In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ in $A_{\epsilon }$, $u_{\epsilon }=0$ on $\partial A_{\epsilon }$, where $A_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^n$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{|}^2$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...

We consider a class of semilinear elliptic equations of the form$$-{\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$$where $\epsilon \>0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if $\epsilon$ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...