We obtain existence, nonexistence and asymptotic results for solutions to cylindrical equations of the form: \[ -\triangle u+\frac{A}{\left| y\right| ^{\alpha }}u= f\left( u\right) ~\textrm{in }\mathbb{R}^{N},~ x=\left( y,z\right) \in \mathbb{R}^{k}\times \mathbb{R}^{N-k},~N>k\geq 2, \] where $A,\alpha>0$ and $f$ is continuous and satisfies power-type growth conditions.

We consider a nonlinear elliptic equation driven by the $p$-Laplacian with Dirichlet boundary condition. Using variational techniques, combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least six nontrivial solutions: two positive, two negative and two nodal (sign-changing) solutions. Our framework of analysis incorporates both coercive and $p-1$-superlinear problems. Also, the result on multiple constant sign solution incorporates the case of concave-convex nonlinearities.

This paper studies existence, regularity and continuous dependence upon the data of solutions to parabolic semilinear problems of the form: $u'(t)=Au(t) +g[u(t)]$, $u(0)=u_0$. Here, $A:D(A)\to X$ generates an analytic semigroup on a Banach space $X$ and $g:D(g)\to X$. It is assumed that $D(g)$ contains a certain interpolation space of $X$ and $D(A)$; this will allow to treat parabolic partial semilinear problems in the cases where the nonlinear term depends also on the gradient of $u$.

In this paper, we prove the existence and nonexistence results for positive solutions to semilinear elliptic boundary value problems, with concave nonlinearities inside a smooth bounded domain and on the boundary. Our approach relies on sub and supersolutions, as well as the Nehari manifold that may contain the critical points for the energy functional associated with the boundary value problem. The fibering method helps us to study the properties of the Nehari manifold.