Physical Sciences and Mathematics Commons^™

A Sign-Changing Solution For A Superlinear Dirichlet Problem, Ii, Alfonso Castro, Pavel Drabek, John M. Neuberger

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In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of -Δ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the k-th and k+1-st eigenvalues there still exists a solution which changes sign at most k times. In particular, when k=1 the sign-changing exactly-once solution persists although one-sign solutions no …

The Effect Of The Domain Topology On The Number Of Minimal Nodal Solutions Of An Elliptic Equation At Critical Growth In A Symmetric Domain, Alfonso Castro, Mónica Clapp

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We consider the Dirichlet problem Δu + λu + |u|^2*−2u = 0 in Ω, u = 0 on ∂Ω where Ω is a bounded smooth domain in R^N, N≥4, and 2* = 2N/(N−2) is the critical Sobolev exponent. We show that if Ω is invariant under an orthogonal involution then, for λ>0 sufficiently small, there is an effect of the equivariant topology of Ω on the number of solutions which change sign exactly once.

On The Number Of Radially Symmetric Solutions To Dirichlet Problems With Jumping Nonlinearities Of Superlinear Order, Alfonso Castro, Hendrik J. Kuiper

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This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem

Δu+f(u)=h(x)+cφ(x)

on the unit ball Ω⊂R^N with boundary condition u=0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u)<μ, where μ is the smallest positive eigenvalue for Δψ+μψ=0 in Ω with ψ=0 on ∂Ω. It is shown that, given any integer k≥0, the value c may be chosen so large that there are 2k+1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

A Minmax Principle, Index Of The Critical Point, And Existence Of Sign Changing Solutions To Elliptic Boundary Value Problems, Alfonso Castro, Jorge Cossio, John M. Neuberger

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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 asymptotically linear, we provide sufficient conditions for:

i) the existence of at least four solutions (one of which changes sign exactly once),

ii) the existence of at least five solutions (two of which change sign), and …

A Sign-Changing Solution For A Superlinear Dirichlet Problem, Alfonso Castro, Jorge Cossio, John M. Neuberger

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We show that a superlinear boundary value problem has at least three nontrivial solutions. A pair are of one sign (positive and negative, respectively), and the third solution changes sign exactly once. The critical level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. If nondegenerate, the one sign solutions are of Morse index 1 and the signchanging solution has Morse index 2. Our results extend and complement those of Z.Q. Wang [12].

Radial Solutions To A Dirichlet Problem Involving Critical Exponents When N=6, Alfonso Castro, Alexandra Kurepa

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In this paper we show that, for each λ>0, the set of radially symmetric solutions to the boundary value problem

-Δu(x) = λu(x) + u(x)|u(x)|, x ε B := {x ε R⁶:|x|<1},

u(x) = 0, x ε ∂B

is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.

Radially Symmetric Solutions To A Dirichlet Problem Involving Critical Exponents, Alfonso Castro, Alexandra Kurepa

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In this paper we answer, for N = 3,4, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem -Δu(x) = λu(x) + u(x)|u(x)|^{4/(N-2)}, x ε B: = x ε R^N:{|x| < 1}, u(x)=0, x ε ∂B, where Δ is the Laplacean operator and λ>0. Indeed, we prove that if N = 3,4, then for any λ>0 this problem has only finitely many radial solutions. For N = 3,4,5 we show that, for each λ>0, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.

Uniqueness And Stability Of Nonnegative Solutions For Semipositone Problems In A Ball, Ismael Ali, Alfonso Castro, Ratnasingham Shivaji

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We study the uniqueness and stability of nonnegative solutions for classes of nonlinear elliptic Dirichlet problems on a ball, when the nonlinearity is monotone, negative at the origin, and either concave or convex.

Radially Symmetric Solutions To A Superlinear Dirichlet Problem In A Ball With Jumping Nonlinearities, Alfonso Castro, Alexandra Kurepa

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Let p,φ :[0,T] → R be bounded functions with φ > 0. Let g:R → R be a locally Lipschitzian function satisfying the superlinear jumping condition:

(i) lim u → - ∞ (g(u)/u) ε R

(ii) lim u → ∞ (g(u)/(u^{1 + ρ} )) = ∞ for some ρ > 0, and

(iii) lim u → ∞ (u/g(u))^N/2(NG(κ u) - ((N - 2)/2)u · g(u)) = ∞ for some κ ε (0,1] where G is the primitive of g.

Here we prove that the number of solutions of the boundary value problem Δu + g(u) = p(|x|) + cφ …

Nonnegative Solutions For A Class Of Radially Symmetric Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

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We consider the existence of radially symmetric non-negative solutions for the boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\qua... ...\\ {u(x) = 0\quad \left\Vert x \right\Vert = 1} \\ \end{array} \end{displaymath}

where $ \lambda > 0,f(0) < 0$ (non-positone), $ f' \geq 0$ and $ f$ is superlinear. We establish existence of non-negative solutions for $ \lambda $ small which extends some work of our previous paper on non-positone problems, where we considered the case $ N = 1$. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.

Nonnegative Solutions To A Semilinear Dirichlet Problem In A Ball Are Positive And Radially Symmetric, Alfonso Castro, Ratnasingham Shivaji

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We prove that nonnegative solutions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular, this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the domain which ensures that nonnegative solutions are positive in the interior.

Multiple Solutions For A Dirichlet Problem With Jumping Nonlinearities Ii, Alfonso Castro, Ratnasingham Shivaji

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No abstract provided for this article.

Critical Point Theory And The Number Of Solutions Of A Nonlinear Dirichlet Problem, Alfonso Castro, A. C. Lazer

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No abstract provided.

A Semilinear Dirichlet Problem, Alfonso Castro

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Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:

Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω

u(x) = 0 ; x є ∂Ω

where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rⁿ⁺¹ → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.