet Problems ◽  
Indirect Approach ◽  
Smooth Curves ◽  
Semilinear Problems ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity

AbstractWe revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in [6]. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approach, and study the evolution of turning points. We give conditions under which the critical (turning) points continue on smooth curves, which allows us to reduce the problem to the easier case of f (0) = 0. We show that the smallest root of f (u) does not have to be restricted.

Strong Maximum Principle for Some Quasilinear Dirichlet Problems Having Natural Growth Terms

2020 ◽  
Vol 20 (2) ◽  
pp. 503-510
Author(s):  
Lucio Boccardo ◽  
Luigi Orsina
Keyword(s):  
Maximum Principle ◽  
Elliptic Equations ◽  
Strong Maximum Principle ◽  
Quasilinear Elliptic Equations ◽  
Quadratic Growth ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Natural Growth Terms ◽  
Quasilinear Elliptic ◽  
Lower Order Terms

AbstractIn this paper, dedicated to Laurent Veron, we prove that the Strong Maximum Principle holds for solutions of some quasilinear elliptic equations having lower order terms with quadratic growth with respect to the gradient of the solution.

Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities

2003 ◽  
Vol 133 (3) ◽  
pp. 705-717 ◽  
Author(s):  
Moxun Tang
Keyword(s):  
Dirichlet Problem ◽  
Unit Ball ◽  
Critical Number ◽  
Dirichlet Problems ◽  
Semilinear Elliptic ◽  
Concave And Convex Nonlinearities ◽  
Image Position ◽  
Exact Multiplicity

Let B be the unit ball in Rn, n ≥ 3. Let 0 < p < 1 < q ≤ (n + 2)/(n − 2). In 1994, Ambrosetti et al. found that the semilinear elliptic Dirichlet problem admits at least two solutions for small λ > 0 and no solution for large λ. In this paper, we prove that there is a critical number Λ > 0 such that this problem has exactly two solutions for λ ∈ (0, Λ), exactly one solution for λ = Λ and no solution for λ > Λ.

Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces

2012 ◽  
Vol 75 (12) ◽  
pp. 4441-4456 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giovanni Molica Bisci ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Sobolev Spaces ◽  
Dirichlet Problems ◽  
Quasilinear Elliptic
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