Abstract
Minimal positive solutions for systems of semilinear elliptic equations
Highlights
We discuss the existence of minimal positive solutions for the following problem
There exists rich literature devoted to similar problems which arise in many applications e.g. in pseudoplastic fluids [6], reaction–diffusion processes or chemical heterogeneous catalysts [3], heat conduction in electrically conducting materials [7]
To prove the above result we will use the ideas described by Kawano in [22] for the case when the elliptic problem contains the gradient terms
Summary
We discuss the existence of minimal positive solutions for the following problem. ∆u(x) + f1(x, u(x), v(x)) + g1( x )x · ∇u(x) = 0, ∆v(x) + f2(x, u(x), v(x)) + g2( x )x · ∇v(x) = 0, for x ∈ GR,. Considering parameters satisfying additional conditions and applying a new type of energy function, the authors investigate the existence of ground states in the case when the system is not variational. The existence result for solutions of elliptic problems under the assumption concerning the existence of subsolutions and supersolutions was first proved by H. To prove the above result we will use the ideas described by Kawano in [22] for the case when the elliptic problem contains the gradient terms. We can follow his steps because of the fact that the differential operator is linear in our problem. The proof of Theorem 1.2 is standard we present the sketch of the reasoning in the Appendix for the reader’s convenience
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