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...

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $$-{\Delta }_{N}u+b{\left|u\right|}^{N-2}u-{\Delta }_N\left(u^2\right)u=h\left(u\right),x\in {ℝ}^N,$$ where ${\Delta }_N$ is the $N$-Laplacian operator, $h\left(u\right)$ is continuous and behaves as $exp\left(\alpha \right|u{|}^{N/(N-1)})$ when $\left|u\right|\to \infty$. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u\left(x\right)\in W^{1,N}\left({ℝ}^N\right)$ with $u\left(x\right)\to 0$ as $\left|x\right|\to \infty$ is established.

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).

The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$-Laplacian $$-{\Delta }_{N}u\equiv -{div\left(\right|\nabla u|}^{N-2}\nabla u)=e(x,u)+h\left(x\right)\text{in}\Omega$$ where $u\in W_0^{1,N}\left({ℝ}^N\right)$, $\Omega$ is a bounded smooth domain in ${ℝ}^N$, $N\ge 2$, $e(x,u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h\left(x\right)\in {\left(W_0^{1,N}\right)}^{*}$ is a small perturbation.

We study solutions of the 2D Ginzburg–Landau equation $-\Delta u+{\epsilon }^{-2}{u\left(\right|u|}^2-1)=0$ subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, $\left|u\right|=1$, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small $\epsilon$. For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy...

The initial boundary-transmission problems for electromagnetic fields in homogeneous and anisotropic media for canonical semi-infinite domains, like halfspaces, wedges and the exterior of half- and quarter-plane obstacles are formulated with the use of complex quaternions. The time-harmonic case was studied by A. Passow in his Darmstadt thesis 1998 in which he treated also the case of an homogeneous and isotropic layer in free space and above an ideally conducting plane. For thin layers and free...

We present critical groups estimates for a functional $f$ defined on the Banach space $W_0^{1,p}\left(\mathrm{\Omega }\right)$, $\mathrm{\Omega }$ bounded domain in ${\mathbb{R}}^N$, $2<p<\mathrm{\infty }$, associated to a quasilinear elliptic equation involving $p$-laplacian. In spite of the lack of an Hilbert structure and of Fredholm property of the second order differential of $f$ in each critical point, we compute the critical groups of $f$ in each isolated critical point via Morse index.