Superlinear Indefinite Elliptic Problems and Pohoz′aev Type Identities

Abstract

We prove the existence of a nontrivial solution for a nonlinear elliptic problem &2u=+u+a(x) g(u) with Dirichlet boundary condition on a bounded domain, where g is superlinear both at zero and at infinity, a(x) changes sign and +>0.1998 Academic Press

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

“…First, compactness conditions are nontrivial, like the Palais-Smale condition for the energy functional Φ ∈ C 1 (H complicated. This can be seen in the paper [42] by Ramos, Terracini and Troestler, for instance. A third major difficulty appears when one wants to prove that a certain solution of (E λ ) changes sign.…”

“…In [42], the existence of nontrivial equilibria of (3.1) is proved for f (x, u) = λu + a(x)g(u) under some assumptions on a, g which are different from those in (A1) and (A3). In particular, those assumptions allow the difference between g(u) and |u| p−1 u to grow faster than linearly so that the superlinearity assumption (3.5) need not be true.…”

“…We refer the reader to [5,21] for sufficient conditions concerning this boundedness assumption. On the other hand, if a ∈ C 2 (Ω), ∇a(x) = 0 whenever a(x) = 0, and some additional hypotheses are satisfied, then the existence of a nontrivial solution was proved by Ramos, Terracini and Troestler in [42], but no information on the nodal properties of the solution was provided.…”

Existence of solutions for semilinear elliptic problems without (PS) condition

“…Denote by Ind(p−σ, u) the Morse index of u with respect to the operator −∆−(p−σ)Q(x)|u| p−1 . The uniform convergence of u n to u on compact sets implies that the Morse index Ind(p − σ, u) of u is finite (see [2] or Lemma 6 of [16]), and then Ind(p, u) is finite. Thus Proposition 2.1 yields that u L p+1 (R N ) and ∇u L p+1 (R N ) are finite.…”

Energy and Morse index of solutions of Yamabe type problems on thin annuli

“…It is also related to the geometric properties of solutions to PDE problems. For details see works of Bahri [2], Bahri-Lions [4], De Figueiredo-Yang [10], Lazer-Solimini [16], Pacella [18], Ramos-Terracini-Troestler [19], Solimini [24] and Yang [26], [27].…”