Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Blow-up Rate Estimates and Blow-up Set for a System of Two Heat Equations with Coupled Nonlinear Neumann Boundary Conditions

This paper deals with the blow-up properties of positive solutions to a parabolic system of two heat equations, defined on a ball in associated with coupled Neumann boundary conditions of exponential type. The upper bounds of blow-up rate estimates are derived. Moreover, it is proved that the blow-up in this problem can only occur on the boundary.

p-Harmonic Functions in ℝN+ with Nonlinear Neumann Boundary Conditions and Measure Data

We study a concept of renormalized solution to the problem\begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases}where {1<p\leq N}, {N\geq 2}, {{\mathbb{R}}^{N}_{+}=\{(x^{\prime},x_{N}):x^{\prime}\in{\mathbb{R}}^{N-1},\,x% _{N}>0\}}, {u_{\nu}} is the normal derivative of u, μ is a bounded Radon measure, and {g:{\mathbb{R}}\rightarrow{\mathbb{R}}} is a continuous function. We prove stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existence of solutions for problems with nonlinear terms of absorption type in both the subcritical and supercritical case. For the problem with source we study the power nonlinearity {g(u)=-u^{q}}, showing existence in the supercritical case, and nonexistence in the subcritical one. We also give a characterization of removable sets when {\mu\equiv 0} and {g(u)=-u^{q}} in the supercritical case.

Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Existence of Heteroclinic Solutions for a Pseudo-relativistic Allen-Cahn Type Equation

We study the existence and uniqueness of heteroclinic solutions to non-linear Allen-Cahn equationwhere G is a double-well potential. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.