scholarly journals Infinitely Many Weak Solutions of thep-Laplacian Equation with Nonlinear Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Feng-Yun Lu ◽  
Gui-Qian Deng
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Nonlinear Boundary ◽  
Smooth Boundary ◽  
Nonlinear Boundary Conditions ◽  
Fountain Theorem ◽  
Laplacian Equation ◽  
Variant Fountain Theorem

We study the followingp-Laplacian equation with nonlinear boundary conditions:-Δpu+μ(x)|u|p-2u=f(x,u)+g(x,u),  x∈Ω,|∇u|p-2∂u/∂n=η|u|p-2uandx∈∂Ω,  whereΩis a bounded domain inℝNwith smooth boundary∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) andf,gdo not need to satisfy the(P.S)or(P.S*)condition.

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 GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR STRONGLY DAMPED WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS AND BALANCED POTENTIALS

2019 ◽  
Vol 99 (03) ◽  
pp. 432-444
Author(s):  
JOSEPH L. SHOMBERG
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Wave Equations ◽  
Polynomial Growth ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Dynamic Boundary Conditions ◽  
Existence Of Weak Solutions ◽  
Strongly Damped

We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. The associated linear operator is $(-\unicode[STIX]{x1D6E5}_{W})^{\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{t}u$ , where $\unicode[STIX]{x1D703}\in [\frac{1}{2},1)$ and $\unicode[STIX]{x1D6E5}_{W}$ is the Wentzell–Laplacian. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth of each potential, as well as mixed dissipative/antidissipative behaviour.

Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions

2010 ◽  
Vol 140 (2) ◽  
pp. 259-272 ◽  
Author(s):  
Nguyen Thanh Chung ◽  
Quôc-Anh Ngô
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Nonlinear Boundary ◽  
Smooth Boundary ◽  
Nonlinear Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Image Position ◽  
Quasilinear Elliptic ◽  
Outer Unit

Using variational methods we study the non-existence and multiplicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the formwhere Ω; is a bounded domain with smooth boundary, n is the outer unit normal to ∂Ω and λ is a parameter. Furthermore, we want to emphasize that g : ∂Ω × [0,∞)→ ℝ is a continuous function that may or may not satisfy the Ambrosetti–Rabinowitz-type condition.

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