Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

Abstract

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.

Nonlinear elliptic equations with an asymptotically linear reaction term

“…The next lemma can be found in Filippakis-Papageorgiou [3]. It establishes certain lattice structure for the sets of upper and of lower solutions of problem (1).…”

“…Note that Proposition 6 was proved using H 2 , which does not involve any condition near the origin. The first nodal solution u 0 ∈ C 1 0 (Z) is obtained an in Filippakis-Papageorgiou [3] (see the proof of Theorem 4.6). Now we shall produce a second nodal solution.…”

“…Eventually we are able to establish the existence of at least five nontrivial smooth solutions and also provide precise information about their sign. Recently, such multiplicity results for equations driven by the p-Laplacian, were proved by Bartsch-Liu [1], Carl-Perera [2], Filippakis-Papageorgiou [3], Liu [4], Liu-Liu [5], Motreanu-Motreanu-Papageorgiou [6], Papageorgiou-Papageorgiou [7], Papageorgiou-Rocha-Staicu [8] and Zhang-Chen-Li [9]. However, in the aforementioned works one or more of the key features of the present work is missing.…”

Robin problems with indefinite, unbounded potential and reaction of arbitrary growth

“…Using this estimate in (9) with ∈ (0, c 3 r ) and because of hypothesis H (ξ ) and since r > 2, we see that ψ is coercive. Moreover, using the Sobolev embedding theorem and the trace theorem, we see that ψ is sequentially weakly lower semicontinuous.…”

“…As in Filippakis and Papageorgiou [9] (see also Motreanu et al [17,p. 421]), we can show that the set S + is downward directed (that is, if u 1 , u 2 ∈ S + , then we can find u ∈ S + such that u u 1 , u u 2 ) and the set…”