Solutions of Semilinear Elliptic Equations in Tubes

Abstract

Given a smooth compact k-dimensional manifold Λ embedded in ℝ m, with m≥2 and 1≤k≤m-1, and given ε{lunate}>0, we define B ε{lunate}(Λ) to be the geodesic tubular neighborhood of radius ε{lunate} about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation {Mathematical expression} when the parameter ε{lunate} is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m-k≤2 or {Mathematical expression} when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ε{lunate} tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for {Mathematical expression}, n≥3, if ε{lunate} is sufficiently small. © 2012 Mathematica Josephina, Inc

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

“…Such a phenomenon is not new and, in the context of singularly perturbed semilinear elliptic equations, was originally found by A. Malchiodi and M. Montenegro in [34]. Since this seminal paper, this phenomenon has also been found in other instances, for example in the study of other semilinear partial differential equations [15,16,31,33,36] or in the study of constant mean curvature surfaces [32,35]. Loosely speaking, it is caused by the presence of the tangential dimension θ along the curve Γ and the fact that the profile in the normal t direction in unstable (see the discussion in the next paragraph).…”

“…The eigenvalues of the linearization, corresponding to (1.13), were · · · < λ 1 = 0 < λ 0 , and the fact that λ 1 = 0 caused a further difficulty that is not present in our case. An elliptic problem, involving resonance, in which the corresponding eigenvalues are as in the present situation, as described below (1.13), can be found in [36]. Remark 1.1.…”

“…The desired result for the full problem (11.1) then follows directly from Schauder's fixed point theorem. In fact, refining the fixed-point region, we can actually get [15], which we use in the present paper, the preferred approach seems to be that of [34] (see [31,33,36]). Let us briefly discuss this point and what are the difficulties in employing the scheme of [34] in the present situation.…”

“…Improving inductively the accuracy of the approximation by adding lower order terms, determined from linear inhomogeneous problems of the form (3.33)-(3.34), seems to be difficult and complicated because each inhomogeneous term and boundary condition grows polynomially instead of decaying exponentially to zero as was the case in [33] or [34], where arbitrarily high-order approximations for other equations were constructed in this manner. Unlike the previous references, the problem at hand has in common with [36] the fact that the linearization of the profile normal to Γ is invertible (recall Proposition 3.1), and another possibility for improving the approximate solution could be to adapt the iteration scheme of that paper. Problem (1.1) posed in N dimensions, as in the previous paragraph, has variational structure, and the stable solution satisfying (1.5) is a local minimizer of the corresponding functional.…”

Bubble Concentration on Spheres for Supercritical Elliptic Problems

“…We also point out that in the supercritical case a result of Bahri-Coron type ( [2]) cannot hold in general nontrivially topological domains as shown by a nonexistence result of Passaseo ([15]), obtained exploiting critical exponents in lower dimensions. Using similar ideas, some results for exponents p which are subcritical in dimension n < d and instead supercritical in dimension d have been obtained in different kind of domains in [1,4,6,8,9,10,11,13].…”