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Positive solutions for some quasilinear elliptic equations with natural growths

Lucio Boccardo (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is - div 1 + u r u + u m - 2 u u 2 = f in Ω u = 0 su Ω .

Positive solutions for sublinear elliptic equations

Bogdan Przeradzki, Robert Stańczy (2002)

Colloquium Mathematicae

The existence of a positive radial solution for a sublinear elliptic boundary value problem in an exterior domain is proved, by the use of a cone compression fixed point theorem. The existence of a nonradial, positive solution for the corresponding nonradial problem is obtained by the sub- and supersolution method, under an additional monotonicity assumption.

Positive solutions of critical quasilinear elliptic equations in R N

Paul A. Binding, Pavel Drábek, Yin Xi Huang (1999)

Mathematica Bohemica

We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in N , where λ , α , 1 < p < N , p * = N p / ( N - p ) , the critical Sobolev exponent, and 1 < q < p * , q p . Let λ 1 + > 0 be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with u 1 + > 0 the associated eigenfunction. We prove that, if N f | u 1 + | p * < 0 , N h | u 1 + | q > 0 if 1 < q < p and N h | u 1 + | q < 0 if p < q < p * , then there exist λ * > λ 1 + and α * > 0 , such that for λ [ λ 1 + , λ * ) and α [ 0 , α * ) , (1) has at least one positive solution.

Positive solutions of inequality with p -Laplacian in exterior domains

Robert Mařík (2002)

Mathematica Bohemica

In the paper the differential inequality Δ p u + B ( x , u ) 0 , where Δ p u = div ( u p - 2 u ) , p > 1 , B ( x , u ) C ( n × , ) is studied. Sufficient conditions on the function B ( x , u ) are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.

Positive solutions of nonlinear elliptic systems

Robert Dalmasso (1993)

Annales Polonici Mathematici

We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, L a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.

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