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Given an open set $\mathrm{\Omega }$ of ${\mathbb{R}}^N$ $\left(N>2\right)$, bounded or unbounded, and a function $w\in L^{\frac{N}{2}}\left(\mathrm{\Omega }\right)$ with $w^{+}\ne 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$-\mathrm{△}u=\lambda w\left(x\right)u,u\in {\mathcal{D}}_0^{1,2}\left(\mathrm{\Omega }\right)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I\left(u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{\left|\nabla u\right|}^{2}dx-{\int }_{\mathrm{\Omega }}G\left(x,u\right)dx,u\in {\mathcal{D}}_0^{1,2}\left(\mathrm{\Omega }\right),$$ when $G$ has a subquadratic growth at infinity.

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $$\Delta u+\rho \left(\frac{e^u}{{\int }_T{e}^u}-\frac{1}{\left|T\right|}\right)=0,{\int }_{T}u=0.$$ It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8\pi$. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting ${\lambda }_1\left(T\right)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le min\{8\pi ,{\lambda }_1\left(T\right)|T\left|\right\}$. Our results hold more generally for solutions that are Steiner...

Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.