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 ${ℝ}^N$ 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 ${ℝ}^N$, that is, $\Omega ={ℝ}^N\setminus D$, where D is a bounded domain in ${ℝ}^N$ 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,...

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic $p(·)$-Laplacian problem with Dirichlet-type boundary conditions and $L^1$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

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 $\Omega \subset {ℝ}^N$, 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.