We discuss a simple and effective technique for obtaining nonexistence results for nonnegative solutions of the following problem:

\begin{equation}\label{eq:D} -\Delta_p u = g_\lambda(x, u, \nabla u), \quad x \in \Omega, \end{equation} where $p>1$, $\lambda \in \mathbb{R}$ is a parameter, and $\Omega \subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain whose boundary $\partial \Omega$ is of class $C^{1}$.

Our main assumption on the nonlinearity $g_\lambda$ is the following:

$(\ast)$ There exists $M \subset \mathbb{R}$ such that $g_\lambda(x, t, \eta) > \lambda_1 t^{p-1}$ for all $\lambda \in M$, $x \in \Omega$, $t > 0$, and $z \in \mathbb{R}^N$.

Here $\lambda_1$ is the first eigenvalue of the $p$-Laplacian with the corresponding eigenfunction $\varphi_1$, i.e.,

\[\left\{ \begin{aligned} -\Delta_p \varphi_1 &= \lambda_1 |\varphi_1|^{p-2}\varphi_1, &&x \in \Omega, \\\ \varphi_1 &= 0 &&x \in \partial \Omega \end{aligned} \right.\]

It is known [¹] that $\lambda_1$ is positive, simple and isolated, and $\varphi_1 > 0$ in $\Omega$. Moreover by the regularity result of [²] we have $\varphi_1 \in C^{1,\beta}(\overline{\Omega})$ for some $\beta \in (0,1)$.

Under a weak solution $u$ of \eqref{eq:D}, we mean a function $u \in W^{1,p}(\Omega)$ where $W^{1,p}(\Omega)$ is the standart Sobolev space, which satisfies \begin{equation} \label{eq:weak} \int_{\Omega} |\nabla u|^{p-2} \left( \nabla u, \nabla \phi \right) dx = \int_{\Omega} g_\lambda(x, u, \nabla u) \phi \, dx, \quad \forall \phi \in W_0^{1,p}(\Omega) \backslash { 0}. \end{equation}

Our main result is the following.

Theorem. Assume $(\ast)$ is satisfied. Then for every $\lambda \in M$ problem \eqref{eq:D} has no weak nonnegative solution $u \in C^{1}(\overline{\Omega})$.

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Particular examples of equation \eqref{eq:D} with $g_\lambda(x, t, \eta)$ satisfying $(\ast)$ are:

1) equations with concave nonlinearity [³]

\[-\Delta_p u = \lambda u^{p-1} + u^{\gamma - 1}, \quad 1 < p < \gamma,\]

2) equations with convex-concave nonlinearity [⁴]

\[-\Delta_p u = \lambda u^{q-1} + u^{\gamma - 1}, \quad 1 < q < p < \gamma,\]

3) example from [⁵]

\[-\Delta_p u = \lambda \frac{u^{\theta - 1}}{1 + u^{\alpha - 1}} + u^{\gamma - 1}, \quad 1 < \alpha < \theta < p < \gamma, \\\]

4) equations with singular nonlinearity [⁶]

\[-\Delta_p u = \lambda u^{\gamma - 1} + \frac{1}{u^{\alpha-1}}, \quad 1 < \alpha,~ 1 < p \leq \gamma,\]

5) the Liouville-Bratu-Gelfand problem [⁷]

\[-\Delta_p u = \lambda e^{u}, \quad 1<p <+\infty,\]

6) equations with polynomial-type reaction term [⁸]

\[-\Delta_p u = \lambda (1 + u)^{\gamma - 1}, \quad 1 < p \leq \gamma.\]

For each example of (1)-(6) there exist $\lambda^* > 0$ such that there are no weak positive $C^1$-solutions of \eqref{eq:D} for any $\lambda > \lambda^*$.

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The main tool for the proof of Theorem is the following Picone inequality, see Theorem 1.1 in [⁹].

Lemma. Let $\phi \in W_0^{1,p}(\Omega) \cap C^1(\overline{\Omega})$ and $u \in C^1(\overline{\Omega})$ be such that $u > 0$, $\phi \geq 0$, and $\partial u(x) /\partial \nu < 0$ whenever $u(x) = 0$, $x \in \partial \Omega$, where $\nu$ is an outward normal to the boundary $\partial \Omega$. Then $\phi^p/u^{p-1} \in W_0^{1,p}(\Omega)$ and

\[\int_{\Omega} |\nabla u|^{p-2} \left( \nabla u, \nabla \left( \frac{\phi^{p}}{u^{p - 1}} \right) \right) dx \leq \int_{\Omega} |\nabla \phi|^{p} \, dx.\]

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Let us now prove our main result.

Proof of Theorem. Let $\lambda \in M$ where $M$ is defined by $(\ast)$. Assume that there exists a weak nonnegative solution $u \in C^1(\overline{\Omega})$ of \eqref{eq:D}. It follows from $(\ast)$ that $g_\lambda(x, t, \eta) > \lambda_1 t^{p-1} > 0$ for all $t>0$ and $x \in \Omega$. Hence $-\Delta_p u \geq 0$, and we apply the Hopf maximum prinicple from [¹⁰] to deduce that $u > 0$ in $\Omega$ and $\partial u(x) /\partial \nu < 0$ if $u(x)=0$ for $x \in \partial \Omega$.

Therefore, by Lemma,

\begin{equation} \label{eq:Il} \int_{\Omega} |\nabla u|^{p-2} \left( \nabla u, \nabla \left( \frac{\varphi_1^{p}}{u^{p - 1}} \right) \right) dx \leq \int_{\Omega} |\nabla \varphi_1|^{p} \,dx = \lambda_1 \int_{\Omega} |\varphi_1|^{p} \, dx. \end{equation}

On the other hand, since $u$ is the weak solution of \eqref{eq:D}, we test \eqref{eq:weak} by $\varphi_1^{p} / u^{p-1}$ and get

\begin{equation} \label{eq:u} \int_{\Omega} |\nabla u|^{p-2} \left( \nabla u, \nabla \left( \frac{\varphi_1^{p}}{u^{p - 1}} \right) \right) dx = \int_{\Omega} g_\lambda(x, u, \nabla u) \frac{\varphi_1^{p}}{u^{p - 1}} \, dx. \end{equation}

From \eqref{eq:Il} and \eqref{eq:u} it follow that

\[\int_{\Omega} g_\lambda(x, u, \nabla u) \frac{\varphi_1^{p}}{u^{p - 1}} \, dx \leq \lambda_1 \int_{\Omega} \varphi_1^{p} \, dx,\]

which reads as

\[\int_{\Omega} \left( g_\lambda(x, u, \nabla u) - \lambda_1 u^{p-1} \right) \frac{\varphi_1^{p}}{u^{p - 1}}\, dx \leq 0.\]

However, since $\lambda \in M$, the assumption $(\ast)$ gives us a contradiction. $\square$

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Remark. Essentially, Theorem is a corollary of the fact that the eigenvalue problem

\[\left\{ \begin{aligned} -\Delta_p u &= \lambda |u|^{p-2}u, &&x \in \Omega, \\\ u &= 0 &&x \in \partial \Omega \end{aligned} \right.\]

does not possess any nonnegative supersolution whenever $\lambda > \lambda_1$. In particular, there are no positive eigenfunctions other than $\varphi_1$. Or, in other words, any higher eigenfunction have to be sign-changing.

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Bibliography