A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function

Abstract

In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (E λ,μ ) involving nonlinear boundary condition and sign-changing weight function. With the help of the Nehari manifold, we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ, μ) belongs to a certain subset of R 2 .

The Nehari Manifold For A Quasilinear Elliptic Equation With Singular Weights And Nonlinear Boundary Conditions

“…Due to this singularity in the weights, the extensions are challenging and nontrivial. Indeed, motivated by [22], and using recent ideas from [11], we shall stablish the existence and multiplicity results for problem (1). In the case when a = 0 and c = p = 2 similar problems (with Dirichlet or Neuman boundary condition ) have been studied by Drabek et al [7,8], Ambrosetti-Brezis-Cerami [2] using variational methods and by Amman and Lopez-Gomez [3] by using global bifurcation theory.…”

“…In the recent paper, Brown and Wu [11] studied the multiplicity results of nontrivial nonnegative solutions for a semilinear elliptic system. Here we focus on further extending the study in [11] for the quasilinear elliptic problem involving the singular weights. Due to this singularity in the weights, the extensions are challenging and nontrivial.…”

A remark on the existence and multiplicity result for a nonlinear elliptic problem involving the p-Laplacian

“…The author have altogether proved that, there exists λ 0 > 0 such that if the parameter λ satisfy 0 < λ < λ 0 , then problem (1) for m ≡ 1, b ≡ 1, p = 2, and 1 < α < 2 < β < 2 * , has at least two positive solutions. In this paper, using the technique of Brown and Wu [10], we discuss the problem (1) again but with m ≡ 1, b ≡ 1, p > 2, and 2 < β < p < α < p * . The change in α completely changes the nature of the solution set of (1).…”

“…In recent years, several authors use the Nehari manifold to solve semilinear and quasilinear problems (see [1,[8][9][10][11][20][21][22]). Brown and Zhang [11] have studied the following subcritical semilinear elliptic equation with a sign-changing weight function…”

“…In this work, we give a variational method which is similar to the fibering method (see [12] or [11]) to prove the existence of at least two nontrivial nonnegative solutions of problem (1). In particular, by using the method of [10], we do this without the extraction of the Plais-Smale sequences in the Nehari manifold as in [1,20]. This paper is divided into three sections, organized as follows.…”