Inequalities of Korn's type involve a positive constant, which depends on the domain, in general. A question arises, whether the constants possess a positive infimum, if a class of bounded two-dimensional domains with Lipschitz boundary is considered. The proof of a positive answer to this question is shown for several types of boundary conditions and for two classes of domains.

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-{\Delta }_{p}u+\lambda \left(x\right){\left|u\right|}^{p-2}u=\mu f(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $\lambda \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{x\in \Omega }\lambda \left(x\right)>0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Si considera una classe di equazioni ellittiche semilineari su ${\mathbb{R}}^N$ della forma $-\mathrm{\Delta }u+u=a\left(x\right){\left|u\right|}^{p-1}u$ con $p>1$ sottocritico (o con nonlinearità più generali) e $a\left(x\right)$ funzione limitata. In questo articolo viene presentato un risultato di genericità sull'esistenza di infinite soluzioni, rispetto alla classe di coefficienti $a\left(x\right)$ limitati su ${\mathbb{R}}^N$ e non negativi all'infinito.

We introduce a method to treat a semilinear elliptic equation in ${ℝ}^N$ (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of ${ℝ}^N$ but requires an oscillatory behavior of the potential b.

This paper is concerned with the following periodic Hamiltonian elliptic system $\{-\Delta \varphi +V\left(x\right)\varphi =G_{\psi }(x,\varphi ,\psi )\text{in}{ℝ}^N,-\Delta \psi +V\left(x\right)\psi =G_{\varphi }(x,\varphi ,\psi )\text{in}{ℝ}^N,\varphi \left(x\right)\to 0\text{and}\psi \left(x\right)\to 0\text{as}|x|\to \infty .$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma (-\Delta +V)$, $G(x,\eta )$ is periodic in x and asymptotically quadratic in $\eta =(\varphi ,\psi )$, existence and multiplicity of solutions are obtained via variational approach.


Under no Ambrosetti-Rabinowitz-type growth condition, we study the existence of infinitely many solutions of the p(x)-Laplacian equations by applying the variant fountain theorems due to Zou [Manuscripta Math. 104 (2001), 343-358].

Under a suitable oscillatory behavior either at infinity or at zero of the nonlinear term, the existence of infinitely many weak solutions for a non-homogeneous Neumann problem, in an appropriate Orlicz--Sobolev setting, is proved. The technical approach is based on variational methods.