Abstract
This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem -Δpu=1/σ(∂F(x,u)/∂u)+λa(x)|u|q-2u for x∈Ω with zero Dirichlet boundary conditions, where Ω is a bounded open set in ℝn, 1<q<p<σ<p*(p*=np/(n-p) if p<n, p*=∞ if p≥n), λ∈ℝ∖{0}, a is a smooth function which may change sign in Ω̅,, and F∈C1(Ω̅ × ℝ,ℝ). The method is based on Nehari results on three submanifolds of the space W01,p(Ω).
Highlights
In this paper, we are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation: −Δ pu 1 σ ∂F (x, u) ∂u +λa (x) |u|q−2u in Ω, (1)u = 0, on ∂Ω, where Ω is a bounded domain of Rn (n ≥ 3), 1 < q < p < σ < p∗(p∗ = np/(n−p) if p < n, p∗ = ∞ if p ≥ n), λ ∈ R\{0}, F ∈C1(Ω × R, R) is positively homogeneous of degree σ; that is, F(x, tu) = tσF(x, u) holds for all (x, u) ∈ Ω × R and the signchanging weight function a satisfies the following condition: (A) and a− a :=∈ C(Ω) with max(−a, 0) ≢
This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem −Δ pu = (1/σ)(∂F(x, u)/∂u) + λa(x)|u|q−2u for x ∈ Ω with zero Dirichlet boundary conditions, where Ω is a bounded open set in Rn, 1 < q < p < σ < p∗(p∗ =
We are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation:
Summary
We are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation:. Jλ(tu), where Jλ is the Euler function associated with the equation) to solve semilinear and quasilinear problems. Brown and Zhang [10] studied the following subcritical semilinear elliptic equation with sign-changing weight function:. The authors in [10] by the same arguments considered the following semilinear elliptic problem:. We use the decomposition of the Nehari manifold as λ vary to prove our main result. Before stating our main result, we need the following assumptions:. Under the assumptions (A), (H1), and (H2), there exists λ0 > 0 such that for all 0 < |λ| < λ0, problem (1) has at least two nontrivial nonnegative solutions.
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