scholarly journals Multiplicity results for some elliptic problems with nonlinear boundary conditions involving variable exponents

2011 ◽  
Vol 62 (9) ◽  
pp. 3464-3471 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Csaba Varga
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Elliptic Problems ◽  
Nonlinear Boundary Conditions ◽  
Variable Exponents ◽  
Multiplicity Results

scholarly journals Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents

2014 ◽  
Vol 33 (2) ◽  
pp. 123-133 ◽  
Author(s):  
Abdellah Ahmed Zerouali ◽  
Belhadj Karim ◽  
Omarne Chakrone ◽  
Aomar Anane
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Critical Points ◽  
Nonlinear Boundary ◽  
Elliptic Problems ◽  
Nonlinear Boundary Conditions ◽  
Variable Exponents ◽  
Three Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity

By applaying the Ricceri's three critical points theorem, we show the existence of at least three solutions to the following elleptic problem:\begin{equation*}\begin{gathered}-div[a(x, \nabla u)]+|u|^{p(x)-2}u=\lambda f(x,u), \quad \text{in }\Omega, \\a(x, \nabla u).\nu=\mu g(x,u), \quad \text{on } \partial\Omega,\end{gathered}\end{equation*}where $\lambda$, $\mu \in \mathbb{R}^{+},$$\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain ofsmooth boundary $\partial\Omega$ and $\nu$ is the outward normalvector on $\partial\Omega$. $p: \overline{\Omega} \mapsto\mathbb{R}$, $a: \overline{\Omega}\times \mathbb{R}^{N} \mapsto\mathbb{R}^{N},$ $f: \Omega\times\mathbb{R} \mapsto \mathbb{R}$and $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ arefulfilling appropriate conditions.

scholarly journals Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jincheng Huang
Keyword(s):  
Boundary Conditions ◽  
Exterior Domain ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Multiplicity Results ◽  
Exterior Problems ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

scholarly journals Homogenization of a class of elliptic problems with nonlinear boundary conditions in domains with small holes

2015 ◽  
Vol 31 (1) ◽  
pp. 77-88
Author(s):  
AMAR OULD-HAMMOUDA ◽  
◽  
RACHAD ZAKI ◽  
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Elliptic Problems ◽  
Nonlinear Boundary Conditions ◽  
Dirichlet Condition ◽  
Second Order ◽  
External Boundary ◽  
Oscillating Coefficients ◽  
Second Order Elliptic Problems

We consider a class of second order elliptic problems in a domain of RN , N > 2, ε-periodically perforated by holes of size r(ε) , with r(ε)/ε → 0 as ε → 0. A nonlinear Robin-type condition is prescribed on the boundary of some holes while on the boundary of the others as well as on the external boundary of the domain, a Dirichlet condition is imposed. We are interested in the asymptotic behavior of the solutions as ε → 0. We use the periodic unfolding method introduced in [Cioranescu, D., Damlamian, A. and Griso, G., Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 99–104]. The method allows us to consider second order operators with highly oscillating coefficients and so, to generalize the results of [Cioranescu, D., Donato, P. and Zaki, R., Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal., Vol. 53 (2007), No. 4, 209–235].

scholarly journals EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO ELLIPTIC PROBLEMS WITH SUBLINEAR MIXED BOUNDARY CONDITIONS

2009 ◽  
Vol 11 (04) ◽  
pp. 585-613 ◽  
Author(s):  
JORGE GARCÍA-MELIÁN ◽  
JULIO D. ROSSI ◽  
JOSÉ C. SABINA DE LIS
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Mixed Boundary ◽  
Elliptic Problems ◽  
Mixed Boundary Conditions ◽  
Nonlinear Boundary Conditions ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Outward Unit

In this work, we consider a class of semilinear elliptic problems with nonlinear boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies on the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x,u) on Γ and u = 0 on Γ', where ν is the outward unit normal to Γ while Γ,Γ' are open, Γ ∩ Γ' = ∅, [Formula: see text], but [Formula: see text] need not be disjoint.

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