Existence and multiplicity of solutions for anisotropic elliptic equation

In this article we study the nonlinear problem $$\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\ u=0 \qquad on \qquad \partial\Omega \end{array} \right.$$ Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.

NEW GLOBAL LOGARITHMIC STABILITY RESULTS ON THE CAUCHY PROBLEM FOR ELLIPTIC EQUATIONS

We prove the global logarithmic stability of the Cauchy problem for $H^{2}$-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.

Some remarks on an eigenvalue problem for an anisotropic elliptic equation with indefinite weight

In this paper, we consider an eigenvalue problem for an anisotropic elliptic equation with indefinite weight, in which the differential operator involves partial derivatives with different variable exponents. Under some suitable conditions on the growth rates of the anisotropic coefficients involved in the problem, we prove some results on the existence and non-existence of a continuous family of eigenvalues by using variational methods.