This paper investigates sub-linear elliptic equations on self-similar fractal sets. With an appropriately defined Laplacian, we obtain the existence of nontrivial solutions of sub-linear elliptic equations  $-Δu=λu-α(x)|u|^{q-1}u-f(x,u)$,  with zero boundary Dirichlet conditions. The results are obtained by using Mountain Pass Lemma and Saddle Point Theorem.

In contrast to the cognate Laplace equation, for which a vast number of Green's functions is available, the field is not that developed for the static Klein-Gordon equation. The latter represents, nonetheless, a natural area for application of some of the methods that are proven productive for the Laplace equation. The perspective looks especially attractive for the methods of images and eigenfunction expansion. This study is based on our experience recently gained on the construction of Green's functions for elliptic partial differential equations. An extensive list of boundary-value problems formulated for the static Klein-Gordon equation is considered. Computer-friendly representations of their Green's functions are obtained, most of which have never been published before.

We study the nonlinear one-dimensional viscoelastic nonlocal problem:  $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].

We give an analogy of Seiberg-Witten monopole equations on flat Euclidian space R^5. For this we used an irreducible representation of complex Clifford algebra Cl_5. For the curvature equation we use a kind of self-duality notion of a 2-form on R^5 which is given in [1].

In this paper, we consider the viscous, micropolar, compressible flow in one dimension. We give the proof of existence and uniqueness of strong solutions for the initial boundary problem that vacuum can be allowed initially.

In this paper we prove the existence and uniqueness of global weak solutions to the weakly dissipative Camassa-Holm equation.

We study the existence of solutions for the Schrödinger-Poisson system $-Δu+u+k(x)φu=α(x)|u|^{p-1}u$, in $R^3$,  $-Δφ=k(x)u^2$, in $R^3$, where 3 ≤ p < 5, α(x) is a sign-changing function such that both the supports of α^+ and α^- may have infinite measure. We show that the problem has at least one nontrivial solution under some assumptions.