We show how a change of variable and peak solution methods can be used to prove that a number of nonlinear elliptic partial differential equations have many solutions.

The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.

We consider the Dirichlet problem for an elliptic system of FitzHugh-Nagumo type. We prove that the problem has solutions with arbitrary many sharp peaks near the boundary but not on the boundary.

The aim of this paper is to study the effect of the domain topology on the existence of solutions with arbitrarily many peaks in the interior of the domain and arbitrarily many peaks on the boundary of the domain for a singularly perturbed mixed boundary value problem.

We prove the Lazer-McKenna conjecture for an elliptic problem of Ambrosetti-Prodi type with critical and supercritical nonlinearities by constructing solutions concentrating on higher dimensional manifolds, under some partially symmetric assumption on the domain.

In this paper, we shall construct multibump solutions for an elliptic problem on this expanding domain, such that all the local maximum points of the solution are close to the set...

This parabolic system in one space dimension is a simplification of the original Hodgkin-Huxley nerve conduction equations. Here u denotes an activator and v acts as its inhibitor. This system has also been studied in many applied areas. The readers can find more references in for the background of the systems of FitzHugh--Nagumo type. Here we mention some early results on the systems of FitzHugh--Nagumo type, obtained by Klaasen and Troy, Klaasen and Mitidieri and DeFigueiredo and Mitidieri.

In this paper, we consider the following elliptic problem... where a(x) is a continuous function satisfying 0 < a(x)<1 for x∈...,Ω is a bounded domain in R^N with smooth boundary, ε > 0 is a small number. A solution of (1.1) can be interpreted as a steady state solution of the corresponding problem: ut=ε²Δu+f(x,u), where f(x,t)=t(t-a(x))(1-t), which arises in a number of places such as population genetics [24,33]. The readers can refer to [22] for more background material for problem (1.1).

We prove the Lazer–McKenna conjecture for a superlinear elliptic problem of Ambrosetti–Prodi type by constructing solutions with sharp peaks.