Edcarlos D. Silva

Quasilinear Schrödinger Equations with Asymptotically Linear Nonlinearities

Abstract We deal with the existence of nonzero solution for the quasilinear Schrödinger equation −Δu + V(x)u − Δ(u2)u = g(x, u), x ∈ ℝN, u ∈ H1(ℝN), where V is a positive potential and the nonlinearity g(x, s) behaves like K0(x)s at the origin and like K∞(x)|s|p, 1 ≤ p ≤ 3, at infinity. In the proofs we apply minimization methods.

Existence of solution for a generalized quasilinear elliptic problem

It establishes existence and multiplicity of solutions to the elliptic quasilinear Schrodinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈ℝN,where g, h, V are suitable smooth functions. The function g is asymptotically linear at infinity and, for each fixed x∈ℝN, the function h(x, s) behaves like s at the origin and s3 at infinity. In the proofs, we apply variational methods.It establishes existence and multiplicity of solutions to the elliptic quasilinear Schrodinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈ℝN,where g, h, V are suitable smooth functions. The function g is asymptotically linear at infinity and, for each fixed x∈ℝN, the function h(x, s) behaves like s at the origin and s3 at infinity. In the proofs, we apply variational methods.

Resonant-Superlinear Elliptic Problems Using Variational Methods

Abstract In this work we establish existence and multiplicity of solutions for resonant-superlinear elliptic problems using appropriate variational methods. The nonlinearity is resonant at −∞ and superlinear at +∞ and the resonance phenomena occurs precisely in the first eigenvalue of the corresponding linear problem. Our main theorems are stated without the well known Ambrosetti-Rabinowitz condition.

Superlinear elliptic problems under the non-quadraticity condition at infinity

We present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems.

Nonquadraticity condition on superlinear problems

It is established existence of weak solution for a semilinear superlinear elliptic problems on bounded domains. The main feature of the paper is to prove that, for superlinear problems, the nonquadraticity condition introduced by Costa and Magalhaes in (Nonlinear Anal. 23:1401–1412, 1994) is sufficient to get the compactness required by minimax procedures.