López-Gómez, Julián Omari, Pierpaolo
Published in
Advanced Nonlinear Studies

The aim of this paper is analyzing the positive solutions of the quasilinear problem - ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ a ( x ) f ( u ) in ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) = 0 , -\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0, where λ ∈...

Mancini, Gabriele Martinazzi, Luca
Published in
Advanced Nonlinear Studies

We prove the existence of extremals for fractional Moser–Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler–Lagrange equation, which requires new sharp estimates obtained via commutator techniques.

García-Huidobro, Marta Manasevich, Raúl Tanaka, Satoshi
Published in
Advanced Nonlinear Studies

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of...

Boccardo, Lucio Orsina, Luigi
Published in
Advanced Nonlinear Studies

In this paper, dedicated to Laurent Veron, we prove that the Strong Maximum Principle holds for solutions of some quasilinear elliptic equations having lower order terms with quadratic growth with respect to the gradient of the solution.

Bueno, Hamilton P. Ercole, Grey Macedo, Shirley S. Pereira, Gilberto A.
Published in
Advanced Nonlinear Studies

Let Ω be a Lipschitz bounded domain of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 2 ${N\geq 2}$ . The fractional Cheeger constant h s ( Ω ) ${h_{s}(\Omega)}$ , 0

Editors, The
Published in
Advanced Nonlinear Studies

Ponce, Augusto C. Wilmet, Nicolas
Published in
Advanced Nonlinear Studies

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator - Δ + V {-\Delta+V} with a nonnegative potential V which merely belongs to L loc 1 ( Ω ) {L_{\mathrm{loc}}^{1}(\Omega)} . More precisely, if u ∈ W 0 1 , 2 ( Ω ) ∩ L 2 ( Ω ; V d x ) {u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\m...

Nguyen, Quoc-Hung Phuc, Nguyen Cong
Published in
Advanced Nonlinear Studies

We characterize the existence of solutions to the quasilinear Riccati-type equation { - div 𝒜 ( x , ∇ u ) = | ∇ u | q + σ in Ω , u = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle...

López-Gómez, Julián
Published in
Advanced Nonlinear Studies

This paper characterizes whether or not Σ ∞ ≡ lim λ ↑ ∞ σ [ 𝒫 + λ m ( x , t ) , 𝔅 , Q T ] \Sigma_{\infty}\equiv\lim_{\lambda\uparrow\infty}\sigma[\mathcal{P}+\lambda m(% x,t),\mathfrak{B},Q_{T}] is finite, where m ⪈ 0 {m\gneq 0} is T-periodic and σ [ 𝒫 + λ m ( x , t ) , 𝔅 , Q T ] {\sigma[\mathcal{P}+\lambda m(x,t),\mathfrak{B},Q_{T}]}...

del Teso, Félix Endal, Jørgen Vázquez, Juan Luis
Published in
Advanced Nonlinear Studies

The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Ste...