Abstract
In this paper we consider a semilinear elliptic system with nonlinearities, indefinite weight functions and critical growth terms in bounded domains. The existence result of nontrivial nonnegative solutions is obtained by variational methods.
Highlights
In this paper we consider the existence results of the following two coupled semilinear equation−∆u = λau − pav2u|u|p−2 + 2au|v|p, x ∈ Ω,−∆v = λav − pau2v|v|p−2 + 2av|u|p, x ∈ Ω, (1) ∂u∂v (1 − α) + αu = (1 − α) + αv = 0, x ∈ ∂Ω, ∂n∂n where α and λ are real parameters, p < 2∗ − 2, for 2∗ =
In this work we extend this studies to classes of Robin boundary conditions
We prove our existence results via variational methods
Summary
In this paper we consider the existence results of the following two coupled semilinear equation. Ω is an open bounded domain in RN , N ≥ 3 with a smooth boundary ∂Ω, and a : Ω → R is a sign changing weight function. This work is motivated by the results in the literature for the single equation case, namely the equation of the form. It is well known that the weak solutions of the system (1) are the critical points of the Euler functional Jλ. Let I be the Euler functional associated with an elliptic problem on a Banach space X. Heidarkhani of I, and so is a solution of the corresponding elliptic problem. The Euler functional Jλ is not bounded below on the whole space H, but is bounded on an appropriate subset, and a minimizer on this set (if it exists) gives a solution to the system (1).
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