We investigate the properties of the operator ∆(σ∆·) : H0(Ω) → H−2(Ω), where σ is a given parameter whose sign can change on the bounded domain Ω. Here, H0(Ω) denotes the subspace of H2(Ω) made of the functions v such that v = ν · ∇v = 0 on ∂Ω. The study of this problem arises when one is interested in some configurations of the Interior Transmission Eigenvalue Problem. We prove that ∆(σ∆·) : H...

Via variational methods, we study multiplicity of solutions for the problem    −∆u = λb(x)|u|q−2u + a u + g(x, u) in Ω , u = 0 on ∂Ω . where a simple example for g(x, u) is |u|p−2u; here a, λ are real parameters, 1 < q < 2 < p ≤ 2∗ and b(x) is a function in a suitable space L. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any λ > 0, and a...

We prove the existence of multiple solutions for following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u h(x) \ \mbox{on} \Omega$ and $ u=\Delta u=\Delta^2 u=0 \partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $\Delta_{p(x)}^3u \operator...

In this paper, we consider the existence of least energy sign-changing solutions for a class of Kirchhoff-type problem [Formula: see text]where [Formula: see text] is a bounded domain in [Formula: see text], [Formula: see text], with a smooth boundary [Formula: see text], [Formula: see text] and [Formula: see text]. By using variational approach and some subtle analytical skills, the existence ...