Nodal solutions for resonant and superlinear (p, 2)‐equations

Abstract

We consider nonlinear, nonhomogeneous elliptic Dirichlet equations driven by the sum of a p‐Laplacian and a Laplacian (so‐called (p, 2)‐equation). We are concerned with both cases 1<p<2 (singular case) and p>2. In the first one, the reaction f(z,x) is linear grow near ±∞ and resonant with respect to a nonprincipal nonnegative eigenvalue. In the second case, the reaction f(z,·) is (p−1)‐superlinear near ±∞ and has z‐dependent zeros of constant sign. Using variational methods together with flow invariance arguments, we establish the existence of nodal solutions.

Multiple Solutions with Sign Information for a Class of Coercive (p, 2)-Equations

“…Also, using the Sobolev embedding we see that ϕ + (•) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u 0 ∈ W 1,p 0 (Ω) such that (9) ϕ…”

“…We mention that recently (p, 2)-equations attracted considerable interest and there have been various existence and multiplicity results for such equations. We mention the works of Aizicovici-Papageorgiou-Staicu [1], Gasiński-Papageorgiou [6], He-Lei-Zhang-Sun [9], Liang-Han-Li [12], Liang-Song-Su [13], Papageorgiou-Rȃdulescu [15,16], Papageorgiou-Vetro-Vetro [19,20,21], Papageorgiou-Zhang [23], Sun-Zhang-Su [25], Zhang-Liang [26]. In particular [1,15] also deal with coercive problems but under more restrictive conditions on the source term f (z, x) which exclude from consideration logistic equations.…”

“…In [12,13,26] the authors prove on existence results for (p, 2)-equations. In [9] the equation is parametric and the approach is based on flow invariant arguments and in [23] the authors deal with equations involving a parametric concave term.…”

On Noncoercive (p, q)-Equations

“…Our work complements that of Gasiński-Papageorgiou [7], where an analogous multiplicity theorem is proved for equations driven by the p-Laplacian only and with a reaction which satisfies more restrictive conditions and no nodal solutions are obtained. Finally we mention the recent works of He-Lei-Zhang-Sun [10] (with q = 2 and (p − 1)-superlinear reaction) and of Papageorgiou-Vetro-Vetro [22] (also with q = 2, parametric concave-convex problems).…”