We construct an upper bound for the following family of functionals ${\left\{E_{\epsilon }\right\}}_{\epsilon \>0}$, which arises in the study of micromagnetics:$$E_{\epsilon }\left(u\right)={\int }_{\Omega }{\epsilon \left|\nabla u\right|}^2+\frac{1}{\epsilon }{\int }_{{ℝ}^2}{\left|H_u\right|}^2.$$Here $\Omega$ is a bounded domain in ${ℝ}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by$$\left\{\begin{array}{cc}\mathrm{div}(\tilde{u}+H_u)=0\hfill & \text{in}{ℝ}^2,\hfill \\ \mathrm{curl}{H}_u=0\hfill & \text{in}{ℝ}^2,\hfill \end{array}\right.$$where $\tilde{u}$ is the extension of $u$ by $0$ in ${ℝ}^2\setminus \Omega$. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

We prove the existence of positive and of nodal solutions for $-\Delta u={\left|u\right|}^{p-2}u+\mu {\left|u\right|}^{q-2}u$, $u\in {\mathrm{H}}_0^1\left(\Omega \right)$, where $\mu \>0$ and $2\<q\<p=2N(N-2)$, for a class of open subsets $\Omega$ of ${ℝ}^N$ lying between two infinite cylinders.

We prove the existence of positive and of nodal solutions for -Δu = |u|p-2u + µ|u|q-2u, $u\in {\mathrm{H}}_0^1\left(\Omega \right)$, where µ > 0 and 2 < q < p = 2N(N - 2) , for a class of open subsets Ω of ${ℝ}^N$ lying between two infinite cylinders.

In this paper we show some results of multiplicity and existence of sign-changing solutions using a mountain pass theorem in ordered intervals, for a class of quasi-linear elliptic Dirichlet problems. As a by product we construct a special pseudo-gradient vector field and a negative pseudo-gradient flow for the nondifferentiable functional associated to our class of problems.

We investigate the following quasilinear and singular problem,$$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\&gt;0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$where $\Omega$ is an open bounded domain with smooth boundary, $1\&lt;p\&lt;\infty$, $p-1\&lt;q\le p^{*}-1$, $\lambda \&gt;0$, and $0\&lt;\delta \&lt;1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\&lt;p\&lt;N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\&gt;N$. 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.