Extend existence theory for weighted cubic elliptic ground states to 3 ≤ n < 4

Establish the existence and non-degeneracy of positive radial ground state solutions u(x) = Q(|x|) to the elliptic problem Δu = u − |x|^{2−n} u^3 on R^3 for 3 ≤ n < 4 in an appropriate function space (potentially different from H^1(R^3)), so that Proposition 2.3 holds in this regime and the corresponding forward-bounded localized solutions to the nonautonomous real Ginzburg–Landau equation (d/ds + n/(2s))^2 A = A − A^3 are guaranteed.

Background

The paper shows that for 0 < n < 3 there exists a positive radial ground state Q solving Δu = u − |x|^{2−n} u^3 on R^3 and that the associated linearization has no nontrivial forward-bounded radial solutions. This result (Proposition 2.3) provides the needed input to construct localized ring and spot B patterns.

However, for n > 3 standard H^1(R^3) techniques fail and H^1(R^3) solutions do not exist. To enable results for 3 ≤ n < 4, the authors posit Hypothesis 2.4, asserting that Proposition 2.3 also holds in this range. They suggest that solutions might exist in a different function space but defer resolving this question. Proving such existence and non-degeneracy would remove the auxiliary hypothesis and extend the rigorous construction of ring and spot B patterns to 3 ≤ n < 4.