Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent (original) (raw)

Abstract

Consider the equation &2 p u=*g(x) |u| p&2 u+f(x) |u| p*&2 u (1) in R N , where 1<p<N and p*=NpÂ(N&p) is the critical Sobolev exponent. Let * + 1 >0 be the principal eigenvalue of &2 p u=*g(x) |u| p&2 u in R N , | R N g(x) |u| p >0, (2) with u + 1 >0 the associated eigenfunction. We prove: (i) Equation (1) has at least one positive solution if * # (0, * + 1). (ii) Suppose R N f (x)(u + 1) p* <0. Then there exists * 0 >* + 1 such that (1) has at least one positive solution for * # [* + 1 , * 0). Moreover, if p 2, there exists * 0 >* + 1 such that (1) has at least two positive solutions for * # (* + 1 , * 0).

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What conditions allow for multiple positive solutions in quasilinear elliptic equations?add

The paper demonstrates that for weight functions g and f meeting specific conditions, multiple positive solutions exist, particularly for values of * in the range (0, * + 1 ). In cases where * > * + 1, two positive solutions are obtained under similar conditions.

How does the Mountain Pass Theorem apply in this context?add

The Mountain Pass Theorem is employed to establish the existence of solutions when * is less than or equal to * + 1 by overcoming challenges presented by the indefinite nature of the associated functional. Notably, for * > * + 1, finding two solutions requires careful analysis due to the functional's indefinite principal part.

What role does the critical Sobolev exponent play in solution existence?add

The paper focuses on the critical Sobolev exponent, showing that distinct solution existence results emerge when * equals or surpasses this exponent. For instance, specific variational approaches yield positive solutions under modified conditions involving the sign-changing nature of f and g.

What challenges arise with indefinite weight functions in solution construction?add

Indefinite weight functions present significant hurdles, particularly concerning the lack of local minimizers in the associated functional when using traditional variational methods. The analysis reveals that without certain bounds on energy levels, classic techniques falter, necessitating modifications to established methodologies.

How does the study extend previous research on elliptic equations?add

This work builds on previous research by addressing sign-changing weight functions g and f, a deviation from conventional approaches that typically assume nonnegative functions. Such an extension opens new avenues for exploring positive solution existence in complex variable contexts.

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