Zero sets of solutions to semilinear elliptic systems of first order

Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n − 2)dimensional submanifolds. Hence it is countably (n − 2)-rectifiable and its Hausdorff dimension ...

Multiplicity of Positive Solutions for Semilinear Elliptic Systems

and Applied Analysis 3 Let Kλ,μ : E → R be the functional defined by Kλ,μ (z) = ∫ Ω (λf (x) |u| q + μg (x) |V| q ) dx ∀z = (u, V) ∈ E. (11) We know that Iλ,μ is not bounded below on E. From the following lemma, we have that Iλ,μ is bounded from below on the Nehari manifoldNλ,μ defined in (9). Lemma 3. The energy functional Iλ,μ is coercive and bounded below onNλ,μ. Proof. If z = (u, V) ∈ Nλ,μ, ...

Minimal positive solutions for systems of semilinear elliptic equations

The paper is devoted to a system of semilinear PDEs containing gradient terms. Applying the approach based on Sattinger’s iteration procedure we use sub and supersolutions methods to prove the existence of positive solutions with minimal growth. These results can be applied for both sublinear and superlinear problems.

Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

Multiple Nontrivial Solutions of Elliptic Semilinear Equations

We find multiple solutions for semilinear boundary value problems when the corresponding functional exhibits local splitting at zero.