Marcelo F. Furtado

Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential

Abstract We deal with the nonlinear Schrödinger equation -Δu + V(x)u = f(u) in ℝN, where V is a (possible) sign changing potential satisfying mild assumptions and the nonlinearity f ∈ C1(ℝ, ℝ) is a subcritical and superlinear function. By combining variational techniques and the concentration-compactness principle we obtain a positive ground state solution and also a nodal solution. The proofs rely in localizing the infimum of the associated functional constrained to Nehari type sets.

SOLUTIONS FOR A RESONANT ELLIPTIC SYSTEM WITH COUPLING IN

ABSTRACT Existence and multiplicity of solutions are established, via the Variational Method, for a class of resonant semilinear elliptic system in under a local nonquadraticity condition at infinity. The main goal is to consider systems with coupling where one of the potentials does not satisfy any coercivity condition. The existence of solution is proved under a critical growth condition on the nonlinearity.

Positive solutions for nonlinear elliptic equations with fast increasing weights

We study the equation − div(K(x)∇u) = K(x)u2−1 + λK(x)|x|α−2u, u > 0 ∈ R , (1.1) where N 3, the nonlinearity is given by the critical Sobolev exponent 2∗ = 2N/(N−2), the weight is K(x) = exp(4 |x|), α 2 and λ is a parameter. According to the function space in which we seek solutions, u is forced to decrease sufficiently fast to infinity. As in [12], for α = 2 and λ = (N − 2)/(N + 2), equation (1.1) occurs when one tries to find self-similar solutions

A note on the number of nodal solutions of an elliptic equation with symmetry

Abstract We consider the semilinear problem − Δ u + λ u = | u | p − 2 u in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded smooth domain and 2 p 2 ∗ = 2 N / ( N − 2 ) . We show that if Ω is invariant under a nontrivial orthogonal involution then, for λ > 0 sufficiently large, the equivariant topology of Ω is related to the number of solutions which change sign exactly once.

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.

MULTIPLE SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS VIA REDUCTION METHOD

We establish the existence of two nontrivial solution for some elliptic systems. In the proofs we apply variational methods and Morse theory.

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.

MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH

We prove the existence of infinitely many solutions

On a Class of Semilinear Elliptic Eigenvalue Problems in ℝ 2

u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})

Positive and nodal solutions for an elliptic equation with critical growth

for the Kirchhoff equation