has a solution «=g(x) for O^x^Xo with g(0)=a and u = h(x) tor xo^x^l with h(l)=b where g(x0)=h(x0). It will be assumed that g'(xo)*h'(xo). The case of (1) with f=l — (y')t and where \a — b\ <1 can be treated explicitly. For small e>0 the solution of (1) tends to the broken line solution of (2) with g(x)=a — x and h = b — 1+x and Xo = (l+a—b)/2. (There is another broken line solution of (2) with...

We consider the equation −ε2∆u+u = u in Ω ⊆ R , where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N ≥ 3 and for k ∈ {1, . . . , N − 2}. We impose Neumann boundary conditions, assuming 1 < p < N−k+2 N−k−2 and ε → 0+. This result settles in full generality a phenomenon previously considered only in the particular case ...

ε > 0 is a small parameter and {u(x),λ} is a solution. For ε 1, the function u(x) has a boundary layer of thickness O(ε) near x = 0. Under the above conditions, there exists a unique solution to problem (1.1), (1.2) (see [7, 12]). An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [6, 7, 8, 9, 12, 13, 15, 16]. In [7,...

There has been much work on various singularly perturbed partial differential equations or systems. Such equations or systems depend on some small parameters ε > 0, solutions denoted as uε. There are at least two types of questions being investigated. The first type is to study possible behavior of uε as ε tends to zero. The second is to actually construct, by various methods, such solutions. I...