### A class of $p$-$q$-Laplacian type equation with potentials eigenvalue problem in ${\mathbb{R}}^{N}$.

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We study a nonlinear elliptic system with resonance part and nonlinear boundary conditions on an unbounded domain. Our approach is variational and is based on the well known Landesman-Laser type conditions.

We shall prove a weak comparison principle for quasilinear elliptic operators $-\mathrm{div}\left(a\right(x,\nabla u\left)\right)$ that includes the negative $p$-Laplace operator, where $a:\Omega \times {\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\to {p}^{-}$, the behavior of positive solutions of the problem $-{\Delta}_{p}u=\lambda {u}^{s-1}+{u}^{q-1}$ in $\Omega $, ${u}_{\mid \partial \Omega}=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.

We deal with a bifurcation result for the Dirichlet problem ⎧$-{\Delta}_{p}u=\mu /{\left|x\right|}^{p}{\left|u\right|}^{p-2}u+\lambda f(x,u)$ a.e. in Ω, ⎨ ⎩${u}_{|\partial \Omega}=0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $\lambda {*}_{\mu}$ such that for every $\lambda \in ]0,\lambda {*}_{\mu}[$ the above problem admits a nonzero weak solution ${u}_{\lambda}$ in $W{\u2080}^{1,p}\left(\Omega \right)$ satisfying $li{m}_{\lambda \to 0\u207a}\left|\right|{u}_{\lambda}\left|\right|=0$.

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system ${\Delta}_{p}u=g(u-\alpha v),{\Delta}_{p}v=f(v-\beta u)$ in a smooth bounded domain of ${\mathbb{R}}^{N}$, where ${\Delta}_{p}$ is the p-Laplacian operator defined by ${\Delta}_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${\mathbb{R}}^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty $. Let $p$ tend to $\infty $. There is a critical radius ${R}_{C}$ of the ball such that the principal eigenvalue goes to $\infty $ for $0<R\le {R}_{C}$ and to $0$ for $R>{R}_{C}$. The critical radius is ${R}_{C}=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and ${R}_{C}=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet...

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-{div\left(\right|x|}^{-ap}{\left|\nabla u\right|}^{p-2}{\nabla u)+h\left(x\right)|u|}^{p-2}u=g\left(x\right){\left|u\right|}^{r-2}u$, x ∈ Ω, ⎨ ⎩ ${\left|x\right|}^{-ap}{\left|\nabla u\right|}^{p-2}\partial u/\partial \nu =\lambda f\left(x\right){\left|u\right|}^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in ${\mathbb{R}}^{N}$, that is, $\Omega ={\mathbb{R}}^{N}\setminus D$, where D is a bounded domain in ${\mathbb{R}}^{N}$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $\Omega \subset {\mathbb{R}}^{N}$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $\rho \left(u\right)={\int}_{\Omega}\left(\Phi \right(\left|\nabla u\right|)+\Phi \left(\right|u\left|\right))dx$, M: [0,∞) → ℝ is a continuous function, $K\in {L}^{\infty}\left(\Omega \right)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.