We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O\left(N\right)$ such that $HG\subset O\left(N\right)$. We denote the orbit of $G$ through $x\in {ℝ}^N$ by $G\left(x\right)$, i.e., $G\left(x\right):=\{gx:g\in G\}$. We prove that if $H\left(x\right)G\left(x\right)$ for all $x\in \overline{\Omega }$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore...

We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.

We investigate the following quasilinear and singular problem,$$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\>0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$where $\Omega$ is an open bounded domain with smooth boundary, $1\<p\<\infty$, $p-1\<q\le p^{*}-1$, $\lambda \>0$, and $0\<\delta \<1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\<p\<N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\>N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions...

We study the existence of positive solutions to ⎧ ${div\left(\right|\nabla u|}^{p-2}\nabla u)+q\left(x\right)u^{-\gamma }=0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is ${ℝ}^N$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

A model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered and the existence and uniqueness theorem of the classic solution is proved.

A mathematical model of dissociative adsorption and associative desorption for diatomic molecules is generalized. The model is described by a coupled system of parabolic and ordinary differential equations. The existence and uniqueness theorem of the classical solution is proved.

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...