A Note on Additional Properties of Sign Changing Solutions to Superlinear Elliptic Equations
نویسندگان
چکیده
We obtain upper bounds for the number of nodal domains of sign changing solutions of semilinear elliptic Dirichlet problems using suitable min-max descriptions. These are consequences of a generalization of Courant’s nodal domain theorem. The solutions need not to be isolated. We also obtain information on the Morse index of solutions and the location of suband supersolutions.
منابع مشابه
On a Class of Semilinear Elliptic Equations with Boundary Conditions and Potentials Which Change Sign
We study the existence of nontrivial solutions for the problem ∆u = u, in a bounded smooth domain Ω ⊂ RN, with a semilinear boundary condition given by ∂u/∂ν = λu− W(x)g(u), on the boundary of the domain, where W is a potential changing sign, g has a superlinear growth condition, and the parameter λ∈ ]0,λ1]; λ1 is the first eigenvalue of the Steklov problem. The proofs are based on the variatio...
متن کاملSuperlinear elliptic problems with sign changing coefficients
Via variational methods, we study multiplicity of solutions for the problem −∆u = λb(x)|u|q−2u + a u + g(x, u) in Ω , u = 0 on ∂Ω . where a simple example for g(x, u) is |u|p−2u; here a, λ are real parameters, 1 < q < 2 < p ≤ 2∗ and b(x) is a function in a suitable space L. We obtain a class of sign changing coefficients b(x) for which two non-negative solutions exist for any λ > 0, and a...
متن کاملA Minmax Principle, Index of the Critical Point, and Existence of Sign Changing Solutions to Elliptic Boundary Value Problems
In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is a...
متن کاملMultiplicity of Positive Solutions of laplacian systems with sign-changing weight functions
In this paper, we study the multiplicity of positive solutions for the Laplacian systems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic system has at least two positive solutions.
متن کاملExistence and multiplicity of nontrivial solutions for $p$-Laplacian system with nonlinearities of concave-convex type and sign-changing weight functions
This paper is concerned with the existence of multiple positive solutions for a quasilinear elliptic system involving concave-convex nonlinearities and sign-changing weight functions. With the help of the Nehari manifold and Palais-Smale condition, we prove that the system has at least two nontrivial positive solutions, when the pair of parameters $(lambda,mu)$ belongs to a c...
متن کامل