Search

In this paper, we study a class of fractional p(x)-Laplacian Dirichlet problems in a bounded domain with Lipschitz boundary. Using variational methods, we prove in different situations the existence and multiplicity of solutions.

In this paper, a quasilinear elliptic system involving a pair of (p,q)-Laplacian operators with Steklov boundary value conditions is studied. Using the Mountain Pass Geometry, we prove the existence of at least one weak solution. For the infinitely many weak solutions, we based on Bratsch’s Fountain Theorem.

In this paper, we study the existence and non-existence of a weak solutions to the following system: $$\left\{ \begin{array}{ll} \Delta_p u=\Delta_q v=0& \mbox{ in }\Omega\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda m|u|^{p-2} u-\varepsilon[(\alpha+1)|u|^{\alpha-1}u |v|^{\beta+1}-f] & \mbox{ on }\partial\Omega\\ |\nabla v|^{q-2}\frac{\partial v}{\partial \nu}=\lambda n|v|^{q-2} v-\varepsilon[(\beta+1)|v|^{\beta-1}v |u|^{\alpha+1}-g] \mbox{ on } \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq2)$ with a smooth boundary $\partial\Omega$, $\Delta_pu=\mbox{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian, $\frac{\partial}{\partial\nu}$ is the outer normal derivative, $\varepsilon\in\{0, 1\}$, $ m, n$, $f$ and $g$ are functions that satisfy some conditions.

We study the existence and non-existence of positive solutions for $(p,q)$-Laplacian Steklov problem with two parameters. The main result of our research is the construction of a continuous curve in plane, which becomes a threshold between the existence and non-existence of positive solutions.

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $10$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.

Using variational methods, we prove in a different cases the existence and multiplicity of $a$-harmonic solutions for the following elleptic problem:\begin{equation*}\begin{gathered}div(a(x, \nabla u))=0, \quad \text{in }\Omega, \\a(x, \nabla u).\nu=f(x,u), \quad \text{on } \partial\Omega,\end{gathered}\end{equation*} where $\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain ofsmooth boundary $\partial\Omega$ and $\nu$ is the outward normalvector on $\partial\Omega$. $f: \partial\Omega\times \mathbb{R} \rightarrow \mathbb{R},$ $a: \overline{\Omega}\times \mathbb{R}^{N} \rightarrow\mathbb{R}^{N},$ are fulfilling appropriate conditions.

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 of smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector 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 $g:\partial\Omega\times\mathbb{R} \mapsto \mathbb{R}$ are fulfilling appropriate conditions.

By applying two versions of Mountain Pass Theorem, we prove two different situations of the existence of solutions for the following Steklov problem $\Delta_{p(x)}u =|u|^{p(x)-2}u$ in $\Omega$, $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}= \lambda |u|^{q(x)-2}u$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq 2)$ with smooth boundary $\partial\Omega$ and $p(.), q(.):\bar{\Omega}\rightarrow (1, +\infty)$ are continuous functions.

In this paper we prove the existence of a weak solution to the following system△_p u = △_q v = 0 in Ω,|∇u|^{p−2} ∂_ν u = f(x,u) − (α + 1)K(x)|u|^{α−1}u|v|^{β+1} + f1 on ∂Ω,|∇v|^{q−2} ∂_ν v = g(x,v) − (β + 1)K(x)|v|^{β−1}v|u|^{α+1} + g1 on ∂Ω,where Ω is a bounded domain in RN (N ≥ 2), f1, g1, f, g and K are functions that satisfy some conditions.

In this paper, we show the existence of a sequences of eigenvalues for an operator homogenous at the infinity, we give his variational formulation and we establish the simplicity of all eigenvalues in the case N = 1. Finally we study thesolvability of the problemA(u) := −div(A(x,∇u)) = f(x, u) + h in Ω,u = 0 on ∂Ω,as well as the spectrum ofG_0'(u) = λm|u|^{p−2}u in Ω,u = 0 on ∂Ω.

In this paper, we prove the existence of weak solutions to the problem △_p u = 0 in Ω, |∇u|^{p−2} ∂_ν u = λ_1 m(x)|u|^{p−2}u + f(x, u) − h on ∂Ω, where Ω is a bounded domain in RN (N ≥ 2), m ∈ L^q(∂Ω) is a weight, λ_1 is the first positive eigenvalue for the eigenvalue Steklov problem △pu = 0 in Ω and |∇u|^{p−2} ∂_ν u=λ m(x)|u|^{p−2}u on ∂Ω. f and h are functions that satisfy some conditions.

abstract: In this paper, we give some properties of the first nonprincipal eigenvalue for an asymmetric Steklov problem with weights, and we study the Fucik spectrum.

In this article, we study the existence of the eigencurves for a Steklov problem and we obtain their variational formulation. Also we prove the simplicity and the isolation results of each point of the principal eigencurve. Also we obtain the continuity and the differentiability of the principal eigencurve.

In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem $$displaylines{ frac{d}{dt}u(t)=A(t)u(t)+f(t), quad 0 leq sleq tleq T, cr L(t)u(t)=Phi(t)u(t)+g(t) , quad 0leq sleq tleq T, cr u(s)=x. }$$ The solution to this problem is obtained by a variation of constants formula.