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Let p and q be locally Hölder functions in R N , p > 0 and q ≥ 0. We study the Emden-Fowler equation −∆u + q(x)|∇u| a = p(x)u −γ in R N , where a and γ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for… (More)

We study the nonlinear elliptic problem −∆u = ρ(x)f (u) in R N (N ≥ 3), lim |x|→∞ u(x) = ℓ, where ℓ ≥ 0 is a real number, ρ(x) is a nonnegative potential belonging to a certain Kato class, and f (u) has a sublinear growth. We distinguish the cases ℓ > 0 and ℓ = 0 and we prove existence and uniqueness results if the potential ρ(x) decays fast enough at… (More)

We study the boundary value problem −div((|∇u| p1(x)−2 + |∇u| p2(x)−2)∇u) = f (x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N. We focus on the cases when f ± (x, u) = ±(−λ|u| m(x)−2 u + |u| q(x)−2 u), where m(x) := max{p 1 (x), p 2 (x)} < q(x) < N ·m(x) N −m(x) for any x ∈ Ω. In the first case we show the existence of infinitely many… (More)

We consider a class of nonlinear Dirichlet problems involving the p(x)–Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.

We are concerned with positive solutions decaying to zero at infinity for the logistic equation −∆u = λ (V (x)u − f (u)) in R N , where V (x) is a variable potential that may change sign, λ is a real parameter, and f is an absorbtion term such that the mapping f (t)/t is increasing in (0, ∞). We prove that there exists a bifurcation non-negative number Λ… (More)

We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the mountain pass lemma for locally Lipschitz func-tionals.… (More)

We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals.… (More)

We establish existence and multiplicity theorems for a Dirichlet boundary value problem at resonance, which is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta. Our framework includes a sign-changing potential and we locate the solutions by using the Mountain Pass lemma and the Saddle Point theorem.

We establish the existence of an entire solution for a class of stationary Schrödinger equations with subcritical discontinuous nonlinearity and lower bounded potential that blows-up at infinity. The abstract framework is related to Lebesgue–Sobolev spaces with variable exponent. The proof is based on the critical point theory in the sense of Clarke and we… (More)

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