"\"We consider the semilinear elliptic equation Δu = W (u) with Dirichlet boundary condition in a Lipschitz, possibly unbounded, domain Ω ⊂ Rn. Under suitable assumptions on the potential W, we deduce a condition on the size of the domain that implies the existence of a positive solution. satisfying a uniform pointwise estimate. Here uniform means that the estimate is independent of Ω. Under some geometric restrictions on the domain, we extend the analysis to the case of mixed Dirichlet–Neumann boundary conditions. As an application of our estimate we give a proof of the existence of potentials such that, independent of the choice of Ω and of the value of λ > 0, the equation Δu = λW(u) has infinitely many positive solutions.\""
A uniform estimate for positive solutions of semilinear elliptic equations
FUSCO G;LEONETTI F;PIGNOTTI C
2011-01-01
Abstract
"\"We consider the semilinear elliptic equation Δu = W (u) with Dirichlet boundary condition in a Lipschitz, possibly unbounded, domain Ω ⊂ Rn. Under suitable assumptions on the potential W, we deduce a condition on the size of the domain that implies the existence of a positive solution. satisfying a uniform pointwise estimate. Here uniform means that the estimate is independent of Ω. Under some geometric restrictions on the domain, we extend the analysis to the case of mixed Dirichlet–Neumann boundary conditions. As an application of our estimate we give a proof of the existence of potentials such that, independent of the choice of Ω and of the value of λ > 0, the equation Δu = λW(u) has infinitely many positive solutions.\""Pubblicazioni consigliate
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