Asymptotically linear elliptic equations with resonance at infinity

Morse Complex, Even Functionals and Asymptotically Linear Differential Equations with Resonance at Infinity

I. Motivation. Let H be a Hilbert space and f : H → R a C-functional. To study critical points of f in the framework of the classical approaches (Morse Theory [39], Ljusternik–Schnirelman theory [40], etc.) one needs to assume, in particular, that f satisfies the Palais–Smale condition (in short, PS-condition): any sequence {xn} ⊂ H with {f(xn)} bounded and ∇f(xn) → 0 contains a convergent subs...

Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

bifurcation problem for biharmonic asymptotically linear elliptic equations

in this paper, we investigate the existence of positive solutions for the ellipticequation $delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $omega$ of $r^{n}$, $ngeq2$, with navier boundary conditions. we show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

Relaxed penalty at infinity for semilinear elliptic equations

The paper gives a different proof for exitence of ground state solutions to the semilinear problem −∆u + u = b(x)up, p > 2, on RN , where b is invariant with respect to a free acting subgroup of O(N) with m elements. While the standard concentration compactness argument suggests the solvability condition inf b(x)/b∞ ≥ 1, symmetric ground state solutions exist whenever inf b(x)/b∞ > δm, where δm...

Semilinear Elliptic Equations Having Asymptotic Limits at Zero and Infinity