Browsing by Subject "critical points"

Now showing 1 - 4 of 4

Access status: Open Access ,
A Neumann boundary value problem for a class of gradient systems
(2014) Pan, Wen-Wu; Li, Lin
In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and in?nitely many solutions of this problem, respectively.
Access status: Open Access ,
Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator
(2014) Tsouli, Najib; Darhouche, Omar
In this paper we study the following nonlinear boundary-value problem $-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,$ $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial u}{\partial\nu}$ is the outer unit normal derivative on $\partial\Omega$, $\lambda, \mu$ are two real numbers such that $\lambda^{2}+\mu^{2}\neq0$, $p$ is a continuous function on $\overline{\Omega}$ with $\inf_{x\in \overline{\Omega}} p(x)\gt 1$, $\beta\in L^{\infty}(\partial\Omega)$ with $\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0$ and $f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}$, $g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous functions. Under appropriate assumptions on $f$ and $g$, we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.
Access status: Open Access ,
Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators
(2016) Kong, Lingju
We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a $p(x)$-biharmonic operator $\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $p\in C(\overline{\Omega})$, $\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator, and $\lambda\gt 0$ is a parameter. We establish sufficient conditions under which there exists a positive number $\lambda^{*}$ such that the above problem has at least two nontrivial weak solutions for each $\lambda\gt\lambda^{*}$. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$.
Access status: Open Access ,
Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions
(Wydawnictwa AGH, 2019) Bonanno, Gabriele; D'Aguì, Giuseppina; Sciammetta, Angela
In this paper, a nonlinear differential problem involving the $p$-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.