Existence of entire positive solutions for semilinear elliptic systems with gradient term

Monotonic sequences of minimal solutions for semipositone elliptic systems

“…Our task is now to show that the vector function 1) and satisfies (12). It is easy to note that the latter fact is an immediate consequence of the definition of ūq,l i and (17). Bearing in mind ( 15) and ( 16), we obtain for all x ∈ R n such that ||x|| ≥ R,…”

“…The existence and multiplicity of solutions for elliptic systems considered also in unbounded domains has been widely discussed by many authors for years (see, e.g., previous studies 3,13–21 and references therein). The results concerning the multiplicity of solutions for systems of similar form and their additional properties can be found, among others, in Conti et al 13 There, applying the variational approach, the authors proved the existence of at least nine solutions (not necessarily positive) in the case when $$$ \Omega $$$ is a bounded and regular domain in $$$ {\mathbb{R}}^n $$$, the right‐hand side is a Carathéodory function and satisfies, among others, some growth conditions.…”

Connected sets of positive solutions of elliptic systems in exterior domains

“…Recently the existence and multiplicity of solutions for such elliptic systems considered also in unbounded domains has been widely discussed in the literature (see e.g. [5,[11][12][13][14][23][24][25]27,34,36] and the references therein).…”

“…vector functions u k 1 , u k 2 . We will show that u k 1 , u k 2 is a solutions of (23). To this effect we apply the standard reasoning, based on the L p -estimates of Agmon-Douglis-Nirenberg ( [1]), which leads to the existence of C > 0 such that for all m ∈ N,…”