Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems

On the existence of stationary solutions for some nonlinear heat equations

“…The similar situation arised in [19] and [22] but as distinct from the present work, the problems treated there were nonlocal. The fixed point technique was used in [14] to estimate the perturbation to the standing solitary wave of the Nonlinear Schrödinger (NLS) equation when either the external potential or the nonlinear term in the NLS were perturbed but the Schrödinger type operator involved in such nonlinear problem possessed the Fredholm property (see Assumption 1 of [14], also [7]).…”

“…The Laplacian operator with drift from the point of view of non Fredholm operators was studied in [20] and linearized Cahn-Hilliard equations in [18] and [21]. Nonlinear non Fredholm elliptic problems were treated in [19] and [22]. Potential applications to the theory of reaction-diffusion equations were explored in [9,10].…”

“…Under Assumption 1 by means of Lemma 7 of [22] problem (1.8) admits a unique solution u 0 (x) ∈ H 2 (R 5 ) and no orthogonality relations are required. As discussed in Lemmas 5 and 6 of [22], in dimensions d < 5 we need certain orthogonality conditions for the solvability of equation (1.8) …”

“…As discussed in Lemmas 5 and 6 of [22], in dimensions d < 5 we need certain orthogonality conditions for the solvability of equation (1.8) …”

“…We do not discuss the problem in dimensions d > 5 to avoid extra technicalities since the argument will rely on similar ideas (see Lemma 7 of [22]). Due to our Assumption 1 …”